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The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(K\) be the rational function field of a smooth plane curve \(C\) of degree \( d\) (\(d\geq 2)\) defined over an algebraic number field \(k\) of characteristic zero. If \(K_m\) denotes a maximal rational subfield of \(K\), the authors of the paper under review answer, in the case where \(C\) is a quartic curve, the following questions. (1) When is the extension \(K/K_m\) Galois? (2) Let \(L\) be the Galois closure of \(K/K_m\). What could we say about \(L\)? (3) What is the Galois group \(\text{Gal}(L/K_m)\)? The characterisation is dependent on the point \(P\), which is the center of the projection of the curve \(C\) to a line \(l\), and of the genus \(g(P)\) of the curve \(\grave{C}\), which corresponds to the field \(L\). plane quartic curves; function field; Galois group K. Miura - H. Yoshihara, Field theory for function fields of plane quartic curves, J. Algebra, 226 (2000), pp. 283-294. Zbl0983.11067 MR1749889 Arithmetic theory of algebraic function fields, Algebraic functions and function fields in algebraic geometry Field theory for function fields of plane quartic curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper is devoted to the study of an intrinsic distribution, called polar, on the space of \(l\)-dimensional integral elements of the higher order contact structure on jet spaces. The main result establishes that this exterior differential system is the prolongation of a natural system of PDEs, named pasting conditions, on sections of the bundle of partial jet extensions. Informally, a partial jet extension is a \(k\)th order jet with additional \((k+1)\)st order information along \(l\) of the \(n\) possible directions. A choice of partial extensions of a jet into all possible \(l\)-directions satisfies the pasting conditions if the extensions coincide along pairwise intersecting \(l\)-directions.
It is further shown that prolonging the polar distribution once more yields the space of \((l,n)\)-dimensional integral flags with its double fibration distribution. When \(l > 1\) the exterior differential system is holonomic, stabilizing after one further prolongation. jet spaces; exterior differential systems; geometry of PDEs Differentiable manifolds, foundations, Exterior differential systems (Cartan theory), Pfaffian systems, Jets in global analysis, Vector distributions (subbundles of the tangent bundles), Natural bundles, Calculus on manifolds; nonlinear operators, Partial differential equations on manifolds; differential operators, Grassmannians, Schubert varieties, flag manifolds, General topics in partial differential equations, Geometric theory, characteristics, transformations in context of PDEs, Differential topology, Local submanifolds Partial extensions of jets and the polar distribution on Grassmannians of non-maximal integral elements | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The main purpose of the article is to construct, and study, Wronski systems on a family \(X/S\) of curves. When \(X/S\) is a family consisting of complete intersection reduced curves, the author constructs a Wronski system consisting of locally free \(\mathcal O_X\)-modules \(Q^i\) that have natural functorial properties and that fit into exact sequences
\[
0\to \omega^{\otimes i} \to Q^i \to Q^{i-1} \to 0,
\]
where \(\omega\) is the dualizing sheaf of the family. The \(Q^i\) are unique with the functorial properties. To show that Wronski systems exist is important because they give rise to Wronski determinants whose zero loci give the Weierstrass points of the family. For a single integral Gorenstein curve Weierstrass points have been defined and studied by Widland in his thesis [see also \textit{R. F. Lax} and \textit{C. Widland}, Pac. J. Math. 50, No. 1, 111-122 (1991; Zbl 0686.14033) and \textit{A. Garcia} and \textit{R. F. Lax}, Commun. Algebra 22, No. 12, 4841-4854 (1994; Zbl 0824.14033)], and Wronski systems have been constructed by \textit{D. Laksov} and \textit{A. Thorup} [Ark. Mat. 32, No. 2, 393-422 (1994; Zbl 0839.14020)]. The latter authors also got results for certain families as a consequence of their work on Wronskian systems on schemes of arbitrary dimension. \textit{E. Esteves} has later used residues to compare his approach with other approaches for families of curves [see Bol. Soc. Bras. Mat., Nova Sér. 26, No. 2, 229-243 (1995; Zbl 0855.14003)]. The present article shows that the previous results, in a natural way, can be extended to families of curves, and that reduced curves can be allowed in the family, as long as the family consists of complete intersections. The construction holds in any characteristic. Wronski systems; Weierstrass points; families of curves; complete intersections Esteves, E.: Wronski algebra systems on families of singular curves. Ann. sci. Éc. norm. Super. (4) 29, No. 1, 107-134 (1996) Riemann surfaces; Weierstrass points; gap sequences, Singularities of curves, local rings, Families, moduli of curves (algebraic), Complete intersections Wronski algebra systems on families of singular curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(G\) be a finite group over a field \(k\) of positive characteristic. A full triangulated subcategory \(\mathcal{C}\) of the stable module category \(StMod \, G\) of possibly infinite-dimensional \(G\)-modules is called a \textit{colocalising subcategory} if it is closed under set-indexed products. It is \(\mathrm{Hom}\) closed if whenever \(M\) is in \(\mathcal{C}\), so is \(\mathrm{Hom}_k(L,M)\) for any \(G\)-module \(L\). The main result of the paper gives a bijection between the Hom closed colocalising subcategories of \(StMod \, G\) and the subsets of \(\mathrm{Proj} \, H^*(G,k)\) where the latter is the set of homogeneous prime ideals not containing \(H^{\geq 1}(G,k)\). This bijection is given by sending \(\mathcal{C}\) to its \(\pi\)-cosupport.
In earlier work [the first author et al., J. Am. Math. Soc. 31, No. 1, 265--302 (2018; Zbl 1486.16011)] the authors had classified the tensor closed localising subcategories of \(StMod \;G\). Combined with the present work the assignment \(\mathcal{C} \mapsto \mathcal{C}^{\bot}\) gives a bijection between the tensor closed localising subcategories of \(StMod \, G\) and the Hom closed colocalising subcategories of \(StMod \, G\). cosupport; stable module category; finite group scheme; colocalising subcategory Benson, Dave; Iyengar, Srikanth B.; Krause, Henning; Pevtsova, Julia, Colocalising subcategories of modules over finite group schemes, Ann. K-Theory, 2379-1683, 2, 3, 387\textendash408 pp., (2017) Modular representations and characters, Derived categories, triangulated categories, Group schemes, Cohomology theory for linear algebraic groups, Cohomology of groups Colocalising subcategories of modules over finite group schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a nonsingular surface over complex numbers and \(S_n\) be the symmetric group on \(n\) letters. Denote the Hilbert scheme of \(n\) points on \(X\) by \(X^{[n]}\). By a result of Haiman \(X^{[n]}\) can be identified with the fine moduli space of \(S_n\)-clusters in \(X^n\). If we denote by \(\mathcal Z \subset X^{[n]}\times X^n\) the universal family of \(S_n\)-clusters and \(X^{[n]}\overset{q}{\leftarrow} \mathcal Z \overset{p}\rightarrow X^n\) be the projections then, the derived McKay correspondence of \textit{T. Bridgeland} et al. [J. Am. Math. Soc. 14, No. 3, 535--554 (2001; Zbl 0966.14028)] in this set up gives an equivalence of derived categories \(\Phi: D(X^{[n]})\overset{\sim}{\rightarrow} D_{S_n}(X^n)\) of (\(S_n\)-equivariant) coherent sheaves, where \(\Phi=Rp_*\circ q^*\).
Scala showed that for any vector bundle \(F\) on \(X\), the image of the tautological bundle \(F^{[n]}\) on \(X^{[n]}\) under \(\Phi\) is given by an explicit complex \(\mathsf C^\bullet_F\) of (\(S_n\)-equivariant) coherent sheaves concentrated in nonnegative degrees.
The paper under review studies the derived McKay correspondence above in the reverse order by means of \(\Psi=q_*^{S_n}\circ Lp^*\) (which is not the inverse of \(\Phi\)). The main result of the paper is that if one replaces \(\Phi\) by \(\Psi^{-1}\) the images of \(F^{[n]}\) and \(\bigwedge^k L^{[n]}\) where \(L\) is a line bundle are (explicitly given) sheaves (instead of complexes of sheaves).
This enables the author to prove new formulas and also give simpler proofs for existing formulas for homological invariants of tautological bundles and their wedge powers. derived McKay correspondence; Hilbert scheme; tautological bundle McKay correspondence, Parametrization (Chow and Hilbert schemes) Remarks on the derived McKay correspondence for Hilbert schemes of points and tautological bundles | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We investigate the linear differential operator with polynomial coefficients whose space of holomorphic solutions is spanned by all the branches of a function defined by a generic algebraic curve. The main result is a description of the coefficients of this operator in terms of their Newton polytopes. algebraic function; minimal differential operator; Newton polytope The Newton polytope of the optimal differential operator for an algebraic curve | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\mu \in \mathbb{C} \setminus \{1,\omega,\omega^2\}\) where \(\omega=\exp(2i\pi/3)\) and \(C(\mu) : t_0^3+t_1^3+t_2^3-3 t_0 t_1 t_2=0\) be the Hesse model of an elliptic curve. The authors do for this model a classical and careful study of periods and relations to theta functions in analogy with the study on the Weierstrass model. They first express a basis of the periods \(\psi_A(\mu),\psi_B(\mu)\) in terms of hypergeometric and beta functions and study the monodromy. Then they reprove a result of [\textit{A. Krazer}, Lehrbuch der Thetafunktionen. New York: Chelsea Publishing Company (1970; Zbl 0212.42901)] relating \(\mu\) to quotients of theta functions with characteristics evaluated at \(\tau=\psi_A(\mu)/\psi_B(\mu)\). They use this result to derive an analogue of Thomae's formula (or Jacobi's formula) relating the value of the previous theta functions at \(\tau\), the periods and \(\mu\). As corollary, they get a relation between the value of theta function at \(\tau=\omega\) and the value of the Gamma function at \(1/3\) which can also be obtained from the Chowla-Selberg formula in [\textit{A. Chowla} and \textit{S. Selberg}, J. Reine Angew. Math. 227, 86--110 (1967; Zbl 0166.05204)]. periods; elliptic curve; monodromy; cubic curve; Thomae formula; theta function; hypergeometric function K. Matsumoto, T. Terasoma and S. Yamazaki, Jacobi's formula for Hesse cubic curves, to appear in Acta Math. Vietnam., Proceedings of the Conference Complex Geometry. Elliptic curves, Theta functions and curves; Schottky problem Jacobi's formula for Hesse cubic curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In computer science, a one-way function is a function that is easy to compute on every input, but hard to invert given the image of a random input. Here, ``easy'' and ``hard'' are to be understood in the sense of computational complexity theory, specifically the theory of polynomial time problems. Not being one-to-one is not considered sufficient of a function for it to be called one-way (see Theoretical Definition hereinafter). A twin prime is a prime number that has a prime gap of two, in other words, differs from another prime number by two, for example the twin prime pair \((5,3)\). The question of whether there exist infinitely many twin primes has been one of the great open questions in number theory for many years. This is the content of the twin prime conjecture, which states: There are infinitely many primes \(p\) such that \(p+2\) is also prime. In this work we define a new notion: ``\(r\)-prime number of degree \(k\)'' and we give a new RSA trap-door one-way. This notion generalized a twin prime numbers because the twin prime numbers are 2-prime numbers of degree 1. RSA; prime number; one-way function; cryptography Algebraic coding theory; cryptography (number-theoretic aspects), Applications to coding theory and cryptography of arithmetic geometry, Cryptography \(r\)-prime numbers of degree \(k\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors investigate the spectral geometry and spectral action functionals associated to 1D supersymmetry algebras, using the classification of these superalgebras in terms of Adinkra graphs, a class of decorated \(N\)-regular, \(N\)-edge colored bipartite graphs related to representations of the \(N\)-extended one-dimensional super Poincaré algebra (see [\textit{C. Doran} et al., Adv. Theor. Math. Phys. 19, No. 5, 1043--1113 (2015; Zbl 1382.14009)]). Other closely related constructions are dessins d'enfant and origami curves. The resulting spectral action functionals are computed in terms of the Selberg (super) trace formula and the Poisson summation formula. Adinkra graph; 1D supersymmetry algebras; dessins d'enfant; origami; spectral action; Selberg supertrace formula; bipartite graphs; super Poincaré algebra Superalgebras, Coloring of graphs and hypergraphs, Dessins d'enfants theory, Applications of Lie (super)algebras to physics, etc., Riemann surfaces, Supersymmetric field theories in quantum mechanics Adinkras, dessins, origami, and supersymmetry spectral triples | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We present the matrix models that are the generating functions for branched covers of the complex projective line ramified over 0, 1, and \(\infty\) (Grotendieck's dessins d'enfants) of fixed genus, degree, and the ramification profile at infinity. For general ramifications at other points, the model is the two-logarithm matrix model with the external field studied previously by the second author and \textit{K. Palamarchuk} [``Two logarithm matrix model with an external field'', Mod. Phys. Lett. A 14, 2229--2243 (1999)]. It lies in the class of the generalised Kontsevich models (GKM) thus being the Kadomtsev-Petviashvili (KP) hierarchy tau function and, upon the shift of times, this model is equivalent to a Hermitian one-matrix model with a general potential whose coefficients are related to the KP times by a Miwa-type transformation. The original model therefore enjoys a topological recursion and can be solved in terms of shifted moments of the standard Hermitian one-matrix model at all genera of the topological expansion. We also derive the matrix model for clean Belyi morphisms, which turns out to be the Kontsevich-Penner model introduced by the authors, \textit{C. F. Kristjansen} and \textit{Yu. Makeenko} [Nucl. Phys., B 404, No. 1--2, 127--172 (1995; Zbl 1043.81636); erratum ibid. 449, No. 3, 681 (1995)]. Its partition function is also a KP hierarchy tau function, and this model is in turn equivalent to a Hermitian one-matrix model with a general potential. Finally we prove that the generating function for general two-profile Belyi morphisms is a GKM thus proving that it is also a KP hierarchy tau function in proper times. Belyi function; topological recursion; tau function; Miwa transform Ambjørn, J.; Chekhov, L., The matrix model for dessins d'enfants, Ann. Inst. Henri Poincaré, Comb. Phys. Interact., 1, 337-361, (2014) String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Exact enumeration problems, generating functions, Random matrices (algebraic aspects), Dessins d'enfants theory, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Arithmetic aspects of dessins d'enfants, Belyĭ theory, Relationships between algebraic curves and integrable systems The matrix model for dessins d'enfants | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems JFM 29.0345.03 Algebraic function fields Algebraic functions and function fields in algebraic geometry Algebraic investigations of the Riemann Roch Theorem. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In a previous paper [Int. Math. Res. Not. 2011, No. 15, 3368--3425 (2011; Zbl 1309.14019)] the author extended techniques of Langlands and Kottwitz to determine the local factors of the Hasse-Weil zeta functions of modular curves at primes of bad reduction. Those methods are extended in this paper to obtain the semisimple Hasse-Weil zeta-function of Shimura varieties associated with unitary groups of real rank \(1\). A key part of the proof is the calculation of the semisimple Lefschetz number in terms of orbital integrals, and thus establishing a special case of a conjecture of Haines and Kottwitz. Shimura variety; Hasse-Weil zeta function; unitary group; orbital integral Scholze, P., The Langlands-Kottwitz approach for some simple Shimura varieties, Invent. Math., 192, 3, 627-661, (2013) Modular and Shimura varieties, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Arithmetic aspects of modular and Shimura varieties, Representations of Lie and linear algebraic groups over local fields The Langlands-Kottwitz approach for some simple Shimura varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We survey a few results concerning the Cremona group in two variables. Cremona group; geometric group theory; holomorphic dynamical systems; Tits alternative Cantat S. , 'The Cremona group in two variables', Proceedings of the Sixth European Congress of Mathematics (European Mathematical Society, Zürich, 2013) 211--225. Birational automorphisms, Cremona group and generalizations, Simple groups, Complex Lie groups, group actions on complex spaces, Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables The Cremona group in two variables | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The aim of this paper is to provide an elliptic analogue to Brownawell's generalization of Gel'fond-Fel'dman measure for algebraic independence of \(\alpha^{\beta}\) and \(\alpha^{\beta^ 2}\) for algebraic numbers \(\alpha\) and \(\beta\) with \(\alpha\neq 0\), \(\alpha\neq 1\) and \(\beta\) cubic [see \textit{W. D. Brownawell}, Compos. Math. 38, 355-368 (1979; Zbl 0402.10039)]. Let \(\wp\) be a Weierstrass elliptic function with algebraic invariants \(g_ 2\), \(g_ 3\), and with complex multiplication. Let \(\beta\) be an algebraic number, which is of degree 3 over the field of complex multiplication of \(\wp\). Finally, let u be a complex number such that u, \(\beta\) u and \(\beta^ 2u\) are not poles of \(\wp.\)
It has been proved by \textit{D. W. Masser} and \textit{G. Wüstholz} [Journées arithmétiques, Exeter 1980, Lond. Math. Soc. Lect. Note Ser. 56, 360-363 (1982; Zbl 0491.10025)], that if \(\wp (u)\) is algebraic, then the two numbers \(\wp (\beta u)\) and \(\wp (\beta^ 2u)\) are algebraically independent. In a previous paper [J. Number Theory 23, 60- 79 (1986; Zbl 0591.10025)], the author provided a quantitative version of this result. Here, he drops the hypothesis that \(\wp (u)\) is algebraic by giving a corresponding quantitative version of the fact that two at least of the three numbers \(\wp (u)\), \(\wp (\beta u)\) and \(\wp (\beta^ 2u)\) are algebraically independent. transcendental numbers; simultaneous diophantine approximations to coordinates of points; product of elliptic curves; measure for algebraic independence; Weierstrass elliptic function Robert Tubbs, A Diophantine problem on elliptic curves, Trans. Amer. Math. Soc. 309 (1988), no. 1, 325 -- 338. Algebraic independence; Gel'fond's method, Elliptic curves A diophantine problem on elliptic curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We introduce a new arithmetic invariant for hermitian line bundles on arithmetic varieties. We use this invariant to measure the variation of the volume function with respect to the metric. We apply the theory developed here to the study of the arithmetic geometry of toric varieties. As an application, we obtain a generalized Hodge index theorem for hermitian line bundles which are not necessarily toric. When the metrics are toric, we recover some results due to Burgos, Phillippon, Sombra and Moriwaki. arithmetic variety; volume function; theta invariants; arithmetic degree; toric varieties Arithmetic varieties and schemes; Arakelov theory; heights The theta invariants and the volume function on arithmetic varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this nice paper, the authors extend work of \textit{C. Huneke} et al. [Math. Res. Lett. 11, No. 4, 539--546 (2004; Zbl 1099.13508)] proving the following: If \((R, \mathfrak{m})\) is an excellent local ring of dimension \(d\) of characteristic \(p>0\) with perfect residue field satisfying the condition that \(R_{\mathfrak{p}}\) is regular for all ideals \(\mathfrak{p}\) with dim \((R/\mathfrak{p})=d-1\), then the Hilbert Kunz function of an \(R\)-module \(M\) with respect to a fixed \(\mathfrak{m}\)-primary ideal \(I\) is of the form
\[
\varphi_n(M)=\alpha(M)q^d+\beta(M)q^{d-1}+O(q^{d-2})
\]
where \(\alpha(M)\) and \(\beta(M)\) are both real. They further show that \(\beta(R)=0\) when such a ring is also the homomorphic image of a regular local ring whose canonical module \(\omega_R\) satisfies \([\omega_R]_{d-1}=0\) in the Chow Group of \(R\). They use a homomorphism \(\tau: A_{d-1}(R) \rightarrow \mathbb{R}\) which is additive on short exact sequences to find the Hilbert-Kunz function of a torsion free \(R\)-module \(M\) of rank \(r\) over an excellent local domain which is regular in codimension one: \(\varphi_n(M)=r\varphi_n(R)+\tau([M]_{d-1})q^{d-1}+O(q^{d-2})\). For a short exact sequence \(0 \rightarrow M_1 \rightarrow M_2 \rightarrow M_3 \rightarrow 0\), they define the additive error of the Hilbert Kunz function to be \(\varphi_n(M_3)-\varphi_n(M_2)+\varphi_n(M_1)\) and determine this additive error as a corollary to the preceding formula regarding the Hilbert-Kunz function of a torsion free module. Hilbert-Kunz function; rational equivalence Chan, C.-Y. Jean; Kurano, Kazuhiko, Hilbert-Kunz functions over rings regular in codimension one, Comm. Algebra, 44, 1, 141-163, (2016) Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure, Integral closure of commutative rings and ideals, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, (Equivariant) Chow groups and rings; motives Hilbert-Kunz functions over rings regular in codimension one | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(B_{p,\sigma}(\mathbb{R}^n)\) be the subspace of the set of entire functions of exponential type \(\sigma\) whose restrictions to \(\mathbb{R}^n\) belong to \(L_{p,\sigma}\), \(p\in (0,\infty[\). The author investigates restrictions of the functions from \(B_{p,\sigma} (\mathbb{R}^n)\) to abelian coverings over compact algebraic manifolds and establishes the generalizations of well-known classical inequalities on \(\mathbb{R}^n\) (Berenstein, Plansherel-Pólya and Levin-Logvinenko-Katznelson inequalities). entire function; inequalities; abelian coverings; compact algebraic manifolds Brudnyi, A.: Inequalities for entire functions. Isr. Math. Conf. Proc. 15, 47--66 (2001) Entire functions of several complex variables, \(H^p\)-spaces, Nevanlinna spaces of functions in several complex variables, Real algebraic sets, Inequalities in approximation (Bernstein, Jackson, Nikol'skiĭ-type inequalities), CR structures, CR operators, and generalizations Inequalities for entire functions. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \({\mathcal H}=\{H_ i\}_{i \in I}\) be a finite collection of hyperplanes in a complex projective space \(\mathbb{P}^ N\), \(U=\mathbb{P}^ N- \bigcup_{H \in {\mathcal H}}H\). Suppose that to each \(H \in {\mathcal H}\) a complex \(r \times r\) matrix \(P(H)\) is assigned, such that \(\sum_{H \in {\mathcal H}} P(H)=0\). This defines a connection \(\nabla=d+\omega\) on a trivial \(r\)-dimensional bundle \(E={\mathcal O}_ U \otimes_ \mathbb{C} \mathbb{C}^ r\), \({\mathcal O}_ U\) being the sheaf of holomorphic functions, and the closed \(\text{End} (E)\)-valued 1-form \(\omega\) is defined as \(\omega=\sum_{i \in I} \eta_{i,i_ 0} \otimes P(H_ i)\) where \(\eta_{ij}:=\text{dlog} (f_{ij})\), \(H_ i-H_ j=\text{div} (f_{ij})\).
Suppose that \(\nabla\) is integrable, i.e. \(\omega \wedge \omega=0\). The sheaf of horizontal sections of \(\nabla\) is a complex local system \({\mathcal L}\) on \(U\). Let \(\Omega_ \nabla\) denote the subcomplex of the complex of global sections of the de Rham complex of \((E,\nabla)\) generated as a \(\mathbb{C}\)-vector space by all forms \(\eta_{i_ 1j_ 1} \wedge \cdots \wedge \eta_{i_ pj_ p} \otimes v\), \(v \in \mathbb{C}^ r\). It is complex of finite dimensional vector spaces. One has a natural map
\[
H^*(\Omega_ \nabla) \to H^*(U,{\mathcal L}) \tag{1}
\]
The following result is proven.
Theorem. Suppose that for any nonempty intersection \(L\) of hyperplanes from \({\mathcal H}\), none of the eigenvalues of the operator \(\sum_{H | L \subset H} P(H)\) is a positive integer. Then the map (1) is an isomorphism.
This establishes a conjecture made by K. Aomoto.
The result is a not difficult consequence of a local result by Deligne, Brieskorn theorem, and mixed Hodge theory. local systems; de Rham cohomology; collection of hyperplanes; connection esv-coh H. Esnault, V. Schechtman, E. Viehweg, Cohomology of local systems on the complement of hyperplanes. Invent. Math. \textbf 109 (1992), 557--561, Invent. Math. \textbf 112 (1993), 447--447. Analytic sheaves and cohomology groups, de Rham cohomology and algebraic geometry, Rational and unirational varieties Cohomology of local systems on the complement of hyperplanes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems An old question, raised by A. A. Albert, asks whether every division algebra of prime index is cyclic. It has variously been suggested that counter-examples might be found among division algebras over a function field (assuming an algebraically closed ground field) that are ramified along a cubic divisor. The author shows that this is not so by proving the following Proposition. Let \(D\) be a non-trivial central division algebra over \(K(P^2_k)\), with ramification divisor \(R\) of degree 3 and assume either (i) \(R\) is singular, or (ii) \(R\) is smooth and \(D\) is of odd period in the Brauer group. Then \(D\) is cyclic of period (i.e. exponent) equal to its index. -- Part (i) was proved by by \textit{T. J. Ford} [New York J. Math. 1, 178-183 (1995; Zbl 0886.16017)]\ while \textit{D. J. Saltman} [in Proc. Symp. Pure Math. 58, Pt. I, 189-246 (1995; Zbl 0827.13003)]\ showed that for smooth \(R\), \(D\) is similar to a tensor product of three cyclic algebras. -- In all this the ground field \(k\) is assumed to be algebraically closed, but an appendix discusses the general situation, when only partial (rather technical) results are obtained. tensor products of cyclic algebras; division algebras of prime index; division algebras over function fields; cubic divisors; central division algebras; ramification divisors; Brauer groups; exponents Michel Van den Bergh, Division algebras on \?² of odd index, ramified along a smooth elliptic curve are cyclic, Algèbre non commutative, groupes quantiques et invariants (Reims, 1995) Sémin. Congr., vol. 2, Soc. Math. France, Paris, 1997, pp. 43 -- 53 (English, with English and French summaries). Finite-dimensional division rings, Arithmetic theory of algebraic function fields, Quaternion and other division algebras: arithmetic, zeta functions, Brauer groups of schemes Division algebras on \(\mathbb{P}^2\) of odd index, ramified along a smooth elliptic curve are cyclic | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The purpose of this paper is to prove, and give some simple applications of a formula relating the Hodge numbers of a variety X, smooth and proper over a perfect field k, and certain numerical invariants that can be extracted from the slope spectral sequence of X. These invariants are of two kinds; the first, \(m^{i,j}(X)\), only depend on the Newton polygon of the crystalline cohomology \(H^{i+j}_{cris}(X)\), while the second, \(T^{i,j}(X)\), describe the torsion in the \(E_ 1\) terms of the slope spectral sequence of X. We will make extensive use of the Illusie-Raynaud structure theory [\textit{L. Illusie} and \textit{M. Raynaud}, Publ. Math., Inst. Haut. Étud. Sci. 57, 73-212 (1983; Zbl 0538.14012)] of this spectral sequence. After proving this formula \((theorem 4\)) we give some simple applications. These all concern a surface X; we give a criterion, in terms of the Hodge and crystalline cohomology of X, for the slope spectral sequence to degenerate, and prove a semicontinuity theorem for \(T^{0,2}\). characteristic p; Hodge numbers; slope spectral sequence; Newton polygon of the crystalline cohomology R. Crew, On torsion in the slope spectral sequence , Compositio Math. 56 (1985), no. 1, 79-86. \(p\)-adic cohomology, crystalline cohomology, Transcendental methods, Hodge theory (algebro-geometric aspects) On torsion in the slope spectral sequence | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\mathcal C\) be a smooth projective algebraic geometrically connected curve over the field \(\mathbb F_q\). Let \(\infty\) be a fixed rational point on \(\mathcal C\) and let \(\mathbb A\) be the ring of functions on \(\mathcal C\) holomorphic away from \(\infty\). Let \(k\) be the quotient field of \(\mathbb A\), \(K=k_{\infty}\) the completion of \(k\) with respect to the \(\infty\)-adic topology and \(K^{\text{ac}}\) the algebraic closure of \(K\). A lattice \(M\) contained in \(K^{\text{ac}}\) is by definition a finitely generated \(\mathbb A\)-submodule which is discrete in the \(\infty\)-adic topology. Associated to \(M\) one has an exponential function and an algebraic object called a Drinfeld module.
Conversely, a Drinfeld module (over \(K^{\text{ac}}\)) is associated to a lattice in \(K^{\text{ac}}\) -- this is all in very close analogy to the classical situation of \(\mathbb Z\)-lattices in \(\mathbb C\). When the Drinfeld module has algebraic (over \(k\)) coefficients one can make the natural conjectures about the rationality of the elements of the associated lattice. In other papers the author has proved these conjectures. The paper being reviewed is concerned with preliminary results (e.g., an analog of Siegel's lemma). However, the author does establish here an elegant analog of the classical result of Gelfond-Schneider on Hilbert's 7th problem. transcendence; exponential function; Drinfeld module; lattice; analog of Siegel's lemma; Hilbert's 7th problem Yu, J.: Transcendence theory over function fields. Duke math. J. 52, 517-527 (1985) Transcendence theory of Drinfel'd and \(t\)-modules, Drinfel'd modules; higher-dimensional motives, etc., Algebraic functions and function fields in algebraic geometry Transcendence theory over function fields | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [This article was published twice, one times in the book Zbl 0527.00017.]
This paper is a sequel to the author's paper ''Fifteen characterizations of rational double points and simple critical points'' [Enseign. Math. 25, 132-163 (1979; Zbl 0418.14020)]. The characterizations of that paper are for complex varieties and complex functions, and involve the Dynkin diagrams \(A_ k\), \(D_ k\) and \(E_ k\). It turns out that the missing Dynkin diagrams \(B_ k\), \(C_ k\), and \(F_ 4\) (but not \(G_ 2)\) correspond to real singularities and real functions, and that a smaller number of similar characterizations are true for these as well. The main theorem of this paper contains four such characterizations: Let \(f:({\mathbb{R}}^ 3,0)\rightsquigarrow(\mathbb{R},0)\) be the germ at the origin of a real analytic function. Then the following are equivalent: (1) The germ f is right-left equivalent to one of the germs given in a certain list. (2) The germ f is simple (in the sense of Arnold). (3) The complexified variety \(f^{-1}(0)\) has a rational singularity at the origin. (4) A resolution of the real variety \(f^{-1}(0)\) is given in a certain list. The proof of the theorem proceeds by direct computation, or by referring to the corresponding theorem in the complex case. germ of a real analytic function; singularities of real varieties; rational double points; simple critical points; Dynkin diagrams; real singularities A. Durfee, 14 characterizations of rational double points (to appear). Singularities in algebraic geometry, Real algebraic and real-analytic geometry, Germs of analytic sets, local parametrization, Singularities of differentiable mappings in differential topology, Local complex singularities Four characterizations of real rational double points | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This is the first of series of articles in which we are going to study the regularized determinants of the Laplacians of Calabi-Yau metrics acting on \((0, q)\) forms on the moduli space of CY manifolds with a fixed polarization. It is well known that in the case of elliptic curves, the Kronecker limit formula gives an explicit formula for the regularized determinant of the flat metric with fixed volume on the elliptic curves. The following formula holds in this case:
\[
\exp \left(\frac dds\zeta (s)\, |\, s=0\right)\, = \det \Delta_{\tau, l}(\tau)\, =\, (\text{Im}\tau )^2 |\eta(\tau)|^4,\tag{1}
\]
where \(\eta (\tau)\) is the Dedekind eta function. It is well known fact that \(\eta (\tau )^{24}\) is a cusp automorphic form of weight 12 related to the discriminant of the elliptic curve. Formula (1) implies that there exists a holomorphic section of some power of the line bundle of the classes of cohomologies of (1, 0) forms of the elliptic curves over the moduli space with an \(L^{2}\) norm equal to det \(\Delta _{(0, 1)} (\tau )\). This section is \(\eta (\tau )^{24}\). The purpose of this project is to find the relations between the regularized determinants of CY metric on \((0, 1)\) forms and algebraic discriminants of CY manifolds. In this paper we will establish the local analogue of the formula (1) for CY manifolds. Calabi-Yau metrics; Dedekind eta function; regularized determinants Calabi-Yau manifolds (algebro-geometric aspects), Determinants and determinant bundles, analytic torsion, Calabi-Yau theory (complex-analytic aspects), Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Real zeros of \(L(s, \chi)\); results on \(L(1, \chi)\) The analogue of the Dedekind eta function for CY manifolds I | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\subset\mathbb{Z}^n(\mathbb{Q})\) be an algebraic variety defined by the vanishing of polynomials \(f_1,\dots, f_r\in \mathbb{Z}[X_1,\dots, X_n]\). Suppose further that the leading forms \(F_i\) of \(f_i\) have degree at least 3, and height at most \(H\), and define a non-singular variety of codimension \(r\). It is then shown that
\[
N(X,B)\ll_{n,d} B^{n-3r+\delta}(\log B)^{n/2}(\log H)^{2r+1},
\]
where
\[
\delta= r^2{13n- 5- 3r\over n^2+ 4nr- n- r- r^2}.
\]
Here \(N(X, B)\) denotes the number of integral points on \(X\), of height at most \(B\), and \(d\) is the maximum of the degrees of the forms \(F_i\).
This result should be compared with that of the reviewer [Proc. Indian Acad. Sci., Math. Sci. 104, No. 1, 13--29 (1994; Zbl 0808.11042)], which covered the case \(r= 1\), in which \(X\) is a hypersurface. The reviewer's estimate had an exponent \(\delta= 15/(n+ 5)\), which is slightly larger than that obtained here. Thus not only does the present paper contain a full generalization to arbitrary complete intersections, but it gives a rather sharper bound. Indeed the present paper also gives an explicit dependence on \(H\) which was not considered in the reviewer's paper.
As with the reviewer's work, the argument depends on estimating the number of solutions of the simultaneous congruences \(f_i(X_1,\dots, X_n)\equiv 0\pmod{pq}\) in the cube \(\max|X_i|\leq B\), for a suitable pair of primes \(p\), \(q\). This is accomplished using exponential sums, and a \(q\)-analogue of van der Corput's ``AB-process''.
The improvement over the reviewer's work stems from a superior estimate for the complete exponential \(\sum e_q({\mathbf a}.{\mathbf x})\), where \({\mathbf x}\) runs over a complete set of solutions of the simultaneous congruences \(f_i({\mathbf x})\equiv 0\pmod q\). algebraic variety; rational points; counting function; complete intersection; non-singular; exponential sum; van der Corput Marmon, O, The density of integral points on complete intersections, Q. J. Math., 59, 29-53, (2008) Varieties over global fields, Rational points, Counting solutions of Diophantine equations, Estimates on exponential sums The density of integral points on complete intersections | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(f:(\mathbb{K}^{2},0)\rightarrow (\mathbb{K},0)\) be an analytic function germ not identically zero, where \(\mathbb{K}=\mathbb{R}\) or \(\mathbb{C}\). The authors study the curvature of the level curves \(f=c,\) \(c\) small. They introduce the notion of concentration of curvature of \(f\) at \(0\in \mathbb{K} ^{2}\), which intuitively can be formulated as follows: \(f\) has concentration of curvature at \(0\in \mathbb{K}^{2}\) if the set of directions (vectors) in \( \mathbb{K}^{2}\) is finite for which there are Puiseux arcs \(\gamma\), centered at \(0\) and with these directions, such that: in the points of \(\gamma \) the curvature of the level set of \(f\) passing through these points attains maximum (see the precise definition in the paper). They characterize completely this phenomenon in complex case in terms of tree models and topological types of \(f\). They also show that the corresponding characterization does not hold in the real case. singularity; curvature; analytic function; tree model; blow-analiticity Local complex singularities, Singularities in algebraic geometry, Critical points of functions and mappings on manifolds Non concentration of curvature near singular points of two variable analytic functions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The straightening law of Doubilet-Rota-Stein tells that the standard bitableaux bounded by a pair \(m=(m(1),m(2))\) give a vector space basis of the polynomial algebra in \(m(1)m(2)\) variables. In an enumerative proof of the straightening law Abhyankar enumerated the set \(\text{stab}(2,m,p,a,V)\) of certain standard bitableaux. The Abhyankar formula gives also the Hilbert polynomial of a class of determinantal ideals \(I(p,a)\).
In the paper under review, the author outlines an alternate proof of the Abhyankar formula for the cardinality of \(\text{stab}(2,m,p,a,V)\) using a recent result on nonintersecting lattice paths obtained independently by Modak, Kulkarni and Krattenthaler. The lattice path approach leads also to some other known results on the numerators of the Hilbert-Poincaré series of \(I(p,a)\), the so-called \(h\)-vector of the associated simplicial complex, and gives better bounds for the degree of the numerator of the Hilbert series of \(I(p,a)\) (and in some cases the exact value of the degree). The author also discusses some related problems concerning possible generalizations to higher dimensions. He indicates as well connections between the Hilbert function for the Schubert varieties in Grassmannians and the Abhyankar formula. Stanley-Reisner ring; straightening law; standard bitableaux; Abhyankar formula; Hilbert polynomial; determinantal ideals; nonintersecting lattice paths; Hilbert series; Hilbert function; Schubert varieties DOI: 10.1016/0378-3758(95)00156-5 Combinatorial aspects of representation theory, Determinantal varieties, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Factorials, binomial coefficients, combinatorial functions, Exact enumeration problems, generating functions, Linkage, complete intersections and determinantal ideals, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Grassmannians, Schubert varieties, flag manifolds Young bitableaux, lattice paths and Hilbert functions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Reacall that a hyperfield is a nonempty set with two binary operations satisfying the same axioms
as for a field, except that one allows addition to be multi-valued (hyperaddition).
A topological hyperfield is a hyperfield \(H\) with a topology having the conditions
(1) The multiplication \(H\times H\longrightarrow H\), where \(H\times H\) is equiped with product topology
is continuous.
(2) \(H^{\times}=H\setminus(0)\) is open, and the inversion map \(H^{\times}\longrightarrow H^{\times}\)
is continuous.
The paper deals with the following three topological hyperfields:
(1) Krasner hyperfield: Let \(\mathbb{K}=\{0,1\}\) be a commutative monoid with multiplication
\(1.0=0\) and \(1.1=1\). The hyperaddition \(0+1=1+0=1\), \(0+0=0\) and \(1+1=\{0,1\}\).
The open subsets of the topology are: \(\emptyset\), \(\{1\}\), \(\mathbb{K}\).
(2) Tropical hyperfield: Let \(\mathbb{T}=\mathbb{R}\cup\{-\infty\}\), where \(\mathbb{R}\) is the set of
real numbers. The multiplication \(\odot\) is the usual addition of \(\mathbb{R}\) such that \(a\odot(-\infty)=-\infty\)
for all \(a\in\mathbb{T}\). The hyperaddition is defined by \(a\oplus b=\max\{a,b\}\) if \(a\not=b\)
and \(a\oplus a=\{c\in \mathbb{T}, c\leq a\}\). Then \(\mathbb{T}\) is endowed with the Euclidean topology.
(3) Hyperfield of signs: Let \(\mathbb{S}=\{-1,0,1\}\) be a commutative monoid with multiplication
\(1.1=1\), \((-1)(-1)=1\), \((-1).1=-1\), \(1.0=(-1).0=0.0=0\). The hyperaddition follows the rule of signs
\(1+1=1\), \((-1)+(-1)=-1\), \(1+0=1\), \((-1)+0=-1\), \(1+(-1)=\{-1,0,1\}\). The open subsets of the topology are
\(\emptyset\), \(\{1\}\), \(\{-1\}\), \(\{-1,1\}\), \(\mathbb{S}\).
In this paper, the author considers a scheme \(X\) over \(\mathbb{Z}\) and a topological hyperfield \(H\).
The set \(X(H)\) of the \(H-\)rational points of \(X\) is endowed with the fine topology. Then he proves
(1) When \(H\) is the Krasner hyperfield, \(X(H)\) is homeomorphic to the underlying space of type \(X\).
(2) When \(H\) is the tropical hyperfield and \(X\) is of finite type over a complete non-Archimedean valued field,
\(X(H)\) is homeomorphic to the underlying space of the Berkovich analytification \(X^{an}\) of \(X\).
(3) When \(H\) is the hyperfield of signs, \(X(H)\) is homeomorphic to the underlying space of the real
scheme \(X_r\) associated with \(X\). hyperfield; Berkovich analytification; real spectrum; real scheme; locally hyperringed space; rational points; fine topology; representable functor Schemes and morphisms, Combinatorial aspects of matroids and geometric lattices, Foundations of tropical geometry and relations with algebra, Ordered fields, Rigid analytic geometry Geometry of hyperfields | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems An efficient algorithm for computing the canonical height of an algebraic point on an elliptic curve \(E\) over a number field has been devised by \textit{G. Everest} and \textit{T. B. Ward} [New York J. Math. 6, 331--342 (2000; Zbl 0973.11062)]. It relies on the division polynomial \(\phi_n\) associated to \(E\) and on a Diophantine approximation result. The authors extend this approach to hyperelliptic Jacobians over a number field. The method involves recurrence relations obtained by \textit{Y. Uchida} [Manuscr. Math. 134, No. 3--4, 273--308 (2011; Zbl 1226.14039)] for the analog of \(\phi_n\) for Jacobians of hyperelliptic curves. It also involves a Diophantine approximation result due to \textit{G. Faltings} [Ann. Math. (2) 133, No. 3, 549--576 (1991; Zbl 0734.14007)]. The authors have implemented the computation of the values of \(\phi_n\) in the case of genus \(2\). They give explicit examples of height computation. canonical heights; divisibility sequences; hyperelliptic curves; division polynomial; hyperelliptic sigma function; local height Heights, Simultaneous homogeneous approximation, linear forms, Jacobians, Prym varieties, Abelian varieties of dimension \(> 1\) Canonical heights and division polynomials | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems New algorithms are presented for solving ideal membership problems with parameters and for computing integral numbers in a ring of convergent power series. It is shown that the ideal membership problems for zero-dimensional ideals in the ring of convergent power series, can be solved in the polynomial ring. The key idea of the algorithms is computing comprehensive Gröbner systems of ideal quotients in a polynomial ring. Furthermore, new methods for computing integral numbers, in the local ring, are introduced as an application. comprehensive Gröbner systems; local rings; ideal membership problems Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Singularities of curves, local rings Solving parametric ideal membership problems and computing integral numbers in a ring of convergent power series via comprehensive Gröbner systems | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(C \subset \mathbb{P}^3\) be a smooth, irreducible curve, not contained in a plane. Let \(I\) (resp., \(I_1)\) be the saturated homogeneous ideal of \(C\) (resp., of its first infinitesimal neighborhood, \(C_1)\). It is easy to see that \(I^2 \subseteq I_1\). The authors prove three types of results:
(i) They show that \(I_1=I^2\) iff \(I^2=I^{(2)}\) (symbolic power) iff \(I^2\) is saturated. -- \textit{C. S. Peterson} [Ph. D. Thesis (Duke Univ. 1994)] has shown that \(I_1 = I^2\) if \(C\) is projectively normal. The authors show the same thing if \(C\) lies on a quadric surface and give examples to show that, in general, \(I_1 \neq I^2\).
(ii) The authors conjecture that \(C_1\) is arithmetically Cohen-Macaulay iff \(C\) is a complete intersection. They prove the conjecture when the degree of \(C\) is large (resp., small) compared to its genus, or under the additional hypothesis that \(I_1=I^2\).
(iii) They show that the numerical character of a general plane section of \(C_1\) has at most one gap and conclude with an application to the Hilbert function of a general set of fat points in \(\mathbb{P}^2\). first infinitesimal neighborhood; arithmetically Cohen-Macaulay; complete intersection; numerical character of a general plane section; Hilbert function; fat points Plane and space curves, Formal neighborhoods in algebraic geometry, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Some remarks on first infinitesimal neighbourhoods of space curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this thoroughly written article the author gives details and full proofs of results already announced in [Proc. Japan Acad., Ser. A 66, 195-200 (1990; Zbl 0723.11020)] without any restrictions on the class number: there is a natural action \(f_ \bullet\) of the Hecke ring \(\text{HR} (\Gamma (N)_ K, G_ 2^ + ({\mathcal O}_ k))\) of a totally real algebraic number field \(K\) of degree \(g\) with ring of integers \({\mathcal O}_ K\) on the singular homology \(H_ \bullet ((\Gamma(N)_ K \setminus \mathbb{H}^ g)^ \sim, \mathbb{Z})\) of a smooth projective toroidal compactification of the Hilbert modular variety \(\Gamma(N)_ K \setminus \mathbb{H}^ g, \Gamma(N)_ K\) being the principal congruence subgroup of level \(N\geq 3\) as well as an \(\ell\)-adic version \(f^ \bullet\) for the \(\ell\)-adic cohomology of \((\Gamma(N)_ K \setminus \mathbb{H}^ g)^ \sim\) resp. \(D_ p(\Gamma (N)_ K)\) where the latter is a suitable proper smooth Hilbert modular variety (in the sense of Rapoport) over the field \(\overline {\mathbb{F}}_ p\), \(p\nmid N\) prime.
The main results are
(1) An estimation (based on the Weil conjecture) of the eigenvalues of the Hecke operators \(f^{(n)} (T_ p{\mathcal O}_ K)\)
\[
|\lambda_ p|\leq p^{n/2}+ p^{(2n-2)/2} \qquad \text{ for } 0\leq n\leq 2g \quad(\text{Theorem 8}).
\]
(2) The determination of the monic polynomial
\[
P_ n(X)= \text{det} (X^ 2- f^{(n)} (T_ p{\mathcal O}_ K)X+ (\sigma^ \sim_{pn}) p^ g)\in \mathbb{Z}[ X]
\]
where \(\sigma^ \sim_{pn}\) is the endomorphism of the \(\ell\)-adic cohomology induced by the element
\[
\sigma_ p\equiv \left( \begin{smallmatrix} p^{-1} &0\\ 0 &p\end{smallmatrix} \right)\bmod N
\]
of \(\text{SL} (2,\mathbb{Z})\), namely
\[
P_ n(X)= \text{det} (X- [\text{Frob} (p)]_ n) \text{ det} (X-[ \text{Frob} (p) ]_{2g-n}) \quad( \text{Theorem }10).
\]
Here \([\text{Frob} (p)]_ \bullet\) is the endomorphism of the \(\ell\)-adic cohomology induced by the \(\overline {\mathbb{F}}_ p\)-linear Frobenius endomorphism of \(D_ p (\Gamma (N)_ K)\).
(3) An expression of the local zeta function in terms of the above Hecke operators
\[
Z(D_ p (\Gamma(N)_ K, X)= \sqrt {\prod_{n=0}^{2g} \bigl[ \text{det} (1- f^{(n)} (T_ p {\mathcal O}_ K)X+ (\sigma^ \sim_{pn}) p^ g X^ 2) \bigr]^{(-1)^{n+1}}} \quad (\text{Theorem }11).
\]
The methods of proof heavily rely on the theory of toroidal compactifications, Deligne's work on the Weil conjecture as well as the author's work on the same problem in the Siegel case. Hecke operators on Hilbert varieties; estimates of eigenvalues; Hecke ring; Hilbert modular variety; \(\ell\)-adic cohomology; local zeta function; toroidal compactifications; Weil conjecture K. Hatada: On the local zeta functions of compactified Hilbert modular schemes and action of the Hecke rings. Sci. Rep. Fac. Ed. Gifu Univ. Natur. Sci., 18, no. 2, 1-34 (1994). Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), \(p\)-adic theory, local fields, Hecke-Petersson operators, differential operators (several variables) On the local zeta functions of compactified Hilbert modular schemes and action of the Hecke rings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this book the author discusses the Hilbert scheme \(X^{[n]}\) of points on a complex surface \(X\). This object is originally studied in algebraic geometry but, as it has been recently realized, it is related to many other branches of mathematics, such as singularities, symplectic geometry, representation theory, and even to theoretical physics. The book reflects this feature on Hilbert schemes and therefore the subjects are analyzed from various points of view. One sees that \(X^{[n]}\) inherits structures of \(X\), e.g., it is a nonsingular complex manifold, it has a holomorphic symplectic form if \(X\) has one, it has a hyper-Kähler metric if \(X= \mathbb{C}^2\), and so on. A new structure is revealed when one studies the homology group of \(X^{[n]}\). The generating function of Poincaré polynomials has a very nice expression. The direct sum \(\bigoplus_n H_* (X^{[n]})\) is a representation of the Heisenberg algebra.
The book, which is nicely written and well-organized, tries to tell the harmony between different fields rather than focusing attention on a particular one. The reader is assumed to have basic knowledge on algebraic geometry and homology groups of manifolds. Some chapters require more background, say spectral sequences, Riemannian geometry, Morse theory, intersection cohomology. symplectic structure; moment map; hyper-Kähler quotients; Dynkin diagrams; vertex algebra; symmetric products; Hilbert scheme of points; Poincaré polynomials; Heisenberg algebra; Morse theory; intersection cohomology H. Nakajima, \textit{Lectures on Hilbert schemes of points on surfaces}, \textit{University Lecture Series}\textbf{18}, American Mathematical Society, Providence RI U.S.A., (1999). Parametrization (Chow and Hilbert schemes), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Research exposition (monographs, survey articles) pertaining to algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Special Riemannian manifolds (Einstein, Sasakian, etc.) Lectures on Hilbert schemes of points on surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For a complex semisimple algebraic group with parabolic subgroup \(P\) of \(G\), let \(m=\dim(H^2(G/P))\). For each \(t\in\mathbb C^m\), Belkale and Kumar defined a product \(\odot_t\). This product degenerates the usual cup product on \(H^\ast(G/P)\), and it gives applications to the eigenvalue problem and to the problem of finding \(G\)-invariants of the tensor products of representations. Previously, the authors have given a new construction of this product, and they have proved that \((H^\ast(G/P,\mathbb C),\odot_t)\) is isomorphic to a relative Lie algebra cohomology ring \(H^\ast(\mathfrak g_t,\mathfrak l_\Delta)\), where \(\mathfrak l_\Delta\) are subalgebras of \(\mathfrak g\times\mathfrak g\). The present article focuses on \((H^\ast(G/P),\odot_t)\). Let \(B\subset P\) be a Borel subgroup, let \(H\subset B\) be a Cartan subgroup, and let \(\alpha_1,\dots,\alpha_m\) be the simple roots with respect to \(B\). Let \(L\) be the Levi factor of \(P\) containing \(H\), let \(\mathfrak l\) be the Lie algebra of \(L\), and let the simple roots of \(\mathfrak l\) be \(I=\{\alpha_{m+1},\dots,\alpha_n\}\) where \(\alpha_1,\dots,\alpha_m\) are roots of the nilradical \(\mathfrak u\) of \(P\). For \(t=(t_1,\dots,t_m)\in\mathbb C^m\), \(J(t)=\{1\leq q\leq m:t_q\neq 0\}\), \(K=J(t)\cup I\). \(\mathfrak l_K\) denotes the Levi subalgebra generated by the Lie algebra \(\mathfrak h\) of \(H\) and the root spaces \(\mathfrak g_{\pm\alpha_i}\), \(i\in K\), \(L_K\) denotes the corresponding subgroup, and \(P_K=BL_K\) denotes the corresponding standard parabolic.
The main result is perfectly and precisely given as follows (direct quote): {Theorem 1.1.} For parabolic subgroups \(P\subset P_K\) of \(G\), with \(P_K\) determined by \(t\in\mathbb C^m\) as above,
(1) \(H^\ast(P_k/P)\) is isomorphic to a graded subalgebra \(A\) of \((H^\ast(G/P),\odot_t)\).
(2) The ring \((H^\ast(G/P_K),\odot_0)\cong (H^\ast(G/P),\odot_t)/I_+\), where \(I_+\) is the ideal of \((H^\ast(G/P),\odot_t)\) generated by positive degree elements of \(A\). \vskip0,2cm So \((H^\ast(G/P),\odot_t)\) has a classical part which is the usual cohomology ring, with the associated quotient given by the degenerate Belkale-Kumar product. The theorem is proved by applying the Hochschild-Serre spectral sequence in relative Lie algebra cohomology, that is, by using the relative Lie algebra cohomology description of the product. There is a need to show that the spectral sequence degenerates at the \(E_2\)-term, to compute the edge morphisms, and to determine the product structure on the \(E_2\)-term. Thus the main content of this article is to fulfill this need, that is carrying out these computations using the original approach of Hochschild and Serre. This approach is generalized to the the relative setting, with a main new point: A Lie algebra \(\mathfrak g\) is given, together with an ideal \(I\) and a subalgebra \(\mathfrak k\) which is reductive in \(\mathfrak g\). To construct the spectral sequence, there is a need for an action of \(\mathfrak g/I\) on the relative cohomology group \(H^\ast((I,I\cap\mathfrak k,M)\), \(M\) a \(\mathfrak g\)-module. If \(I\cap\mathfrak k\) is nonzero, the Lie algebra \(\mathfrak g/I\) acts in a special way on the space of cochains \(C^\ast(I,I\cap\mathfrak k)\). The authors construct a an action on the cohomology group by a formula involving cochains. Then they verify that this yields the \(d_1\) differential in the spectral sequence . The confirmation of the last fact is the main technical complication of the article.
The authors give a nice introduction to Lie algebra cohomology, and they generalize this to the relative setting. This involves explicit formulas for the differentials, needed in the relative computations. Then they give a rather complete module on spectral sequences, including the computation of general edge morphisms. Finally they apply the spectral sequences on the relative Hochscild Serre cohomology. This means that they give explicit conditions of convergence and descriptions of the edge morphisms. Finally, the article apply the Hochschild-Serre spectral sequence to prove the main theorem, and this proves that the cohomology of a generalized flag variety, equipped with the Belkale-Kumar product, has a structure analogous to the cohomology of a fibre bundle.
The article also includes some interesting theorems on the structure of cohomology rings in the relative case. The article is detailed and precise, and in addition to giving nice results, it is also a nice introduction to the techniques of Lie algebra cohomology rings. Belkale-Kumar product; relative Lie cohomology; Hochschild-Serre spectral sequence; Lie cohomology ring Evens, S; Graham, W, The relative Hochschild-Serre spectral sequence and the belkale-kumar product, Trans. Amer. Math. Soc., 365, 5833-5857, (2013) Cohomology of Lie (super)algebras, Grassmannians, Schubert varieties, flag manifolds, Representation theory for linear algebraic groups The relative Hochschild-Serre spectral sequence and the Belkale-Kumar product | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We study the invariant rational 1-forms which correspondences of curves admit. We show that in the case of unequal degrees of the morphisms, the number of poles and zeroes of an invariant rational 1-forms is uniformly bounded. This bound is explicitly given in terms of the degrees of the morphisms and the genus of the curves and this bound is independent of the characteristic of the base field. This result is however not true in the case of equal degrees of the morphisms. In the case of correspondences of the projective line \(\mathbb{P}^1\), when both the morphisms are conjugates to polynomial maps, under an assumption of sufficient separateness of the degrees, we prove that there are only two kinds of invariant rational 1-forms that they can admit. In one of the above two cases, the morphisms arise from the multiplicative maps. correspondence of curves; arithmetic dynamical systems; number theory; algebraic geometry Arithmetic aspects of modular and Shimura varieties, Modular correspondences, etc., Arithmetic ground fields for curves Invariant rational forms for correspondences of curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Notation: k a field of characteristic 0, often algebraically closed; \(k\{\) u] the ring of power series in the arguments \(u_ 1,...,u_ n,\) with coefficients in k. In a previous paper [Sympos. Math., Roma 3, 247- 277 (1970; Zbl 0194.522)] the author gave a definition of (holomorphic) theta type functions \(\vartheta\) (u)\(\in k\{u]\), as formal power series, which need not be repeated here, and which obviously did not mention periods; theta functions are theta types whose two invariants dim and transc coincide; they reduce to classical thetas when k is the complex field. In the paper under review the definition is further simplified: \(\vartheta\) (u)\(\in k\{u]\) is a theta type when the prostaferesis formula \(\vartheta (u+v)\vartheta (u-v)\in k\{u]\otimes k\{v]\) obtains; the relation between theta types and thetas is also clarified: theta types are what thetas become after replacing their arguments with linear combinations, with coefficients in k, of a lesser number of arguments. Now the main result: for any \(\vartheta\) (u)\(\in k\{u]\) (just one argument u), and for \(i=1,2,...\), set \(\vartheta_ i(u)=(i!)^{- 1}(d/du)^ i\log \vartheta (u);\) call \(P_{2j}\), for \(j=0,1,...\), the universal polynomials in j indeterminates such that \(\vartheta (u+v)\vartheta (u-v)=2\vartheta^ 2(u)\sum^{\infty}_{i=0}P_{2i}(\vartheta_ 2(u),\vartheta_ 4(u),...,\vartheta_{2i}(u))v^{2i};\) then \(\vartheta\) satisfies the prostaferesis, hence is a theta type, if and only if the vector space P generated over k by the \(P_{2i}'s\) is finite-dimensional. Thus, theta types in one argument (resp. thetas), or better their second logarithmic derivatives (which are abelian functions) are exhibited for the first time as solutions of certain standard systems of nonlinear ordinary (resp. partial) differential equations. Several geometric consequences on commutative group varieties (c.g.v. henceforth) are discussed in detail; main sample: if \(\vartheta\) (u) (one u only) is a theta type, then \(k[\vartheta_ 2,\vartheta_ 3,...,\vartheta_ s]\) is, for a suitable s, an affine ring of a c.g.v. over k; and any c.g.v. over k, of any dimension, can be reached in this manner, with the exception of those having a vectorial direct factor of dimension \(>1\). Proofs lean heavily on properties of hyperfields as given by \textit{G. Gerotto} [Ann. Mat. Pura Appl., IV. Ser. 115, 349-379 (1977; Zbl 0385.14013)]; an appendix proposes a refinement of the definition of c.g.v. which rehabilitates the degenerate points \((=\) points of the variety but not of the group). This being a self-review, I inform that a sequel is about to appear in vol. 9 of the same periodical, with corrections to a few details; the necessity of corrections stems from previous unawareness of the existence of thetas whose space P has smaller dimension than the P of the generic element of the complete linear system containing that theta. abelian variety; abelian function; soliton; theta type; commutative group varieties; hyperfields I. Barsotti , Le equazioni differenziali delle funzioni theta , Rend. Accad. Naz. XL , 101 , 1983 , p. 227 . Theta functions and abelian varieties, Group varieties, Elliptic functions and integrals Differential equations of theta functions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [See also volume I (1972; Zbl 0234.14001) and volume III (1978; Zbl 0446.14001).]
The papers collected in this second volume (mainly from the years 1946 to 1962) are divided in two groups: Part I (entitled ''Holomorphic functions'' with an introduction by M. Artin): theory of formal holomorphic functions on algebraic varieties (in any characteristic), meaning primarily analytic properties of an algebraic variety V, either in the neighborhood of a point (strictly local theory) or - and this is the deeper aspect of the theory - in the neighborhood of an algebraic subvariety of V (semi- global theory); part II with an introduction by D. Mumford): ''Linear systems, the Riemann-Roch theorem and applications'' (again in any characteristic), the applications being primarily to algebraic surfaces (minimal models, characterization of rational or ruled surfaces, etc.). Mathematicians; formal holomorphic functions; Linear systems; Riemann-Roch theorem Zariski, O.: M.artind.mumfordcollected papers, vol. II. Collected papers, vol. II (1973) Foundations of algebraic geometry, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Conference proceedings and collections of articles, Divisors, linear systems, invertible sheaves, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Riemann-Roch theorems Collected papers. Volume II: Holomorphic functions and linear systems. Ed. by M. Artin and D. Mumford | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The Handbook of homotopy theory has been reviewed in [Zbl 1468.55001].
Summary: Motivic homotopy theory was developed by Morel and Voevodsky in the 1990s. The original motivation for the theory was to import homotopical techniques into algebraic geometry. This chapter introduces the motivic Adams spectral sequence, which is one of the key tools for computing stable motivic homotopy groups. The precise relationship is that the motivic stable homotopy groups are the global sections of the motivic stable homotopy sheaves. For cellular motivic spectra, the motivic stable homotopy groups do detect equivalences, and the most commonly studied motivic spectra are typically cellular. So a thorough understanding of motivic stable homotopy groups over arbitrary fields leads back to complete information about the sheaves as well. The chapter considers motivic stable homotopy groups over larger classes of fields. Naturally, specific information is harder to obtain when the base field is allowed to vary widely. stable motivic homotopy group; stable motivic homotopy theory; Adams spectral sequence; Adams-Novikov spectral sequence; effective spectral sequence Stable homotopy of spheres, Motivic cohomology; motivic homotopy theory, Adams spectral sequences, Research exposition (monographs, survey articles) pertaining to algebraic topology Motivic stable homotopy groups | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(K\) be a field which is complete with respect to a discrete valuation. Let \(S\) be a scheme of finite type over \(K\) or an affinoid space over \(K\). Suppose that an abelian scheme \(X\to S\) is such that for every closed point \(s\in S\) the fibre \(X_ s\) is potentially toric. The main result of the paper states that there exists a surjective (finite) étale map \(S'\to S\) such that \(X'=X\times_ SS'\) has an analytic presentation \(\mathbb{G}^ d_{m,S'}/M\) where \(M=\mathbb{Z}^ d_{S'}\) is a constant \(S'\)-subgroup of \(\mathbb{G}^ d_{m,S'}\). A consequence is the following rigidity result: ``If \(S\) is a connected \(K\)-scheme of finite type, then \(X\to S\) is constant in the sense that the map from \(S\) to the coarse moduli space of abelian varieties of dimension \(d\) over \(K\) is constant.''
Some proofs in the paper are rather sketchy. \(p\)-adic tori; abelian scheme; rigid analytic spaces; uniformization; affinoid space; rigidity Local ground fields in algebraic geometry, Toric varieties, Newton polyhedra, Okounkov bodies Algebraic families of \(p\)-adic tori | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper studies compatible systems of \(\ell\)-adic Galois representations (\(\ell\) a varying prime), as arise from the étale cohomologies of algebraic varieties. The purpose seems to be to derive as much as possible from general principles. The main results are as follows: Let \(G_ \ell\) denote the connected reductive part of \(\text{Im}(\rho_ \ell)\). Then there exists a finite Galois extension \(E\) of \(\mathbb{Q}\) such that, except for \(\ell\) in a set of density zero, \(G_ \ell\) splits over \(E\mathbb{Q}_ \ell\) and its Weyl group depends only on the conjugacy class of \(\text{Frob}_ \ell\) in \(\text{Gal}(E/\mathbb{Q})\). Furthermore, if all \(\rho_ \ell\) are absolutely irreducible, the root- datum and the representation of \(G_ \ell\) also depend only on the conjugacy class of \(\text{Frob}_ \ell\). compatible systems of \(\ell\)-adic Galois representations; étale cohomologies of algebraic varieties Larsen, M.; Pink, R., On \textit{l}-independence of algebraic monodromy groups in compatible systems of representations, Invent. Math., 107, 3, 603-636, (1992) Varieties over global fields, Semisimple Lie groups and their representations, Étale and other Grothendieck topologies and (co)homologies On \(\ell\)-independence of algebraic monodromy groups in compatible systems of representations | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems It was conjectured by M. E. Rossi that the Hilbert function of a one-dimensional Gorenstein local ring is non-decreasing. This conjecture is open even for Gorenstein local rings corresponding to numerical semigroups, namely to monomial curves in affine 4-space. In the paper under review, the authors give a method to construct new large families of monomial curves in affine 4-space with corresponding Gorenstein local rings in embedding dimension 4 supporting Rossi's conjecture. By starting with any monomial curve in affine 2-space, this constructive method provides monomial curve families in affine 4-space having non-Cohen-Macaulay tangent cones with non-decreasing Hilbert functions. monomial curve; symmetric numerical semigroup; gluing; Gorenstein; Hilbert function of a local ring; Rossi's conjecture Arslan, F.; Sipahi, N.; Şahin, N., Monomial curve families supporting Rossi's conjecture, J. symbolic comput., 55, 10-18, (2013) Singularities of curves, local rings, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Semigroups, Symbolic computation and algebraic computation Monomial curve families supporting Rossi's conjecture | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper under review is concerned with the general problem of constructing dual objects in the category of projective schemes, meaning that a given scheme should be canonically isomorphic to its associated double dual. Prominent examples in algebraic geometry are provided by the dual of an abelian variety, by S. Mukai's duality theory of polarized \(K3\) surfaces, or by certain moduli spaces of stable vector bundles over a smooth projective curve [à la \textit{M. S. Narasimhan} and \textit{S. Ramanan}, Ann. Math. (2) 101, 391--417 (1975; Zbl 0314.14004)]. With these classical examples in mind, the author's underlying philosophy is that certain stacks of vector bundles should be natural categorical candidates for dual objects in algebraic geometry. Indeed, pursuing this strategy, he is able to construct dual objects for quotient stacks of schemes in the following way.
Let \({\mathcal X}= [X/G]\) be the quotient stack associated with a scheme \(X\) acted on by an affine group scheme \(G\), and denote by \(\text{VB}({\mathcal X})\) the category of all vector bundles on the stack \({\mathcal X}\). For any affine base scheme \(S\), a \(S\)-valued fiber functor on \(\text{VB}({\mathcal X})\) is defined to be an exact additive tensor functor from \(\text{VB}({\mathcal X})\) to \(\text{VB}(S)\), and these fiber functors form a fibered category denoted by \(\text{FIB}(\text{VB}({\mathcal X}))\). The basic technical framework for this construction is explained in the first sections of the present paper. Then it is shown that there is a natural morphism of fibered categories \(D_{{\mathcal X}}:{\mathcal X}\to\text{FIB} (\text{VB}({\mathcal X}))\), and the author's main theorem establishes the following fact:
Assume that the scheme \(X\) is quasi-compact and that there is a line bundle on the quotient stack \({\mathcal X}= [X/G]\) whose underlying line bundle on \(X\) is very ample. Then the associated ``duality functor'' \(D_{{\mathcal X}}:{\mathcal X}\to\text{FIB} ({\mathcal VB}({\mathcal X}))\) is a categorical equivalence. In particular, if the affine group \(G\) is trivial, i.e., if \(G= \text{Spec}(k)\) for a field \(k\), then one is dealing with true schemes, in which case the author's main theorem yields that for a quasi-compact \(k\)-scheme \(X\) admitting an ample line bundle, the duality morphism \(D_X\) provides a duality construction in the ordinary sense. The author's main result may be regarded as a stack-theoretic analogue of the classical Tannaka duality for affine groups in the sense of \textit{P. Deligne} and \textit{J. S. Milne} [in: Hodge cycles, motives, and Shimura varieties, Lect. Notes Math. 900, 101--228 (1982; Zbl 0477.14004)]. On the other hand, the author's result allows to derive a just as interesting and important conclusion, which makes the analogy to the classical duality theorems immediately recognizable. This second main result ensures the existence of another categorical equivalence, the so-called bi-duality morphism, between the quotient stack \({\mathcal X}\) and some tensor category fibered in groupoids. This construction uses the stack of all vector bundles on \({\mathcal X}=[X/G]\) as well as the existing universal bundle for them. In the course of the proofs, the framework of sheaves on fibered tensor categories appears as an essential ingredient. Accordingly, the technical details are thoroughly developed and explained, thereby making this rather advanced work easily accessible even for non-experts. algebraic stacks; group actions on schemes; tensor categories; Grothendieck topologies; duality theory; vector bundles; scheme; fibered categories; Tannakian categories Savin, V, Tannaka duality for quotient stacks, Manuscripta math., 119, 287-303, (2006) Generalizations (algebraic spaces, stacks), Group actions on varieties or schemes (quotients), Algebraic moduli problems, moduli of vector bundles, Étale and other Grothendieck topologies and (co)homologies, Fibered categories, Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) Tannaka duality on quotient stacks | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(I\) denote a homogeneous ideal in a polynomial ring \(S=k[x_0,x_1,\dots,x_r]\). After fixing a monomial order, one may consider the initial ideal in\((I)\). It is an interesting question to determine how much information about \(I\) may be gleaned from in\((I)\). The authors consider this problem for the reverse lexicographic order. (This monomial order has certain nice properties, such as in\((I:x_r)=\) in\((I):x_r\).) First, the authors define a sequence of ideals \(I^j\) (not the \(j\)-th power of \(I\)) by
\[
I^j=\text{im}((I:x_r^j)\rightarrow S\rightarrow S/(x_r)).
\]
The Hilbert functions of the \(I^j\) constitute a more refined invariant of \(I\) than the Hilbert function of \(I\) and are closely related to in\((I)\). Using these algebraic preliminaries, the authors go on to consider the information (mainly of a cohomological nature) encoded in the generic initial ideal in several geometric examples, such as space curves, zero-dimensional subschemes of 3-space, and the intersection of a curve and a surface in 3-space.
In the last section, they generalize these ideas to the generic higher initial ideal gin\(_1(I)\) in the case of a space curve \(C\). As a consequence, it is shown that if \(C\) is linked to a curve \(X\), then knowing the generic higher initial ideal of \(I_C\) is equivalent to knowing the generic initial ideal of \(I_X\). Higher initial ideals were introduced previously by the first author [\textit{G. Fløystad}, ``Higher initial ideals of homogeneous ideals'', Mem. Am. Math. Soc. 638 (1998; Zbl 0931.13013)], while the second author has previously written an expository article on generic initial ideals [\textit{M. L. Green}, in: Six lectures on commutative algebra. Lect. Summer School, Bellaterra 1996, Prog. Math. 166, 119-186 (1998; Zbl 0933.13002)]. generic initial ideal; initial module; Hilbert function; Hartshorne-Rao module; reverse lexicographic order; monomial order; space curves; 3-space; generic higher initial ideal G. Fløystad, M. Green. The information encoded in initial ideals. To appear in Transactions of the American Mathematical Society. Polynomial rings and ideals; rings of integer-valued polynomials, Plane and space curves, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Ideals and multiplicative ideal theory in commutative rings The information encoded in initial ideals | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(E\) be an elliptic curve over a function field \(K\) of a nice curve \(C\) over a field \(k\) of characteristic \(p\). In this paper the author deals with a sequence of points \(nP+Q\) where \(P\) is a point of infinite order and \(Q\) is a torsion point. Every point \(R\) on \(E\) defines a section \(\sigma\) on the associated elliptic surface \(\mathcal{E}\). We consider for the morphism \(\sigma:C\rightarrow\mathcal{E}\) a divisor \(\sigma^{*}_{R}(\mathcal{O})\) where \(\mathcal{O}\) is the image of the zero section. For any point \(nP+Q\) one has an effective divisor \(D_{nP+Q}:= \sigma^{*}_{nP+Q}(\mathcal{O})\). The author studies the primitive divisors in this sequence given a fixed order \(r\) of the torsion point \(Q\) and the characteristic of the algebraically closed field \(k\).
The results of this paper are direct analogues of the statements from [\textit{M. Verzobio}, Res. Number Theory 7, No. 2, Paper No. 37, 29 p. (2021; Zbl 1479.11099)], however the methods are quite different. If the field \(k\) characteristic is \(0\) or at least \(17\), the author proves that for a torsion point of order \(r\geq 2\) there exists a primitive divisors in the sequence \(D_{nP+Q}\). For characteristics \(5,7,11\) and \(13\) some additional restrictions are imposed on the order of the point \(Q\).
The main proof method uses techniques developed in the paper [\textit{P. Ingram} et al., J. Aust. Math. Soc. 92, No. 1, 99--126 (2012; Zbl 1251.11008)] and their extension to positive characteristics developed in [\textit{B. Naskręcki}, New York J. Math. 22, 989--1020 (2016; Zbl 1417.11110)]. elliptic curves; function fields; elliptic surfaces; elliptic divisibility sequences; primitive divisors; Zsigmondy bound Elliptic curves, Elliptic curves over global fields, Special sequences and polynomials, Elliptic surfaces, elliptic or Calabi-Yau fibrations Primitive divisors of sequences associated to elliptic curves over function fields | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For a prime \(p\), let \(\mathbb{Q}_p\) denote the field of \(p\)-adic numbers and \(\mathbb{Z}_p\), the ring of \(p\)-adic integers. Let \(\chi\) be a character of \(\mathbb{Z}_p\), the set of units of \(\mathbb{Z}_p\). Let \(f(x)= f(x_1,\dots, x_n)\in \mathbb{Z}_p[x_1,\dots, x_n]\). Igusa's local zeta-function \(Z_f(s)\) is directly related to the number of solution of the congruence \(f(x)\equiv 0\bmod p^m\), \(m= 1,2,3,\dots\)\ . Using resolutions of singularities, \textit{J.-I. Igusa} [Trans. Am. Math. Soc. 245, 419--429 (1978; Zbl 0401.12013)] proved that \(Z_f(s,\chi)\) is a rational function of \(p^{-s}\). \textit{J. Denef} [Invent. Math. 77, 1--23 (1984; Zbl 0537.12011)] obtained an entirely different proof using \(p\)-adic cell decomposition.
In this paper, a very explicit formula has been obtained for \(Z_f(s,\chi)\) where \(f\) is non-degenerate over the finite field \(F_p\) with respect to all the faces of its Newton polyhedron. This study was first started by \textit{B. Lichtin} and \textit{D. Meuser} [Compos. Math. 55, 313--332 (1985; Zbl 0606.14022)] for polynomials in two variables. Using the formula for \(Z_f(s,\chi)\) a set of possible poles for \(Z_f(s,\chi)\) together with upper bounds for their orders have also been obtained. In particular, it gives information on the largest real pole of \(Z_f(s)\) and its order. Moreover, the formula implies that \(Z_f(s)\) has always at least one real pole. \(p\)-adic integers; zeta function; Newton polyhedron; character; compact face; Igusa's local zeta-function Hoornaert, K.: Newton Polyhedra and the Poles of Igusa's Local Zeta Function. Bull. Belg. Math. Soc. Simon Stevin 9, 589--606 (2002) Zeta functions and \(L\)-functions, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) Newton polyhedra and the poles of Igusa's local zeta function | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Given a smooth projective variety \(X\), the Hilbert scheme \(X^{[n]}\) of \(n\) points on \(X\) is again a smooth projective variety of dimension \(\dim X^{[n]} = n\, \dim X\). Each vector bundle \(E\) on \(X\) defines the \emph{vector bundle} \(E^{[n]} = p_{2,*}p_1^*E\). Here \(p_k\) is the projection to the \(k\)-factor from the universal subscheme \(\Pi_n \subset X \times X^{[n]}\). The bundle \(E^{[n]}\) is called the Fourier-Mukai transform of \(E\) (with respect to \(\Pi_n\)). By work of Lehn and Lehn-Sorger, these transforms are important tools to study the topology and geometry of Hilbert schemes. Conversely, they are useful to study bundles on \(X\) itself e.g. by work of Voison, Ein-Lazarsfeld and Agostini.
The present article enhances the Fourier-Mukai transform to so-called V-cotwisted Hitchin pairs \((E, \theta)\). Here \(E\) and \(V\) are vector bundles on \(X\) and \(\theta\colon E \otimes V \to E\) is a section. The outcome of the enhanced Fourier-Mukai transform are \(V^{[n]}\)-cotwisted Hitchin pairs \((E^{[n]}, \theta^{[n]})\) on \(X^{[n]}\). Note here that if \(V = T_X\), then \(V^{[n]}\cong T_{X^{[n]}}(-\log B_n)\) (by a result of Stapleton) where \(B_n\subset X^{[n]}\) is the locus of non-reduced sub-schemes of \(X\). In particular, the enhanced Fourier-Mukai transforms of Higgs bundles (i.e. \(T_X\)-cotwisted Hitchin pairs) are logarithmic Higgs bundles on \(X^{[n]}\).
After establishing basic results on the enhanced Fourier-Mukai transform, which are of independent interest, the authors prove various interesting results on the relationship between Hitchin pairs on \(X\) and their enhanced Fourier-Mukai transforms (similar results for vector bundles were already obtained by the second author), for example:
\begin{itemize}
\item If \((E, \theta)^{[n]} \cong (F, \eta)^{[n]}\) on \(X^{[n]}\), then \((E,\theta) \cong (F, \eta)\) on \(X\) where \(X\) is any smooth projective curve of genus \(\geq 1\) or any smooth quasi-projective variety of \(\dim X \geq 2\);
\item relationship between the stability conditions for \((E, \theta)\) on \(X\) and \((E, \theta)^{[n]}\) on \(X^{[n]}\) for any smooth projective curve \(X\).
\end{itemize} logarithmic Higgs bundle; Hilbert scheme; Fourier-Mukai transformation; stability Stacks and moduli problems, Algebraic moduli problems, moduli of vector bundles, Coverings of curves, fundamental group, Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Parametrization (Chow and Hilbert schemes) Fourier-Mukai transformation and logarithmic Higgs bundles on punctual Hilbert schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We study special ``discriminant'' circle bundles over two elementary moduli spaces of meromorphic quadratic differentials with real periods denoted by \(\mathcal{Q}_0^{\mathbb{R}}(-7)\) and \(\mathcal{Q}_0^{\mathbb{R}}([-3]^2)\). The space \(\mathcal{Q}_0^{\mathbb{R}}(-7)\) is the moduli space of meromorphic quadratic differentials on the Riemann sphere with one pole of order seven with real periods; it appears naturally in the study of a neighborhood of the Witten cycle \(W_5\) in the combinatorial model based on Jenkins-Strebel quadratic differentials of \(\mathcal{M}_{g,n}\). The space \(\mathcal{Q}_0^{\mathbb{R}}([-3]^2)\) is the moduli space of meromorphic quadratic differentials on the Riemann sphere with two poles of order at most three with real periods; it appears in the description of a neighborhood of Kontsevich's boundary \(W_{1,1}\) of the combinatorial model. Applying the formalism of the Bergman tau function to the combinatorial model (with the goal of analytically computing cycles Poincaré dual to certain combinations of tautological classes) requires studying special sections of circle bundles over \(\mathcal{Q}_0^{\mathbb{R}}(-7)\) and \(\mathcal{Q}_0^{\mathbb{R}}([-3]^2)\). In the \(\mathcal{Q}_0^{\mathbb{R}} (-7)\) case, a section of this circle bundle is given by the argument of the modular discriminant. We study the spaces \(\mathcal{Q}_0^{\mathbb{R}}(-7)\) and \(\mathcal{Q}_0^{\mathbb{R}}([-3]^2)\), also called the spaces of Boutroux curves, in detail together with the corresponding circle bundles. moduli space; quadratic differential; Boutroux curve; tau function; Jenkins-Strebel differential; ribbon graph Differentials on Riemann surfaces, Families, moduli of curves (analytic), Differential algebra Discriminant circle bundles over local models of Strebel graphs and Boutroux curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For the entire collection see Zbl 0665.00004.]
The goal of the note under review is the description of a construction which connects the ring of invariants of a connected, commutative, affine algebraic group scheme G over a field k acting polynomially on an affine scheme X and the ring of invariants of a certain cyclic group acting on an extension of X and such that the initial ring of invariants is a subring of the ring of invariants of this cyclic group. This construction provides some methods for investigation of invariants of the action of G on X. group scheme; ring of invariants Geometric invariant theory, Group actions on varieties or schemes (quotients), Group schemes Invariants of affine group schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Nous allons donner ici une classification complète des surfaces rationnelles réelles, c'est-à-dire, des surfaces réelles propres et lisses \(X\) dont la complexifiée \(X_ \mathbb{C}=X\times_ \mathbb{R}\mathbb{C}\) est birationnellement équivalente, sur \(\mathbb{C}\), à \(\mathbb{P}^ 2_ \mathbb{C}\). -- Une autre façon de formuler les choses est de dire que nous allons donner une classification complète des corps \(K\) de degré de transcendance 2 sur \(\mathbb{R}\) tels que \(K\times_ \mathbb{R}\mathbb{C}=\mathbb{C}(x,y)\), l'extension transcendante pure de degré 2 sur \(\mathbb{C}\).
On peut résumer les résultats connus de la manière suivante: Les surfaces rationnelles réelles se divisent en première approximation en deux classes, l'une formée de surfaces \(\mathbb{R}\)-réglées et l'autre de celles qui ne le sont pas.
Les surfaces \(\mathbb{R}\)-réglées ont un corps des fonctions rationnelles \(K\), isomorphe au corps des fractions d'un anneau de la forme \(\mathbb{R}[x,y,z]/(x^ 2+y^ 2-\sum^{2m}_{i=1}(z-a_ i))\) où les \(a_ i\) sont des réels tous distincts; ou de la forme \(\mathbb{R}[x,y,z]/(x^ 2+y^ 2-(\pm 1))\) (cas \(m=0\)).
Pour les surfaces non-réglées et relativement minimales on peut assez facilement déduire qu'elles sont isomorphes, sur \(\mathbb{C}\), à des surfaces de Del Pezzo de degré 1 ou 2. Nous montrerons que la partie réelle de telles surfaces a 4 composantes connexes si le degré est 2 ou 5 si le degré est 1. On en déduira que le corps des fonctions rationnelles est isomorphe dans ce cas soit, si le degré est 2, au corps des fractions de \(\mathbb{R}[x,y,z]/(z^ 2-f(x,y))\); soit, si le degré est 1, au corps des fractions de \(\mathbb{R}[x,y,z,w]/((w^ 2- f(x,y,z))\), \((x^ 2+y^ 2-1))\) où \(f\) est un polynôme.
D'un résultat d'Iskovskikh nous déduirons de plus des critères d'isomorphie pour ces corps. birational classification of real rational surfaces; classification of function fields; ruled surface Silhol, R., Classification birationnelle des surfaces rationnelles réelles, 308-324, (1990), Berlin Special surfaces, Topology of real algebraic varieties, Rational and birational maps, Families, moduli, classification: algebraic theory, Arithmetic theory of algebraic function fields Birational classification of real rational surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(K\) be an algebraically closed field of characteristic \(0,\) \(S=K[x_0,\dots,x_n]\) and \({\mathbb P}^n_K=\mathrm{Proj} S.\) Let us consider on the set of the terms of \(S\) a degreverse order. The authors prove that in the Hilbert scheme of points in \({\mathbb P}^n_K,\) the point corresponding to a segment ideal, with respect to a degreverse order, is singular. Unfortunately this result cannot be generalized to Hilbert schemes with a Hilbert polynomial of a positive degree.
Moreover they provide an algorithm for computing all the saturated Borel ideals with a given Hilbert polynomial. Hilbert scheme of points; Borel ideal; segment ideal; Gröbner stratum; degrevlex term order; Gotzmann number Cioffi, F.; Lella, P.; Marinari, M. G.; Roggero, M., Segments and Hilbert schemes of points, Discrete Math., 311, 20, 2238-2252, (2011) Parametrization (Chow and Hilbert schemes), Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) Segments and Hilbert schemes of points | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors generalize two conjectures (now proved) about Hurwitz numbers, the ELSV formula, which relates them to Hodge integrals, and the Bouchard-Mariño (BM) conjecture, which relates them to the topological recursion of \textit{L. Chekhov} and \textit{B. Eynard} [J. High Energy Phys. 2006, No. 12, 026, 29 p. (2006; Zbl 1226.81138)], \textit{B. Eynard} and \textit{N. Orantin} [Commun. Number Theory Phys. 1, No. 2, 347--452 (2007; Zbl 1161.14026)]. The generalization concerns ``Hurwitz numbers with completed cycles'' that enumerate ramified coverings of the sphere of a certain kind. Okounkov and Pandharipande showed that these numbers express the Gromov-Witten invariants of \(r\)-spin structures over smooth curves. What the authors call the \(r\)-ELSV formula is the conjectural equality of these numbers with Chiodo's \(r\)-spin integrals, an analog of the Hodge integrals. The \(r\)-BM conjecture is that the \(n\)-point functions for completed Hurwitz numbers satisfy the topological recursion with the spectral curve \(x=-y^r+\log y\). The main result is a rigorous proof that the two conjectures are equivalent. The proof uses Frobenius algebra techniques, Givental's \(R\)-matrix, and leaf forms generated by the Bergman kernel, it identifies the Givental-Teleman semi-simple cohomological field theories with the topological recursion.
Ample independent evidence for both conjectures is presented. The \(r\)-ELSV formula is known in genus zero, and in genus one with one marked point. It was also successfully tested numerically in other cases. \textit{V. Bouchard} and \textit{M. Mariño} [Proc. Symp. Pure Math. 78, 263--283 (2008; Zbl 1151.14335)] (BM) gave a ``physics proof'' of their conjecture (before it was rigorously derived from the ELSV formula), which carries over to the \(r\)-BM case. Unfortunately, it can not be easily converted into a mathematical proof, although according to the authors ``it will probably convince a mathematical physicist''. One issue seems to be that the potential in the matrix model is neither a rational function, nor independent of a certain parameter, both of which are required for deriving the topological recursion. Another issue is that the infinite contour in the ``physics proof'' has to be replaced by a finite one for the integral to make sense, but then it does not remain invariant under the performed change of variables.
Additionally, for the completed Hurwitz numbers the spectral curve has been computed and it coincides with the conjectured one. The quantization of that curve leads to an operator annihilating the principal specialization of the generating function for Hurwitz numbers according to another conjecture \textit{S. Gukov} and \textit{P. Sulkowski} [J. High Energy Phys. 2012, No. 2, Paper No. 070, 57 p. (2012; Zbl 1309.81220)]. Finally, the \(r\)-BM conjecture is in line with the general philosophy that the spectral curve of an enumerative problem should be given by its ``\((0,1)\) geometry'', namely the genus zero \(1\)-pointed generating function in the case of Hurwitz numbers. Hurwitz numbers; topological recursion; spectral curve; r-spin structures; Frobenius algebra; Givental-Teleman semi-simple cohomological field theory; Givental R-matrix; Bergman kernel; \((0,1)\) geometry Shadrin, S.; Spitz, L.; Zvonkine, D., Equivalence of ELSV and bouchard-mariño conjectures for \(r\)-spin Hurwitz numbers, Math. Ann., 361, 611-645, (2015) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds Equivalence of ELSV and Bouchard-Mariño conjectures for \(r\)-spin Hurwitz numbers | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We give an overview of Smale's 17th problem describing the context in which Smale proposed it, the ideas that led to its solution, and the extensions and subsequent progress after this solution. polynomial systems; homotopy methods; approximate zero; complexity; polynomial time Global methods, including homotopy approaches to the numerical solution of nonlinear equations, Complexity and performance of numerical algorithms, Symbolic computation and algebraic computation Smale's 17th problem: advances and open directions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In the article under review, the authors expand upon the techniques of \textit{A. Levin} [Invent. Math. 215, No. 2, 493--533 (2019; Zbl 1437.11094)] and establish upper bounds for the counting function of common zeros of two meromorphic functions. In doing so, they establish a greatest common divisor conjecture, for multiplicatively independent meromorphic functions, from \textit{H. Pasten} and \textit{J. T. Y. Wang} [Int. Math. Res. Not. 2017, No. 1, 47--95 (2017; Zbl 1405.30024)].
Further, the authors solve an analogue, for entire functions, of the ``Hadamard quotient theorem'' for recurrence sequences. This result complements the related work of \textit{N. Grieve} and \textit{J. T. Y. Wang} [Trans. Am. Math. Soc. 373, No. 11, 8095--8126 (2020; Zbl 1464.11075)].
The main result, of the present article, pertains to the counting function of \(Y\) a closed subscheme of the algebraic torus \((\mathbb{C}^\times)^n\), with codimension at least equal to two. In more precise terms, the authors prove that if
\[\mathbf{g} \colon \mathbb{C} \rightarrow (\mathbb{C}^\times)^n\]
is a holomorphic map with Zariski dense image, then
\[
N_{\mathbf{g}}(Y,r) \leq_{\mathrm{exc}} \epsilon T_{\mathbf{g}}(r) \text{,}
\]
for each given \(\epsilon > 0\).
Here, \(N_{\mathbf{g}}(Y,r)\) and \(T_{\mathbf{g}}(r)\) are, respectively, the Nevanlinna counting and characteristic functions.
In addition to being one aspect to the proofs of the above mentioned applications, this result gives a new proof, for the special case of algebraic tori, of the main theorem of \textit{J. Noguchi} et al. [Forum Math. 20, No. 3, 469--503 (2008; Zbl 1145.32009)]. meromorphic functions; common zeros; counting function; algebraic tori Global ground fields in algebraic geometry, Recurrences, Value distribution theory in higher dimensions, Diophantine inequalities, Value distribution of meromorphic functions of one complex variable, Nevanlinna theory, Nevanlinna theory; growth estimates; other inequalities of several complex variables Greatest common divisors of analytic functions and Nevanlinna theory on algebraic tori | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(H_{d,g}\) be the Hilbert scheme of locally Cohen-Macaulay space curve of degree \(d\) and genus \(g\). The genus \(g=(d-3)(d-4)/2\) is the biggest value for which the study of \(H_{d,g}\) is nontrivial. According to the techniques developed by \textit{M. Martin-Deschamps} and \textit{D. Perrin} [``Sur la classification des courbes gauches'', Astérisque 184-185 (1990; Zbl 0717.14017)], in the present paper the author studies \(H_{d,g}\) in the case \(g=(d-3)(d-4)/2\). Hilbert scheme; locally Cohen-Macaulay space curve; degree; genus Aït Amrane, S, Sur le schéma de Hilbert des courbes de degré \(d\) et genre\((d-3)(d-4)/2\) de \(\mathbf{P}^3_k\), C. R. Acad. Sci. Paris Sér. I Math., 326, 851-856, (1998) Special algebraic curves and curves of low genus, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Parametrization (Chow and Hilbert schemes) On the Hilbert scheme of curves of degree \(d\) and genus \(\frac{(d-3)(d-4)}{2}\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \({\mathcal N}^0_{d,g}\) be the number of irreducible component of the Hilbert scheme parametrizing smooth arithmetically Cohen-Macaulay curves in the projective space \(\mathbb{P}^3\), of degree \(d\) and genus \(g\). The author constructs families of couples \((d,g)\) such that \({\mathcal N}^0_{d,g}\) increases at least exponentially. space curve; Hilbert scheme; arithmetically Cohen-Macaulay curves Parametrization (Chow and Hilbert schemes), Plane and space curves, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) On the number of components of the Hilbert scheme of ACM space curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this ``Docteur en troisième cycle'' thesis the author extends results of \textit{C. Andradas} [Commun. Algebra 13, 1151-1169 (1985; Zbl 0563.12021) and ``Specialization chains of a real valuation rings'' (Preprint, Madrid 1983)], and of \textit{L. Bröcker} and \textit{H.-W. Schülting} [``Valuations of function fields from the geometrical point of view'' (Preprint, 1984); see also J. Reine Angew. Math. 365, 12-32 (1986; Zbl 0577.12021)] dealing with valuations of function fields over real closed fields.
A prime cone P of a commutative ring A is a subset P of A, additively and multiplicative closed, containing all \(a^ 2, a\) in A, with -1\(\not\in P\) and with the property that xy\(\in P\) implies ``x\(\in P''\) or ``-y\(\in P''\). The support of P, supp(P), is the prime ideal \(P\cap -P\) of A. After a very readable summary of other work in this area, the following main theorem is proved:
Let W be an irreducible affine variety over a real closed field. Denote the coordinate ring of W by A, the function field of W by K, and the dimension of W by n. Let \(Q_{m-1}\subset...\subset Q_ 1\subset Q_ 0\) denote a chain of prime ideals of A with \(\dim (Q_ i)=d_ i\), such that there is a chain of prime cones in A, \(P_{m-1}\subset...\subset P_ 1\subset P_ 0\) with \(\sup p(P_ i)=Q_ i\). Denote further by \(f_ 1,...,f_ t\) elements of A and by \((s_ 0,...,s_{m-1})\) and \((r_ 0,...,r_{m-1})\) two sequences of integers such that, for \(0\leq i\leq (m-1): \) \(P_{m-1}\supset (\sum K^ 2[f_ 1,...,f_ t] \cap A)\), \(0\leq s_ 0<...s_{m-1}\leq n\), \(0<r_{m-1}<...<r_ 0\), and \(s_ i+r_ i\leq n\); \(r_ i\geq m_ i\), \(d_ i\leq s_ i\). Then there exists an order B of K in which the \(f_ i\) are positive, and a chain \(\{V_ i\}\), \(0\leq i\leq m-1\), of valuation rings of K with formally real residue class fields, all compatible with B and containing A such that: \(\dim (V_ i)=s_ i\), \(rank(V_ i)=m-i\), \(rational\quad rank(V_ i)=r_ i,\) the centre of \(V_ i\) in A is \(Q_ i\). valuation rings of function fields; coordinate ring of affine; variety over a real closed field; prime cone Real algebraic and real-analytic geometry, Fields related with sums of squares (formally real fields, Pythagorean fields, etc.), Valuations and their generalizations for commutative rings Constructions de places réelles dans géométrie semialgébrique. (Constructions of real places in semialgebraic geometry). (Thèse) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems There is currently a growing interest in understanding which lattice simplices have uni-modal local \( h^*\)-polynomials (sometimes called box polynomials); specifically in light of their potential applications to unimodality questions for Ehrhart \( h^*\)-polynomials. In this note, we compute a general form for the local \( h^*\)-polynomial of a well-studied family of lattice simplices whose associated toric varieties are weighted projective spaces. We then apply this formula to prove that certain such lattice simplices, whose combinatorics are naturally encoded using common systems of numeration, all have real-rooted, and thus unimodal, local \( h^*\)-polynomials. As a consequence, we discover a new restricted Eulerian polynomial that is real-rooted, symmetric, and admits intriguing number theoretic properties. box polynomial; local \( h^*\)-polynomial; Eulerian polynomial; Ehrhart theory; lattice simplex; weighted projective space; numeral systems; simplices for numeral systems; factoradics; real-rooted; unimodal; symmetric; log-concave Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Toric varieties, Newton polyhedra, Okounkov bodies Local \(h^*\)-polynomials of some weighted projective spaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This long and carefully written paper is the fourth in the author's impressive series on the structure of sets of solutions to systems of equations in a free group [for part II cf. Isr. J. Math. 134, 173-254 (2003; Zbl 1028.20028)].
The structural results obtained in the first papers in the sequence are applied to analyze \(\forall\exists\) sentences. An iterative procedure is associated with an \(\forall\exists\) sentence to produce a sequence of varieties and formal solutions defined over them. The author proves that the complexity of Diophantine sets associated with these varieties produced along the procedure strictly increases. It follows that the procedure terminates after finitely many steps.
The outcome of the iterative procedure can be viewed as a stratification theorem generalising Merzlyakov's known result from positive sentences to general \(\forall\exists\) ones. On every stratum an \(\forall\exists\) sentence can be validated using a finite set of formal solutions. equations over groups; solutions to systems of equations; free groups; elementary sets; positive sentences; quantifier elimination; Diophantine geometry over groups; Makanin-Razborov diagrams: Diophantine sets Z. Sela, ''Diophantine Geometry over Groups. IV: An Iterative Procedure for Validation of a Sentence,'' Isr. J. Math. 143, 1--130 (2004). Word problems, other decision problems, connections with logic and automata (group-theoretic aspects), Free nonabelian groups, Quasivarieties and varieties of groups, Applications of logic to group theory, Diophantine equations in many variables, Noncommutative algebraic geometry, Decidability of theories and sets of sentences, Basic properties of first-order languages and structures, Model-theoretic algebra Diophantine geometry over groups. IV: An iterative procedure for validation of a sentence. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems ''This article contains a proof of an important theorem in soliton mathematics. The theorem, stated roughly by \textit{I. M. Krichever} [Funkts. Anal. Prilozh. 11, No.1, 15-31 (1977; Zbl 0346.35028)], contains necessary conditions for the existence of a vector function \(\psi (t,p)=(\psi_ 1(t,p),...,\psi_{\ell}(t,p))\), \(t\in {\mathbb{C}}^ g\), \(p\in R\), with prescribed poles and \(\ell\) essential singularities on a compact Riemann surface R of genus g. \(\psi\) is called a Baker function. This report clarifies Krichever's description of \(\psi\) for \(\ell >1\) essential singularities. The divisors \(\delta_{{\hat \alpha}}\), defined in the paper, are the key to the \(\ell >1\) construction. \textit{E. Previato} applied our characterization of the \(\delta_{{\hat \alpha}}\) to construct the finite gap solutions to the nonlinear Schrödinger equation, see her Thesis [''Hyperelliptic curves and solitons'', Harvard Univ. (1983), see also Duke Math. J. 52, 329-377 (1985; Zbl 0578.35086)].'' Jacobi variety; Riemann surface; Baker function; divisors; Schrödinger equation R. J. Schilling, ''Baker functions for compact Riemann surfaces,'' Proc. Amer. Math. Soc., vol. 98, iss. 4, pp. 671-675, 1986. Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Jacobians, Prym varieties, Riemann surfaces, Ergodic theorems, spectral theory, Markov operators Baker functions for compact Riemann surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(k\) be a number field, \(S\) a smooth irreducible curve, \(\mathcal{A} \rightarrow S\) an abelian scheme of relative dimension \(g \geq 2\), and \(C\) an irreducible curve in \(\mathcal{A}\) not containing in a proper subgroup scheme of \(\mathcal{A}\), even after a finite base extension.
Suppose that \(S\), \(\mathcal{A}\) and \(C\) are defined over \(k\).
Then, in this paper, it is proved that the intersection of \(C\) with the union of all flat subgroup schemes of \(\mathcal{A}\) of codimension at least 2 is a finite set. Furthermore, if the intersection of \(C\) with the union of all flat subgroup schemes of \(\mathcal{A}\) of codimension at least \(m\), where \(1 \leq m \leq g\), is infinite, it is proved, using the above result, that there exists a finite cover \(S^{\prime} \rightarrow S\) such that \(C\times_S S^{\prime}\) is contained in a flat subgroup scheme of \(\mathcal{A}\times_S S^{\prime}\) of codimension at least \(m-1\).
This result has an interesting application in the study of solvability of almost Pell-equations in polynomials. More precisely, let \(S\) be a smooth irreducible curve defined over a number field \(k\), \(k(S)\) its function field and \(\overline{k(S)}\) the algebraic closure of \(k(S)\). Further, let \(D,F\in k(S)[X]\setminus \{0\}\) with \(D\) squarefree of even degree \(\geq 6\). Suppose that the jacobian of the hyperelliptic curve defined by \(Y^2 = D(X)\) contains no one-dimensional abelian subvariety over \(\overline{k(S)}\). Then, if the equation \(A^2-DB^2 = F\) does not have a non-trivial solution in \(\overline{k(S)}[X]\), it is proved that there are at most finitely many \(s_0\in S(\mathbb{C})\) such that the specialized equation \(A^2-D_{s_0}B^2 = F_{s_0}\) has a solution \(A,B \in \mathbb{C}[X]\) with \(B \neq 0\). abelian variety; polynomial Pell equation; abelian scheme; flat subgroup scheme; elliptic surface Algebraic theory of abelian varieties, Elliptic curves over global fields, Global ground fields in algebraic geometry, Curves of arbitrary genus or genus \(\ne 1\) over global fields, Heights, Model theory (number-theoretic aspects) Unlikely intersections in families of abelian varieties and the polynomial Pell equation | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A celebrated result of Mumford states that a complex normal surface germ \((X,x)\) is smooth if and only if its local fundamental groups is trivial. The same conclusion holds if the topological fundamental group is replaced by the étale fundamental group of the punctured neighbourhood \(U\) of the singularity. This result breaks down in positive characteristic. \textit{M. Artin} [Complex Anal. algebr. Geom., Collect. Pap. dedic. K. Kodaira, 11--22 (1977; Zbl 0358.14008)] asked whether, if \(\pi^{\text{ét}}(U)\) is finite, there always is a finite morphism from a smooth scheme.
This note answers the question with the étale fundamental group replaced by the local Nori fundamental group scheme \(\pi_{\text{loc}}^N(U,X,x)\), which the authors define here by adapting Nori's original construction. The main result is that a surface singularity over an algebraically closed field is a rational singularity, if \(\pi_{\text{loc}}^N(U,X,x)\) is a finite group scheme, and if \(\pi_{\text{loc}}^N(U,X,x)=0\), then it is a rational double point. Using Artin's list it follows that in the last case \(X\) admits a finite morphism from a smooth scheme, and in fact \((X,x)\) is smooth except possibly in the cases \(E_8^1\) (\(p=2,3\)) and \(E_8^3\) (\(p=2\)). covering, local fundamental group; Nori group scheme; rational singularity Esnault, H.; Viehweg, E.: Surface singularities dominated by smooth varieties, J. reine angew. Math. 649, 1-9 (2010) Singularities in algebraic geometry, Coverings in algebraic geometry Surface singularities dominated by smooth varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Recent progress in the structure theory of higher-dimensional varieties has developed into different directions. One of them concentrates on the study of the geometry of curves on varieties: Based on classical geometric ideas, the book under review investigates rational curves on varieties of arbitrary dimension. Main techniques are the use of Hilbert and Chow schemes. This is worked out in chapter I in the necessary generality, taking into account especially the case of positive characteristic where the Chow-functor does not behave as well as in characteristic 0. Also, there are hints and references to the cases of algebraic and analytic spaces, respectively, as well as an appendix with some background from commutative algebra.
Chapter II applies the preceding techniques to study curves on varieties and morphisms from curves to varieties: rational curves yield a closed subvariety of the Chow variety. Néron-Severi's theorem is presented as a result of the investigation of the cone of curves. The chapter culminates in Mori's ``bend-and-break'' technique which is applied to show that for a projective variety \(X\) containing a curve having negative intersection with the canonical class, there exists such rational curve as well. Other applications are vanishing results in positive characteristic obtained using a construction of Ekedahl. The final section of this chapter is devoted to smoothing of morphisms of curves and contains results for reduced curves which are ``combs'' with rational ``teeth''.
The next chapter III of the book aims to give a short illustration of the ideas of Mori's minimal model program in some special cases. It contains a proof of the cone theorem following Mori's original arguments. There is a section devoted to the case of surfaces which presents the classical theory from the viewpoint of the minimal model theory, accompanied by an introduction to del Pezzo surfaces \(X\) with \((K_X^2)\leq 4\) over an arbitrary field. The following chapters constitute the main part of the book: Chapter IV on ``Rationally connected varieties'' studies birational properties of varieties covered by rational curves. There are important parallels between the class of uniruled and rationally connected varieties, some of their numerical properties being stable under smooth deformations (at least if we exclude the case of characteristic \(>0\); this is necessary for some parts of this chapter). The following topics are covered in different sections:
Ruled and uniruled varieties;
Minimizing families of rational curves;
Rationally connected varieties;
Growing chains of rational curves;
Maximal rationally connected fibrations;
Rationally connected varieties over nonclosed fields.
Chapter V on Fano varieties considers the higher-dimensional generalizations of del Pezzo surfaces which are of increasing interest for Mori's program. Though the singular case is not considered, this should be important for a theory of Fano varieties with terminal singularities as well. There are various assertions to support the following principle: ``The geometry of a Fano variety is governed by rational curves of low degree''. It is explained how (in principle) a list of all Fano varieties in any given dimension could be obtained. Further, there is a proof of Mori's theorem characterizing the projective space as the only algebraic variety with ample tangent bundle. The final sections study concrete cases; especially section 5 is devoted to nonrational Fano varieties.
The appendix (chapter VI) gives a proof of Abhyankar's theorem which says that any birational morphism \(X\to Y\) with \(Y\) smooth has only ruled exceptional divisors (without using resolution of indeterminacies). Further, there is an introduction to the intersection theory of divisors (following an unpublished paper of \textit{S. Kleiman}) which is applied to obtain the asymptotic Riemann-Roch theorem for nef line bundles.
This book will be of continuing interest as well as a source of reference as for its presentation of the material which is encouraging the reader to get through even with the technical demanding parts. It is (in a reasonable sense) self-contained and accompanied by many exercises. Mori theory; minimal models; Fano variety; Hilbert scheme; rational curves; Chow schemes; vanishing; positive characteristic; Mori's minimal model program; cone theorem; del Pezzo surfaces; Fano varieties Kollár, J., Rational Curves on Algebraic Varieties, (1995), Springer: Springer Berlin Minimal model program (Mori theory, extremal rays), Rational and ruled surfaces, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Parametrization (Chow and Hilbert schemes), Fano varieties, Rational and unirational varieties, Riemann-Roch theorems, Families, moduli of curves (algebraic), Families, moduli, classification: algebraic theory Rational curves on algebraic varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems For fixed (and admissible) values of the positive integers \(g\), \(d\) and \(n\), the author proves the existence of real smooth algebraic curves embedded into the \(n\)-dimensional real projective space, having genus \(g\) and degree \(d\), with prefixed number of real inflexional points, and prefixed number of nonreal inflexional points.
This beautiful result of real algebraic geometry is proved by using the good behaviour of some irreducible component of the Hilbert scheme of the complex projective \(n\)-dimensional space with respect to the canonical involution induced by complex conjugation. real inflexional points; Hilbert scheme; ramification point; deformations Plane and space curves, Real algebraic and real-analytic geometry, Projective techniques in algebraic geometry Real and non-real inflexional points of real curves in projective spaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let F be the field of real meromorphic functions on a compact connected real analytic surface. Given a positive integer n, we characterize those elements of F which can be written as a sum of 2n-th powers of elements of F. formally real function field; sum of 2n-th powers; real meromorphic functions; real analytic surface Fields related with sums of squares (formally real fields, Pythagorean fields, etc.), Complex spaces, Real algebraic and real-analytic geometry Sums of 2n-th powers of real meromorphic functions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(G\) be a reductive group and \(X\) a (smooth and complete) curve. Let \(\text{Bun}_G\) denote the moduli stack of \(G\)-torsors on \(X\); let \(D^b(\text{Bun}_G\)) be the appropriately defined derived category of constructible sheaves. Let \(\check G\) denote the Langlands dual group of \(G\). The nature of \(\check G\) depends on the sheaf-theoretic context one works in.
Over any ground field, one can work with \(\ell\)-adic sheaves (where \(\ell\) is different from the ground field); in this case \(\check G\) is an algebraic group over \(\overline{\mathbb Q_\ell}\). Let \(E_{\check G}\) be a \(\check G\)-local system on \(X\), thought of as a tensor functor \(V\mapsto V_{E_{\check G}}\) from the category \(\text{Rep}(\check G)\) of finite-dimensional \(\check G\)-representations to that of local systems (= lisse sheaves) on \(X\). In this case one introduces the notion of Hecke eigensheaf, which is an object \(\mathcal S(E_{\check G}) \in D^b(\text{Bun}_G)\), satisfying \(H^V(\mathcal S(E_{\check G} ))\simeq\mathcal S(E_{\check G} )\boxtimes V_{E_{\check G}} ,\) where \(H^V : D^b(\text{Bun}_G) \to D^b(\text{Bun}_G \times X)\) are the Hecke functors, defined for each \(V \in \text{Rep}(\check G)\).
A basic (but in general unconfirmed, and perhaps even imprecise) expectation is that for every \(E_{\check G}\) there corresponds a non-zero Hecke eigensheaf \(\mathcal S(E_{\check G})\). This is a weak form of the geometric Langlands conjecture. A stronger form of the conjecture, which only makes sense in the context of D-modules, says that the assignment \(E_{\check G}\mapsto\mathcal S(E_{\check G})\) should work in families. In other words, if \(E_{\check G,Y}\) is a \(Y\)-family of \(\check G\)-local systems, where \(Y\) is a scheme over a ground field \(k\), then to it there should correspond a \(Y\)-family \(\mathcal S(E_{\check G,Y})\). The necessity to use D-modules here, as opposed to any other sheaf-theoretic context, is that it is only in this case that we have a reasonable notion of \(Y\)-families of objects of \(D^b(\mathcal X)\) on a scheme (or stack) \(\mathcal X\).
The strongest (and wildest) form of the geometric Langlands conjecture says that the above assignment should give rise to an equivalence between the category \(D^b(\text{Bun}_G)\) and the appropriately defined derived category of quasi-coherent sheaves on the stack \(\text{LocSys}_{\check G}\), classifying \(\check G\)-local systems on \(X\).
Let us consider the following intermediate case. Let \(E_{\check G}\) be a fixed local system, and let \(E_{\check G,Y}\) be its formal deformation.
Suppose we have found \(\mathcal S(E_{\check G})\) which is a Hecke eigensheaf with respect to \(E_{\check G}\). Can we extend \(\mathcal S(E_{\check G})\) to a \(Y\)-family \(\mathcal S(E_{\check G,Y} )\) of eigensheaves? This is the question that V. Drinfeld asked on several occasions.
Let us take \(Y\) to be \(\text{Def}(E_{\check G})\) -- the base of the universal deformation of \(E_{\check G}\) as a \(\check G\)-local system. In this case, \(Y\) is indeed quasi-isomorphic to the standard complex of a DG-algebra canonically attached to \(E_{\check G}\). Unfortunately, even the assignment \(E_{\check G}\to \mathcal S(E_{\check G})\) has been constructed only in few cases. One such case is when \(G = \text{GL}_n\) and \(E_{\check G} = E_n\) is an \(n\)-dimensional irreducible local system.
The case that the authors study, is, in some sense, the opposite one. The authors take \(\check G\) to be arbitrary, but \(E_{\check G}\) is assumed ``maximally reducible'', i.e. \(E_{\check G}\) is induced from a local system \(E_{\check T}\) with respect to the Cartan group \({\check T} \subset {\check G}\). In this case the corresponding Hecke eigensheaf was earlier constructed by the authors, under the name ``geometric Eisenstein series''. Denote it by \(\overline{\text{Eis}}(E_{\check T})\). Along with the geometric Eisenstein series \(\overline{\text{Eis}}(E_{\check T})\) there exists a more naive object called ``classical'' Eisenstein series and denoted by \({\text{Eis}}_!(E_{\check T})\). When we work over the finite ground field and \(\ell\)-adic sheaves, \({\text{Eis}}_!(E_{\check T})\) goes over under the \textit{faisceaux-fonctions} correspondence to the usual Eisenstein series as defined in the theory of automorphic forms.
Drinfeld's conjecture is that the family \(\mathcal S(E_{\check G,\text{Def}_{\check B} (E_{\check T})})\) is nothing but (a certain completion of) \({\text{Eis}}_!(E_{\check T})\). This statement has an ideological significance also for the classical (i.e. function theoretic vs. sheaf-theoretic) Langlands correspondence:
\textit{The classical Eisenstein series correspond not to homomorphisms \(\mathrm{Galois} \longrightarrow \check G\) that factor through \(\check T\), but rather to the universal family of homomorphisms \(\mathrm{Galois}\longrightarrow {\check B}\) with a fixed composition \(\mathrm{Galois}\longrightarrow {\check B} \longrightarrow\!\!\!\to \check T\).}
The present paper is devoted to the proof of Drinfeld's conjecture, under a certain simplifying hypothesis on \(E_{\check T}\). Namely, the authors assume that \(E_{\check T}\) is regular, i.e. that it is as non-degenerate as possible. This means that for every co-root \(\check\alpha\) of \(G\), which is the same as a root of \(\check G\), the induced 1-dimensional local system \(\check\alpha(E_{\check T})\) is non-trivial.
This regularity assumption is equivalent to requiring that the DG formal scheme Def\(_ {\check B} (E_{\check T})\) be an ``honest'' scheme.
Moreover, the authors show that in this case both versions of Eisenstein series, namely, \(\overline{\text{Eis}}(E_{\check T})\) and \({\text{Eis}}_!(E_{\check T})\) are perverse sheaves. This fact and the simplified nature of Def\(_{\check B} (E_{\check T})\) makes life significantly easier, since one can avoid a lot of complications of homotopy-theoretic nature.
Proving the above conjecture amounts to the following:
(i) Exhibiting the action of the commutative algebra \(\mathcal O_{\text{Def}_ {\check B} (E_{\check T} )}\) of functions on Def\(_{\check B} (E_{\check T})\) on \({\text{Eis}}_!(E_{\check T})\).
(ii) Establishing an isomorphism \(\mathbb C\overset {L} {\underset \mathcal O_{\text{Def}_ {\check B}(E_{\check T})} \otimes} {\text{Eis}}_!(E_{\check T})\simeq \overline{\text{Eis}}(E_{\check T})\).
(iii) Verifying the Hecke property \(\text{H}^V (\text{Eis}_!(E_{\check T}))\simeq \text{Eis}_!(\check T) \underset{\mathcal O_{\text{Def}_ {\check B}(E_{\check T})}}\boxtimes V_{E_{\check G,\text{Def}_ {\check B}(E_{\check T})}}\), where \(V_{E_{\check G,\text{Def}_ {\check B}(E_{\check T})}}\) is the canonical family of \(\check G\)-local systems over \(\text{Def}_{\check B} (E_{\check T})\).
As is to be expected, the verification of these properties is a nice simple exercise when \(G = \text{GL}_2\), that the authors perform, but not altogether trivial for other groups. Eisenstein series; Hecke eigensheaf; Hecke property; geometric Langlands correspondence; Drinfeld's conjecture; Drinfeld's compactifications; space of deformations; IC sheaves; Koszul complex; Koszul duality; deforming local systems; extension by zero Braverman, A.; Gaitsgory, D., \textit{deformations of local systems and Eisenstein series}, Geom. Funct. Anal., 17, 1788-1850, (2008) Langlands-Weil conjectures, nonabelian class field theory, Geometric Langlands program (algebro-geometric aspects), Geometric class field theory Deformations of local systems and Eisenstein series | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We consider the question of irreducibility of the Hilbert scheme of points \(\mathcal H\mathrm{ilb}_d\mathbb P^n\) and its Gorenstein locus. This locus is known to be reducible for \(d \geq 14\). For \(d \leq 11\) the irreducibility of this locus was proved in the series of papers [\textit{G. Casnati} and \textit{R. Notari}, J. Pure Appl. Algebra 213, No. 11, 2055--2074 (2009; Zbl 1169.14003); ibid. 215, No. 6, 1243--1254 (2011; Zbl 1215.14009); ibid. 218, No. 9, 1635--1651 (2014; Zbl 1287.13013)] and Iarrobino conjectured that the irreducibility holds for \(d \leq 13\). In this paper, we prove that the subschemes corresponding to the Gorenstein algebras having Hilbert function \((1,5,5,1)\) are smoothable, i.e. lie in the closure of the locus of smooth subschemes. This result completes the proof of irreducibility of the Gorenstein locus of \(\mathcal H\mathrm{ilb}_{12}\mathbb P^n\), see Theorem 2. smoothability; zero-dimensional schemes; Gorenstein algebras; Hilbert scheme J. Jelisiejew, \textit{Local finite-dimensional Gorenstein k-algebras having Hilbert function (1,5,5,1) are smoothable}, J. Algebra App., 13 (2014), 1450056. Deformations and infinitesimal methods in commutative ring theory, Parametrization (Chow and Hilbert schemes), Commutative Artinian rings and modules, finite-dimensional algebras, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) Local finite-dimensional Gorenstein \(k\)-algebras having Hilbert function (1,5,5,1) are smoothable | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We classify the linearly reductive finite subgroup schemes \(G\) of \(\mathrm{SL}_2=\mathrm{SL}(V)\) over an algebraically closed field \(k\) of positive characteristic, up to conjugation. As a corollary, we prove that such \(G\) is in one-to-one correspondence with an isomorphism class of two-dimensional \(F\)-rational Gorenstein complete local rings with the coefficient field \(k\) by the correspondence \(G\mapsto \left ((\mathrm {Sym}V)^{G}\right)\widehat {}\). group scheme; Kleinian singularity; invariant theory Has1 M.~Hashimoto, Classification of the linearly reductive finite subgroup schemes of \(SL_2\), Acta Math. Vietnam. \textbf 40 (2015), no. 3, 527--534. Group schemes, Actions of groups on commutative rings; invariant theory Classification of the linearly reductive finite subgroup schemes of \(\mathrm{SL}_2\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper studies the wall-chamber structure of the Bridgeland stability manifold on \(\mathbb{P}^3\), and obtains several results on the relevant moduli spaces and Hilbert schemes. By the work of \textit{E. Macrì} [Algebra Number Theory 8, No. 1, 173--190 (2014; Zbl 1308.14016)], Bridgeland stability conditions on \(\mathbb{P}^3\) are constructed, and this paper can be regarded as a natural sequel.
As for the wall-chamber analysis, a general statement (Theorem 1.3, 6.1) is shown, which enables one to compute tilt stabilities in the sense of
\textit{A. Bayer} et al. [J. Algebr. Geom. 23, No. 1, 117--163 (2014; Zbl 1306.14005)], and thus to do a similar analysis as in the case of surfaces. As an application, in the case of complete intersections of two hypersurfaces or twisted cubics, it is shown that there are two chambers in the stability manifold where the moduli space is a smooth projective irreducible variety and the Hilbert scheme respectively. Also obtained are all the walls and moduli spaces on a path between the two chambers in the case of twisted cubics (Theorem 1.1, 7.1). In particular, a new proof (Theorem 1.2,7.2) is given for the global description of the Hilbert scheme of twisted cubics by \textit{G. Ellingsrud} et al. [Lect. Notes Math. 1266, 84--96 (1987; Zbl 0659.14027)]. Bridgeland stability conditions; wall-crossings; moduli spaces of stable objects, Hilbert scheme of twisted cubics Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Algebraic moduli problems, moduli of vector bundles, Parametrization (Chow and Hilbert schemes) Bridgeland stability on threefolds: some wall crossings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Two kinds of triangular systems are studied: normalized triangular polynomial systems (a weaker form of Lazard's triangular sets [\textit{D. Lazard}, Discrete Appl. Math. 33, No. 1-3, 147-160 (1991; Zbl 0753.13013)] and constructible triangular systems (involved in the dynamic constructible closure programs of \textit{T. Gómez-Díaz} [Quelques applications de l'évaluation dynamique, Ph.D. Thesis, Université de Limoges (1994)]. This paper shows that these notions are strongly related. In particular, combining the two points of view (constructible and polynomial) on the subject of square-free conditions, it allows us to effect dramatic improvements in the dynamic constructible closure programs. normalized triangular polynomial systems; Lazard's triangular sets; constructible triangular systems Dellière, S.: On the links between triangular sets and dynamic constructible closure. J. pure appl. Algebra 163, No. 1, 49-68 (2001) Symbolic computation and algebraic computation, Computational aspects in algebraic geometry On the links between triangular sets and dynamic constructible closure | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems As the title indicates, the book under review provides the basic material for an introductory course on the theory of functions of one complex variable and its applications. The text primarily aims at beginners in the field, particularly at advanced undergraduate students in mathematics, physical sciences, informatics, and engineering. Accordingly, the prerequisites are appropriately modest, thereby assuming just a basic knowledge of real analysis and elementary linear algebra, whereas the necessary rudiments of general topology are developed in the course of the text. Taking into account that complex analysis is a completely new experience for most undergraduates, who are used to think in terms of real calculus, the author has set a high value on translating into action the specific didactic aspects of teaching the subject. In fact, being an utmost experienced teacher and expository writer in mathematics, the author has summoned up all his expertise to present an introduction to complex function theory that captivates by its lucidity, comprehensibility, coherence, and optical appeal likewise. Numerous applications of the concepts, methods, and results of complex analysis accompany each single chapter of the book, thereby demonstrating the ubiquitous significance of complex-analytic methods in various areas of mathematics, illustrating its intrinsic beauty, and serving as an essential pedagogical means to propel both the readers motivation and her or his thirst for knowledge in complex function theory.
The current text is composed of five chapters, each of which is divided in several sections. While the first three chapters form the core of the introductory part of the book, the remaining two chapters treat some more advanced and specific topics building upon the previously developed basics. Chapter 1 introduces the field of complex numbers, complex differentiable functions, and the complex logarithm. Apart from a thorough comparison of the concepts of real and complex differentiability, this chapter discusses first applications of complex methods to computations of trigonometric sums as well as to differential equations, complex numbers in geometry, complex numbers in electricity theory, harmonic functions, and dynamics. Chapter 2 treats curvilinear integrals of holomorphic functions, including Cauchy's integral formula, Cauchy's theorem, and its various consequences. The applications of this material concern the Dirichlet problem, Poisson kernels, Green's function, and some more plane dynamics.
Chapter 3 is devoted to isolated singularities, Laurent series, winding numbers, and the important residue calculus. This is then applied to computations of real integrals, to integral transforms à la Fourier and Laplace, and to other related standard topics. Chapter 4, building upon the residue theorem, focuses on the construction of holomorphic functions with prescribed zeros and poles. This discussion includes the concept of meromorphic functions, analyticity at infinity, normal families of holomorphic functions, the Mittag-Leffler theorem, the Weierstrass product formula, the Gamma function, and an introduction to elliptic functions, together with all the fundamental classic theorems in this context. The applications of this rich material are very versatile and instructive, ranging from explicit computations of infinite series to such important topics like asymptotic expansions of holomorphic functions, fractal geometry, non-euclidean geometry, and Riemann's zeta function. Chapter 5 turns to more advanced topics in geometric function theory. Linear-fractional transformations (Möbius transformations), which are used as a crucial ingredient throughout the entire text, are here investigated more closely. They are then used to derive further fundamental theorems in complex analysis, among those being the marvellous Riemann's mapping theorem, the monodromy theorem, Carathéodory's theorem on the extension of holomorphic functions, and Schwarz's reflection principle.
The applications of the methods and results described in this final chapter are again far-reaching and highly instructive. The reader meets here the Schwarz-Christoffel formula for domains bounded by a polygon, elliptic integrals and Jacobi's elliptic functions, and analytic elliptic curves via the Weierstrass \(\wp\)-function.
The whole text is enhanced by a wealth of illustrating examples and a large number of carefully selected exercises, through which the masterly text is substantially enriched from the didactic point of view. In addition to the many outstanding features of the current student-friendly primer of complex function theory, the multi-coloured layout of the book, which represents a special service to the novice in the field, deserves particularly laudable mention, just as the distinguished hints for further reading by means of the carefully compiled bibliography at the end of the book. Also, there are a helpful list of symbols and an extensive index.
No doubt, this purposefully composed basic course in complex function theory is a didactic masterpiece, which offers much more than the modest title might suggest. Everyone interested in the subject can profit a great deal from the author's expertise reflected by this outstanding textbook of remarkably multifarious character, including instructors and grammar-school masters. functions of a complex variable; holomorphic functions; meromorphic functions; geometric function theory; elliptic functions; elliptic integrals; elliptic curves Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functions of a complex variable, Analytic continuation of functions of one complex variable, Maximum principle, Schwarz's lemma, Lindelöf principle, analogues and generalizations; subordination, Meromorphic functions of one complex variable (general theory), Analytic theory of abelian varieties; abelian integrals and differentials Foundations of function theory. An introduction to complex analysis and its applications. For Bachelor and diploma degrees | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper is motivated by the search of a satisfactory notion of \textit{irreducibility} for semialgebraic sets. Usually, a geometric object is irreducible if it is not the union of two proper geometric objects of the same nature. This definition does not work in the semialgebraic seting because otherwise any semialgebraic set with at least two points would be reducible. The authors thus propose an algebraic definition of irreducibility: A semialgebraic set \(S\subset \mathbb R^n\) is irreducible if the neotherian ring \(\mathcal N(S)\) of Nash functions on \(S\) is an integral domain.
A satisfactory theory of irreducible components of semialgebraic sets is developed. The authors characterize the families of semialgebraic sets for which several classical problems, including Hilbert's 17th Problem and real Nullstellensatz, have an affirmative solution. The solutions to these classical questions were only known previously in case \(S=M\) is an affine Nash manifold. irreducible semialgebraic set; irreducible components of a semialgebraic set; Nash function; Nash set; w-Nash set; q-Nash set; substitution theorem; positivstellensätze; 17th Hilbert problem and real nullstellensatz Fernando, José F.; Gamboa, J. M., On the irreducible components of a semialgebraic set, Internat. J. Math., 23, 4, 1250031, 40 pp., (2012) Semialgebraic sets and related spaces, Nash functions and manifolds, Real-analytic and Nash manifolds On the irreducible components of a semialgebraic set | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We consider the Berglund-Hübsch-Henningson-Takahashi duality of Landau-Ginzburg orbifolds with a symmetry group generated by some diagonal symmetries and some permutations of variables. We study the orbifold Euler characteristics, the orbifold monodromy zeta functions and the orbifold E-functions of such dual pairs. We conjecture that we get a mirror symmetry between these invariants even on each level, where we call level the conjugacy class of a permutation. We support this conjecture by giving partial results for each of these invariants. invertible polynomial; group action; dual pairs; orbifold Euler characteristic; orbifold monodromy zeta function; E-function Group actions on affine varieties, Mirror symmetry (algebro-geometric aspects), Topology and geometry of orbifolds, Monodromy on manifolds Mirror symmetry on levels of non-abelian Landau-Ginzburg orbifolds | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be an abelian scheme over a scheme \(B\). The Fourier-Mukai transform gives an equivalence between the derived category of \(X\) and the derived category of the dual abelian scheme. We partially extend this to certain schemes \(X\) over \(B\) (which we call degenerate abelian schemes) whose generic fiber is an abelian variety, while special fibers are singular.
Our main result provides a fully faithful functor from a twist of the derived category of \(\text{Pic}^{\tau}(X/B)\) to the derived category of \(X\). Here \(\text{Pic}^{\tau}(X/B)\) is the algebraic space classifying fiberwise numerically trivial line bundles.
Next, we show that every algebraically integrable system gives rise to a degenerate abelian scheme and discuss applications to Hitchin systems. Fourier-Mukai transform; abelian scheme; Picard space; integrable system; Hitchin system; Langlands duality Geometric Langlands program (algebro-geometric aspects), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Picard schemes, higher Jacobians, Group schemes Partial Fourier-Mukai transform for integrable systems with applications to Hitchin fibration | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors derive properties mainly of the lemniscatic sine function \(\text{sl}:{\mathbb C}\to{\mathbb C}\cup\{\infty\}\) from two applications:
(1) a conformal mapping from an open square \(\mathcal S\) to a unit disc \(D\);
(2) a parameterization of an elliptic curve.
The article provides an arclength parameterization of the lemniscate wherefrom the authors deduce among others: ``The family of lemniscate parallels is self-orthogonal.'' Moreover, the lemniscate is parametrizable by a simple rational expression in the lemniscatic sine function. ``The beautiful structure of this parametrization becomes fully visible only when complex values of the arclength parameter are allowed and the lemniscate is viewed as complex curve. To visualize such hidden structure'', the authors show how ordinary chessboards are turned to lemniscatic chessboards. lemniscatic sine function; lattice of periods; Riemann surface of genus \(1\); angle addition formulas for lemniscatic functions; constructible number; Abel's result Joel Langer and David Singer, The lemniscatic chessboard, Forum Geometricorum, Vol. 11 (2011), 183--199. Elliptic functions and integrals, Plane and space curves, Tilings in \(2\) dimensions (aspects of discrete geometry), Polyhedra and polytopes; regular figures, division of spaces The lemniscatic chessboard | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let Z be a closed set of a noetherian scheme X. We define the connectedness dimension c(Z) of Z as the minimal value of dim(W), where W runs through all closed subsets of Z such that Z-W is disconnected. Using standard tools from local cohomology (Mayer-Vietoris sequences and vanishing theorems) we give bounds for c(Z) and connectedness criteria for Z. This gives a unified approach to study connectivity of intersections and of fibers. In particular one gets easy proofs for the connectedness theorem of Fulten-Hansen, and for (sharpened versions) of Zariski's connectedness theorem for projective morphisms and also for the connectedness statements of Bertini's theorems. Bertini theorems; connectedness theorems; noetherian scheme; connectedness dimension; local cohomology; intersections; fibers DOI: 10.1007/BF02621928 Topological properties in algebraic geometry, Local cohomology and algebraic geometry, Dimension theory in general topology, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Local cohomology and the connectedness dimension in algebraic varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\mathcal I_{d,g,r}\) denote the Hilbert scheme parametrizing smooth curves of degree \(d\) and genus \(g\) in complex projective space \(\mathbb P^{r}_{\mathbb C}\).
\textit{L. Ein} proved \(\mathcal I_{d,g,r}\) is irreducible if \(d \geq \frac{(2r-2)g+(2r+3)}{r+2}\) [Ann. Sci. Éc. Norm. Supér. (4) 19, No. 4, 469--478 (1986; Zbl 0606.14003); Proc. Symp. Pure Math. 46, 83--87 (1987; Zbl 0647.14012)], which proves Severi's claim of irreducibility for \(d \geq g+r\). He also gave examples showing that Severi's claim fails for \(r \geq 6\).
In the paper under review, the author expands the known irreducibility range in case \(r=5\), showing that \(\mathcal I_{d,g,r}\) is irreducible for \(d \geq \max\{\frac{11}{10}g+2,g+5\}\). The outline of the proof runs as follows. If \(V \subset \mathcal I_{d,g,r}\) is any irreducible component, then \(V\) is generically a fibre bundle over a closed subset of a component \(\mathcal G \subset \mathcal G^{r,d}\) with fibre \(\text{Aut}(\mathbb P^{r})\) and there is \(\alpha \geq r\) such that \(\mathcal G\) is generically a fibre bundle over a closed subset of an irreducible component \(\mathcal W \subset \mathcal W^{\alpha,d}\) with fibre \({\roman Gr}(r,\alpha)\), thus we arrive at the inequality
\[
(r+1)d-(r-3)(g-1) \leq \mathcal W + r^{2}+2r+(r+1)(\alpha-r)
\]
(the spaces \(\mathcal G^{r,d}\) and \(\mathcal W^{\alpha,d}\) are the global versions of \(G^{r,d}(C)\) and \(W^{\alpha,d}(C)\) from the book of \textit{E. Arbarello, M. Cornalba, P. A. Griffiths} and \textit{J. Harris} [Geometry of Algebraic Curves, Grundlehren der Mathematischen Wissenschaften, 267. New York etc.: Springer-Verlag (1985; Zbl 0559.14017)]).
Now if the expected dimension \(\rho(d,g,r)=g-(r+1)(g-d+r)\) is positive, one constructs a unique irreducible component \(V \subset \mathcal I_{d,g,r}\) which dominates the moduli space \(\mathcal M_{g}\). If \(V_{1}\) were a different irreducible component, we would produce \(\mathcal W\) as above which does not dominate \(\mathcal M_{g}\), but then a series of estimates contradicts the inequality above. Hilbert scheme; Brill-Noether theory; linear series H. Iliev, On the irreducibility of the Hilbert scheme of curves in \(\mb{P}^5\), Comm. Algebra 36 (2008), no. 4, 1550--1564. Families, moduli of curves (algebraic), Parametrization (Chow and Hilbert schemes) On the irreducibility of the Hilbert scheme of curves in \(\mathbb P^{5}\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(k\) be a global field. The first author, \textit{B. Fein} and \textit{M. Schacher} [Bull. Am. Math. Soc., New Ser. 1, 766-768 (1979; Zbl 0421.12016)] determined the Ulm invariants of the Brauer group \(\text{Br}(k(t))\) of a rational function field in one variable over \(k\). Now let \(K\) be a non-rational function field of genus zero over \(k\), that is, \(K\) is the quotient field of \(k[x,y]/\langle 1-cx^2-dy^2\rangle\) with \(c,d\) in \(k^*\) where the quaternion algebra \((c,d)\) is not split over \(k\). Then the same authors showed [in J. Algebra 114, No. 2, 479-483 (1988; Zbl 0658.12006)] that all but one of the Ulm invariants of \(\text{Br}(K)\) coincide with those of \(\text{Br}(k(t))\). The main theorem of this paper is to give a class of examples of non-rational function fields of genus zero where the last remaining Ulm invariant is the same as that of \(\text{Br}(k(t))\). In particular, if \(k\) is a totally real Abelian number field not containing \(\sqrt 2\) and \(K\) is a genus zero non-rational extension of \(k\) split by \(k(\sqrt 2)\), then \(\text{Br}(K)\cong\text{Br}(k(t))\). Brauer groups; genus zero function fields; Ulm invariants; Leopoldt conjecture Brauer groups (algebraic aspects), Algebraic functions and function fields in algebraic geometry, Galois cohomology, Arithmetic theory of algebraic function fields Brauer groups of genus zero extensions of number fields. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper, one can find the construction of a \textit{slit analytic fibre space of complex Lie groups}, according to the terminology of \textit{K. Kato} and \textit{S. Usui} [Classifying spaces of degenerating polarized Hodge structures.Annals of Mathematics Studies 169. Princeton, NJ: Princeton University Press. (2009; Zbl 1172.14002)], which \textit{graphs admissible normal functions with no singularities}, in a similar way as the classical Néron model [\textit{S. Bosch, W. Lütkebohmert} and \textit{M. Raynaud}, Néron models. Ergebnisse der Mathematik und ihrer Grenzgebiete, 3. Folge, 21. Berlin etc.: Springer-Verlag. (1990; Zbl 0705.14001)] graphs admissible normal functions arising from families of curves (the definition of admissible normal function can be found in: [\textit{M. Saito}, J. Algebr. Geom. 5, No.2, 235--276 (1996; Zbl 0918.14018)].
The contents of this paper are best presented by the authors' abstract: ``We show that the limit of a one-parameter admissible normal function with no singularities lies in a non-classical sub-object of the limiting intermediate Jacobian. Using this, we construct a Hausdorff slit analytic space, with complex Lie group fibres, which `graphs' such normal functions. For singular normal functions, an extension of the sub-object by a finite group leads to the Néron models. When the normal function comes from geometry, that is, a family of algebraic cycles on a semistably degenerating family of varieties, its limit may be interpreted via the Abel-Jacobi map on motivic cohomology of the singular fibre, hence via regulators on \(K\)-groups of its substrata. Two examples are worked out in detail, for families of 1-cycles on CY and abelian 3-folds, where this produces interesting arithmetic constraints on such limits. We also show how to compute the finite `singularity group' in the geometric setting.'' Néron model; slit analytic space; Abel-Jacobi map; admissible normal function; variation of Hodge structure; limit mixed Hodge structure; motivic cohomology; unipotent monodromy; semistable reduction; algebraic cycle; higher Chow cycle; Ceresa cycle; Clemens-Schmid sequence; polarization; slit analytic space Green, Mark; Griffiths, Phillip; Kerr, Matt, Néron models and limits of Abel-Jacobi mappings, Compos. Math., 0010-437X, 146, 2, 288-366, (2010) Algebraic cycles, Transcendental methods, Hodge theory (algebro-geometric aspects), Applications of methods of algebraic \(K\)-theory in algebraic geometry, Fibrations, degenerations in algebraic geometry, Variation of Hodge structures (algebro-geometric aspects), Motivic cohomology; motivic homotopy theory, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Néron models and limits of Abel-Jacobi mappings | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a projective curve over an algebraically closed field \(k\) of positive characteristic \(p\), and let \(\mathcal L\) be a base point free line bundle on \(X\). Set \(B=\bigoplus_{n\geq 0} H^{0}(X,{\mathcal L}^{\otimes n})\) and \(B_1= H^{0}(X,{\mathcal L})\). Then the Hilbert-Kunz (for short HK) multiplicity of the section ring \(B\) with respect to the ideal \(B_{1}B\) is denoted by \(\text{HKM}(X,\mathcal L)\). The HK multiplicity of \(B\) with respect to the ideal generated by \(W\subseteq H^0(X,\mathcal L)\) where \(W\) is a base point free linear system, is denoted by \(\text{HKM}(X,\mathcal L,W)\). Let \(V_{\mathcal L}(W)\) denote a vector bundle of rank \(r=\) vector space dimension of \(W -1\) and be the kernel of the surjective map \(W\times {\mathcal O}_X \rightarrow \mathcal L\). If \(W=H^0(X,\mathcal L)\), then \(V_{\mathcal L}(W)\) is denoted by \(V_{\mathcal L}\,.\) The author proves that if \(V_{\mathcal L}\) is strongly semistable then \(\text{HKM}(X,\mathcal L)\) is equal to the HK multiplicity of the section ring with respect to its graded maximal ideal (it may not be true in general). For an arbitrary base-point free line bundle \(\mathcal L\) on a nonsingular curve \(X\) of genus \(g\), the author finds an expression of \(\text{HKM}(X,\mathcal L, W)\) in terms of the ranks and degrees of the vector bundles occuring in a strongly stable Harder-Narasimham filtration. Even if this seems difficult to use, this result implies that the HK multiplicity of an irreducible projective curve is a rational number [\textit{H. Brenner}, Math. Ann. 334, No. 1, 91--110 (2006; Zbl 1098.13017) got the same result independently]. Section \(5\) is devoted to plane curves. Theorem \(5.3\) gives a formula for the HK multiplicity of an arbitrary plane curve \(C\) of degree \(d\) over a field of characteristic \(p>0\); corollary \(5.4\) gives a formula in the case that \(X\) is a nonsingular plane curve of degree \(d\). At the end of the paper, the author recalls some results of Monsky about nonsingular quartics of a certain type. Hilbert-Kunz function; projective curves; vector bundles Trivedi V, Semistability and Hilbert-Kunz multiplicities for curves, J. Algebra 284 (2005) 627--644 Vector bundles on curves and their moduli, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Singularities of curves, local rings Semistability and Hilbert-Kunz multiplicities for curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In the 20th century, algebraic geometry has undergone several revolutionary changes with respect to its conceptual foundations, technical framework, and intertwining with other branches of mathematics. Accordingly the way it is taught has gone through distinct phases. The theory of algebraic schemes, together with its full-blown machinery of sheaves and their cohomology, being for now the ultimate stage of this evolution process in algebraic geometry, had created -- around 1960 -- the urgent demand for new textbooks reflecting these developments and (henceforth) various facets of algebraic geometry. The famous volumes ``Éléments de géométrie algébrique'' as a series in Publ. Math., Inst. Hautes Étud. Sci. (1960-1967) by \textit{A. Grothendieck} and \textit{J. Dieudonné} were entirely written in the new language of schemes, without being linked up with the classical roots, and the so far existing textbooks just dealed with classical methods. It was \textit{David Mumford}, who at first started the project of writing a textbook on algebraic geometry in its new setting. His mimeographed Harvard notes ``Introduction to algebraic geometry: Preliminary version of the first three chapters'' (bound in red) were distributed in the mid 1960's, and they were intended as the first stage of a forthcoming, more inclusive textbook. For some years, these mimeographed notes represented the almost only, however utmost convenient and abundant source for non-experts to get acquainted with the basic new concepts and ideas of modern algebraic geometry. Their timeless utility, in this regard, becomes apparent from the fact that two reprints of them have appeared, since 1988, as a proper book under the title ``The red book of varieties and schemes'' [cf. Lect. Notes Math. 1358 (1988; Zbl 0658.14001)]. In the process of exending his Harvard notes to a comprehensive textbook, the author's teaching experiences led him to the didactic conclusion that it would be better to split the book into two volumes, thereby starting with complex projective varieties (in volume I), and proceeding with schemes and their cohomology (in volume II). -- In 1976, the author published the first volume under the title ``Algebraic geometry. I: Complex projective varieties'' (1976; Zbl 0356.14002; corrected second edition 1980; Zbl 0456.14001), where the corrections concerned the wiping out of some misprints, inconsistent notations, and other slight inaccuracies.
The book under review is an unchanged reprint of this corrected second edition from 1980. Although several textbooks on modern algebraic geometry have been published in the meantime, Mumford's ``Volume I'' is, together with its predecessor ``The red book of varieties and schemes'', now as before, one of the most excellent and profound primers of modern algebraic geometry. Both books are just true classics!
As to the intended volume II of the book under review, the author planned to publish it in collaboration with \textit{D. Eisenbud} and \textit{J. Harris}. This would have been based on existing but unpublished notes of the author (partially revised by \textit{S. Lang}), but then the author and his co-authors came to the conclusion that such a second volume was not really what was needed anymore, because \textit{R. Hartshorne}'s famous book ``Algebraic geometry'' (1977; Zbl 0367.14001) already covered a good part of the material they had planned to include. Instead, D. Eisenbud and J. Harris published what they felt is needed more: a brief introduction to schemes [cf. \textit{D. Eisenbud} and \textit{J. Harris}, ``Schemes: The language of modern algebraic geometry'' (1992; Zbl 0745.14002)]. Their booklet may be regarded as a bridge between D. Mumford's thorough classic (under review) and the now existing several textbooks on ``scheme- theoretic'' algebraic geometry, including Hartshorne's book as well as Mumford's other classic, the ``Red book of varieties and schemes''. affine varieties; projective varieties; correspondences; linear systems; algebraic curves; algebraic surfaces; birational algebraic geometry David Mumford, \textit{Algebraic geometry. I}, Classics in Mathematics, Springer-Verlag, Berlin, 1995, Complex projective varieties, Reprint of the 1976 edition. Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Varieties and morphisms, Curves in algebraic geometry, Relevant commutative algebra, Rational and birational maps, Surfaces and higher-dimensional varieties Algebraic Geometry. I: Complex projective varieties. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \((X,x)\subset(\mathbb{C}^ n,x)\) be a germ of an analytic space and \(f:(X,x)\to(\mathbb{C},0)\) be an analytic map germ. Using polar curves and a decomposition technique for fibered graph multilinks a new approach is given for the computation of the zeta function of the monodromy of \(f\). singularities; map germs; zeta function; monodromy András Némethi, The zeta function of singularities, J. Algebraic Geom. 2 (1993), no. 1, 1 -- 23. Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Milnor fibration; relations with knot theory, Singularities in algebraic geometry The zeta function of singularities | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper is a short survey on the motivic Milnor fibre and motivic vanishing cycles introduced by Denef and Loeser, with emphasis on a motivic analogue of a conjecture of Steenbrink on the spectrum of certain hypersurface singularities.
Consider a smooth algebraic variety \(X\), in this review taken for simplicity over \(\mathbb C\), and a regular function \(f\) on \(X\). Recall [\textit{J. Denef} and \textit{F. Loeser}, J. Algebr. Geom. 7, No. 3, 505--537 (1998; Zbl 0943.14010)] that the motivic Milnor fibre of \(f\) at \(x \in \{f=0\}\subset X\) lives in a suitable Grothendieck ring of algebraic varieties, and that Steenbrink's Hodge spectrum \(sp(f,x)\) of \(f\) at \(x\) can be recovered from it. Suppose now that the singular locus of \(f\) is a curve and take a generic linear form \(g\) vanishing on \(x\). Then \(f+g^N\) has an isolated singularity at \(x\) for \(N\) large enough. There is an explicit formula, conjectured by Steenbrink and proven by Saito, for the difference \(sp(f+g^N)-sp(f,x)\).
The author, \textit{F. Loeser} and \textit{M. Merle} [Duke Math. J. 132, No. 3, 409--457 (2006; Zbl 1173.14301)] proved a motivic analogue and generalization of this formula. For this they first extend the motivic vanishing cycles construction of Denef-Loeser to the whole relative Grothendieck ring over \(X\), and they introduce a motivic iterated vanishing cycle class of \(f\) and \(g\), where now \(f\) and \(g\) are \textsl{arbitrary} regular functions on \(X\). The main theorem of [Zbl 1173.14301] is an equality in the appropriate Grothendieck ring relating these constructions, specializing to a formula for \(sp(f+g^N)-sp(f,x)\). In particular, when \(f\) and \(g\) are as in Steenbrink's original setting, this yields Saito's result. motivic Milnor fibre; iterated vanishing cycles; Hodge spectrum Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Applications of methods of algebraic \(K\)-theory in algebraic geometry, Deformations of complex singularities; vanishing cycles, Mixed Hodge theory of singular varieties (complex-analytic aspects), Zeta functions and \(L\)-functions, Singularities in algebraic geometry Motivic vanishing cycles and applications | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We study the generalized hypergeometric system introduced by \textit{I. M. Gel'fand, A. V. Zelevinskij} and \textit{M. M. Kapranov} [Funct. Anal. Appl. 23, No. 2, 94--106 (1989; Zbl 0721.33006)] and its relationship with the toric Deligne--Mumford (DM) stacks recently studied by \textit{L. Borisov, L. Chen} and \textit{G. G. Smith} [J. Am. Math. Soc. 18, No. 1, 193--215 (2005; Zbl 1178.14057)]. We construct series solutions with values in a combinatorial version of the Chen--Ruan (orbifold) cohomology and in the \(K\)-theory of the associated DM stacks. In the spirit of the homological mirror symmetry conjecture of Kontsevich, we show that the \(K\)-theory action of the Fourier--Mukai functors associated to basic toric birational maps of DM stacks are mirrored by analytic continuation transformations of Mellin--Barnes type. Mellin--Barnes integrals; Fourier--Mukai transforms; Gelfand--Kapranov--Zelevinsky hypergeometric systems; Toric Deligne--Mumford stacks; \(K\)-theory; mirror symmetry L. A. Borisov and R. Paul Horja, Mellin-Barnes integrals as Fourier-Mukai transforms, Adv. Math., 207 (2006), 876--927. Toric varieties, Newton polyhedra, Okounkov bodies, Calabi-Yau manifolds (algebro-geometric aspects), (Equivariant) Chow groups and rings; motives, Applications of methods of algebraic \(K\)-theory in algebraic geometry, \(K\)-theory of schemes, Other hypergeometric functions and integrals in several variables, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Mellin--Barnes integrals as Fourier--Mukai transforms | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Singular sectors \(\mathcal{Z}_{\mathrm{sing}}\) (loci of zeros) for real-valued non-positively defined partition functions \(\mathcal{Z}\) of \(n\) variables are studied. It is shown that \(\mathcal{Z}_{\mathrm{sing}}\) have a stratified structure where each stratum is a set of certain hypersurfaces in \(\mathbb{R}^n\). The concept of statistical amoeba is introduced and the properties of a family of statistical amoebas are studied. The relation with algebraic amoebas is discussed. Tropical limits of statistical amoebas are considered too. Applications of the concept of statistical amoeba to the analysis of singular sectors for integrable equations and properties of macroscopic systems with multiple equilibria, including frustrated systems, are discussed. partition function; singular sector; amoeba Classical or axiomatic geometry and physics, Classical equilibrium statistical mechanics (general), Extremal set theory, Applications of tropical geometry Zeros and amoebas of partition functions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems A projective equivariant completion j: \(G\to V\) of a semisimple algebraic group G over an algebraically closed field k of arbitrary characteristic is called a semisimple variety if it is further assumed that V is normal, and that equivariance refers to the two sided action of G. It is shown that there is an algebraic monoid M containing G as the group of units such that the semisimple varieties V and \((M\setminus \{0\})/k^*\) are isomorphic, where \(k^*\) is the identity component of the center of G. Semisimple algebraic monoids have been classified by the author [Trans. Am. Math. Soc. 287, 457-473 (1985; Zbl 0545.20054)] in terms of combinatorial data, the so-called polyhedral root systems (X,\(\Phi\),C), where X is the character group of a maximal torus T of G, where \(\Phi\) denotes the set of roots, and where C is the monoid of those characters which extend to the Zariski closure of T. Indeed the author established the existence of a bijection between the isomorphy classes of the M and the isomorphy classes of the (X,\(\Phi\),C). The polyhedral root systems occurring here are integral, i.e. there is a homomorphism \(\nu\) : \(X\to {\mathbb{Z}}\) mapping \(\Phi\) to \(\{\) \(0\}\) and the set \({\mathcal F}\) of fundamental generators (spanning a subgroup of finite index in X) to \(\{\) \(1\}\). Each fundamental generator \(\chi\in {\mathcal F}\) and each \(\rho\in X\) gives a self-map \(\sigma_{\chi,\rho}\) via \(\sigma_{\chi,\rho}(\chi)=\rho\) and \(\sigma_{\chi,\rho}(\alpha)=\alpha\) for \(\alpha\in \Phi\). Now two integral polyhedral root systems \((X,\Phi,C_ 1)\) and \((X,\Phi,C_ 2)\) are equivariant iff there is a bijection \(\gamma\) : \({\mathcal F}_ 1\to {\mathcal F}_ 2\) which preserves chambers and with which for each \(\chi\in {\mathcal F}_ 1\), the map \(\sigma_{\chi,\gamma (\chi)}\) restricts to an isomorphism of monoids \(C_ 1[1/\chi]\to C_ 2[1/\gamma (\chi)]\). The main result is now readily formulated: Let \(M_ n\), \(n=1,2\) be semisimple algebraic monoids with the same group G of units and with embeddings \(j_ n: G\to M_ n\). Then the varieties \((G/k^*,(M_ n\setminus \{0\})/k^*,j_ n)\), \(n=1,2\) are isomorphic iff the associated polyhedral root systems \((X,\Phi,C_ n)\), \(n=1,2\) are equivariant. projective equivariant completion; semisimple algebraic group; group of units; semisimple varieties; Semisimple algebraic monoids; polyhedral root systems; character group; maximal torus; fundamental generators L. Renner, \textit{Classification of semisimple varieties}, J. Algebra \textbf{122} (1989), no. 2, 275-287. Linear algebraic groups over arbitrary fields, Semigroups, Simple, semisimple, reductive (super)algebras, Special varieties, Semigroups of transformations, relations, partitions, etc. Classification of semisimple varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper, the authors, in Section 2, introduce tower sets and tower schemes and they prove that all these schemes are aCM. Then, they show that every tower scheme has the same Hilbert function as a corresponding tower scheme supported on a left segment whose Hilbert function was computed in [\textit{A. Ragusa} and \textit{G. Zappalà}, Beitr. Algebra Geom. 44, No. 1, 285--302 (2003; Zbl 1033.13004)]. The authors, in Section 3, give a combinatorial characterization for aCM squarefree monomial ideals of codimension 2. Moreover, they give a slight generalization of tower sets and tower schemes and prove numerous preparatory results about these sets and schemes. Finally, they prove the stated characterization. tower set; tower scheme; aCM squarefree monomial ideal Favacchio, G.; Ragusa, A.; Zappalà, G.: Tower sets and other configurations with the Cohen-Macaulay property. J. pure appl. Algebra 219, 2260-2278 (2015) Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Configurations and arrangements of linear subspaces, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes Tower sets and other configurations with the Cohen-Macaulay property | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [The articles of this volume will not be indexed individually.]
Contents: \textit{Takahiro Kawai}, On \({\mathfrak R}\)-holonomic complexes (English) (pp. 1-2); \textit{Hideyuki Majima}, Cohomological characterization of regular singularities in several variables (pp. 3- 21); \textit{Masaaki Yoshida}, Orbifold-uniformizing differential equations (pp. 22-23); \textit{B. Malgrange}, On the work of J. Ecalle (English) (pp. 24-25); \textit{Masafumi Yoshino}, On small divisors of equations with regular singular points (pp. 26-39); \textit{Kazuo Ueno}, Multiplicative Jordan decomposition and exponential maps in \(Aut_{{\mathbb{C}}}{\mathbb{C}}[[ x_ 1,...,x_ n]]\) (pp. 40-61); \textit{Masatoshi Noumi}, Gauss-Manin system and the flat coordinate system. Connection with the expansion of the solutions at \(\infty\) (English) (pp. 62-72); \textit{T. Suwa}, Unfoldings and determinacy of analytic foliation singularities (pp. 73-85); \textit{Takesi Yagami}, Gevrey strong asymptotic solutions of completely integrable linear Pfaff equations with singularities (pp. 86-107); \textit{Kunio Yoshino}, Liouville type theorem for hyperfunctions and its applications (English) (pp. 108- 117); \textit{Masao Yamazaki}, Propagation of microlocal singularities for the solutions of the Euler equations of fluids (pp. 118-140); \textit{Kōji Kubota}, On the propagation of singularities near gliding points of solutions of hyperbolic systems (pp. 141-164); \textit{Kiyômi Kataoka}, Quasipositive pseudodifferential operators (pp. 165-177); \textit{Masayoshi Nagase}, Sheaf theoretic \(L^ 2\)-cohomology (English) (pp. 178-187); \textit{Nobuyuki Ikeda}, The asymptotic behavior of the fundamental solution of the heat equation (pp. 188-219); \textit{Kanehisa Takasaki}, On the structure of solutions to the self-dual Yang-Mills equations (English) (pp. 220-231); \textit{Toshinori \(\hat Oa\)ku}, F-mild hyperfunctions and characteristic boundary value problems (pp. 232-251); \textit{Takashi Aoki}, Logarithms of pseudodifferential operators (English) (pp. 252-255); \textit{Akira Kaneko}, On the analyticity of the loci of singularities of real analytic solutions with minimal dimension (pp. 256-271). Algebraic analysis; Proceedings; Symposium; Kyoto; RIMS; algebraic analysis; regular singularities; hyperfunctions; solutions of hyperbolic systems; pseudodifferential operators; Yang-Mills equations Proceedings, conferences, collections, etc. pertaining to several complex variables and analytic spaces, Proceedings, conferences, collections, etc. pertaining to global analysis, Local complex singularities, Hyperfunctions, Analytic sheaves and cohomology groups, Singularities in algebraic geometry, Complex singularities, Relations of PDEs on manifolds with hyperfunctions, Heat and other parabolic equation methods for PDEs on manifolds, Pseudodifferential and Fourier integral operators on manifolds, Hyperbolic equations on manifolds Algebraic analysis. Proceedings of a Symposium held at the Research Institute for Mathematical Sciences, Kyoto University, Kyoto, October 17- 20, 1983 | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\Gamma\) be an integral plane curve of degree \(d>k\geq 1\) with \(\delta\) ordinary nodes and cusps as its singularities, and let p: \(C\to \Gamma\) be its normalization. Let \({\mathbb{P}}_ k\) be the projective space parametrizing effective divisors of degree k on \({\mathbb{P}}^ 2\). The authors generalize a result of \textit{M. Namba} [``Families of meromorphic functions on compact Riemann surfaces'', Lect. Notes Math. 767 (1979; Zbl 0417.32008)]) concerning linear systems on a smooth curve to the case with ordinary nodes and cusps as follows.
Let \(g^ 1_ n\) be a fixed point free linear system on C with \(n+\delta <k(d-k)\) for some integer \(k>0\). Then there exists a pencil \({\mathbb{P}}\subset {\mathbb{P}}_{k-1}\) including \(g^ 1_ n\) on C. From this main lemma the authors deduce several theorems, and give examples to show the sharpness of the theorems. The results are used by the first author in Math. Ann. 289, No.1, 89-93 (1991; Zbl 0697.14019).
The proof of the main lemma is corrected in Manuscr. Math. 71, No.3, 337- 338 (1991)]. integral plane curve; linear systems Marc Coppens and Takao Kato, The gonality of smooth curves with plane models, Manuscripta Math. 70 (1990), no. 1, 5 -- 25. , https://doi.org/10.1007/BF02568358 Marc Coppens and Takao Kato, Correction to: ''The gonality of smooth curves with plane models'', Manuscripta Math. 71 (1991), no. 3, 337 -- 338. Divisors, linear systems, invertible sheaves, Singularities of curves, local rings The gonality of smooth curves with plane models | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The article is devoted to algebraic geometry related with a Noetherian ring. Dimension and codimension functions are studied, examples are given. Chains of irreducible closed subsets are investigated. Specific features of algebraic schemes over a ring containing an infinite set of elements are elucidated. In particular, the spectrum of the Noetherian ring is considered and topology is used. biequidimensionality; spectrum of a Noetherian ring; dimension formula; codimension function Dimension theory, depth, related commutative rings (catenary, etc.), Commutative Noetherian rings and modules, Relevant commutative algebra, Elementary questions in algebraic geometry Some remarks on biequidimensionality of topological spaces and Noetherian schemes | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(F\) be a totally real number field with \(r_1\) real embeddings and with ring of integers \(\mathcal O_F\). Write \(R=\mathcal O_F[{1\over 2}]\) for the ring of \(2\)-integers in \(F\). Then for all even \(i>0\) one has
\[
2^{r_1}\cdot{{\# K_{2i-2}(R)\{2\}}\over{\# K_{2i-1}(R)\{2\}}}= {{\# H^2_{\text{ét}}(R;{\mathbb Z}_2(i))}\over{\# H^1_{\text{ét}}(R;{\mathbb Z}_2(i))}},
\]
where, for an abelian group \(A\), \(A\{2\}\) denotes its \(2\)-primary torsion subgroup. This result confirms Lichtenbaum's conjecture for the \(2\)-primary \(K\)-theory and its relation with étale cohomology of \(F\). This follows from an explicit description of the algebraic \(K\)-groups of \(\mathcal O_F\) (or \(R\)) in terms of étale cohomology, modulo odd finite groups. This is done, using Borel's famous calculation of \(\dim_{\mathbb Q}(K_n(R)\otimes_{\mathbb Z}{\mathbb Q})\), for all \(n\geq 2\) and \(F\) being a number field with at least one real embedding. Using results of Voevodsky-Suslin and of Bloch-Lichtenbaum, Dwyer-Friedlander theory already gives an explicit expression for \(K_n(R)\) up to an odd finite group for totally imaginary number fields. One also has local analogs, i.e., for a finite extension \(E\) of \({\mathbb Q}_p\) (of characteristic zero), a so-called \(p\)-local field (of characteristic zero), with valuation ring \(\mathcal O_E\) one can describe \(K_n(\mathcal O_E;{\mathbb Z}_2)\simeq K_n(E;{\mathbb Z}_2)\).
Another (partial) result for a long standing conjecture of Lichtenbaum relating the values \(\zeta_F(1-i)\) of the zeta-function of \(F\) to étale cohomology (thus also to \(K\)-theory) can also be stated: Let \(F\) be a totally real abelian number field, then for all even \(i\geq 0\),
\[
\zeta_F(1-i)\sim_22^{r_1}\cdot{{\#K_{2i-2}(R)\{2\}}\over{\#K_{2i-1}(R)\{2\}}},
\]
where \(\sim_2\) means that both sides are rational numbers having the same \(2\)-adic valuation. This result follows from a theorem of A. Wiles on the Main Conjecture of Iwasawa theory, as explained in an appendix by M. Kolster.
Although the (Quillen-)Lichtenbaum conjectures have undergone much study from their beginnings, the results obtained in the present paper could only be proved using quite recent results of Bloch-Lichtenbaum and of Suslin and Voevodsky. Basic to the setup is the Bloch-Lichtenbaum third quadrant spectral sequence (which holds for any field \(F\)):
\[
E_2^{p,q}=CH^{-q}(F,-p-q)\Rightarrow K_{-p-q}(F),
\]
where \(CH^i(F,n)\) denote Bloch's higher Chow groups of \(\text{Spec}(F)\). Actually a version with coefficients \({\mathbb Z}/m\) is needed (and proved in an appendix):
\[
E_2^{p,q}=CH^{-q}(F,-p-q;{\mathbb Z}/m)\Rightarrow K_{-p-q}(F;{\mathbb Z}/m).
\]
Then results of Suslin and Voevodsky relate the \(CH^i(F;{\mathbb Z}/m)\) to motivic cohomology \(H_{\mathcal M}\) of \(F\), and then also (in characteristic zero) to étale cohomology \(H_{\text{ét}}\) of \(F\). For the Bloch-Lichtenbaum spectral sequence this leads to:
\[
E_2^{p,q}=\begin{cases} H^{p-q}_{\text{ét}}(F;{\mathbb Z}/2^{\nu}(-q)),&q\leq p\leq 0\\ 0,&\text{ otherwise} \end{cases}\Rightarrow K_{-p-q}(F;{\mathbb Z}/2^{\nu}).
\]
One may pass to the colimit over \(\nu\), then with \(W(i)\) denoting the union of the étale sheaves \({\mathbb Z}/2^{\nu}(i)\), one obtains:
\[
E_2^{p,q}=\begin{cases} H^{p-q}_{\text{ét}}(F;W(-q)),&q\leq p\leq 0\\ 0,&\text{ otherwise} \end{cases}\Rightarrow K_{-p-q}(F;{\mathbb Z}/2^{\infty}).
\]
Several interesting cases follow easily: (i) \(F\) totally imaginary; (ii) \(F=E\) a \(p\)-local field and \(\nu=1\).
The paper consists of: (i) an introduction with some history of Lichtenbaum's conjecture and a presentation of the results to be proved in subsequent sections; (ii) the Bloch-Lichtenbaum spectral sequence, as described above; (iii) a review of étale cohomology, in particular the description of \(H^0\), \(H^1\) and \(H^2\) of number fields and of \(p\)-local fields; (iv) two-primary algebraic \(K\)-theory of \(2\)-local fields (of characteristic zero); (v) étale cohomology of global fields, in particular various maps between the cohomology of a number field and of its localizations; (vi) the spectral sequence for the real numbers \({\mathbb R}\), in particular \(K_n({\mathbb R};{\mathbb Z}/2)\) and \(K_n({\mathbb R};{\mathbb Z}/2^{\infty})\); (vii) two-primary \(K\)-theory of number fields, describing \(K_n(F;{\mathbb Z}/2^{\infty})\), \(K_n(R;{\mathbb Z}/2^{\infty})\) and \(K_n(R)\{3\}\); (viii) \(\pmod 2\) \(K\)-groups, in particular the \(K_n(R;{\mathbb Z}/2)\). Many interesting side results emerge on the fly. The paper closes with two appendices (and references), the first on the cohomological version of the Lichtenbaum conjecture at the prime \(2\) by \textit{M.~Kolster}, and the second on the Bloch-Lichtenbaum spectral sequence with coefficients. Concluding, one may say that Lichtenbaum's conjectures are still of great interest and intrigue, and one may hope that new methods will give a further breakthrough. two-primary algebraic \(K\)-theory; number fields; Lichtenbaum-Quillen conjectures; étale cohomology; motivic cohomology; Bloch-Lichtenbaum spectral sequence; Lichtenbaum conjecture J Rognes, C Weibel, Two-primary algebraic \(K\)-theory of rings of integers in number fields, J. Amer. Math. Soc. 13 (2000) 1 Computations of higher \(K\)-theory of rings, \(K\)-theory of global fields, \(K\)-theory of local fields, Étale and other Grothendieck topologies and (co)homologies, Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects), Motivic cohomology; motivic homotopy theory Two-primary algebraic \(K\)-theory of rings of integers in number fields (with an appendix by M.~Kolster) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Following similar ideas of Tits and Manin, the author defines a variety over a field \({\mathbb F}_1\) of one element having an extension to \(\mathbb Z\). These varieties are examined in considerable detail and properties of their zeta functions obtained. algebraic variety; toric variety; Euclidean lattice; zeta function; \(J\)-homomorphism Soulé, C., LES variétés sur le corps à un élément, Mosc. Math. J., 4, 1, 217-244, (2004) Foundations of algebraic geometry, Toric varieties, Newton polyhedra, Okounkov bodies, Zeta and \(L\)-functions: analytic theory, \(J\)-morphism, Global ground fields in algebraic geometry Varieties over the one-element field | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper deals with the problem of conjugacy of Cartan subalgebras for affine Kac-Moody Lie algebras. Unlike the methods used by Peterson and Kac, our approach is entirely cohomological and geometric. It is deeply rooted on the theory of reductive group schemes developed by Demazure and Grothendieck, and on the work of J. Tits on buildings. affine Kac-Moody Lie algebra; conjugacy; reductive group scheme; torsor; Laurent polynomials; non-abelian cohomology Chernousov, V.; Gille, P.; Pianzola, A.; Yahorau, U.: A cohomological proof of Peterson-Kac's theorem on conjugacy of Cartan subalgebras for affine Kac-Moody Lie algebras. J. algebra 399, 55-78 (2014) Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Galois cohomology of linear algebraic groups, Group schemes, Coverings in algebraic geometry A cohomological proof of Peterson-Kac's theorem on conjugacy of Cartan subalgebras for affine Kac-Moody Lie algebras | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The original goal of this paper was to prove transcendence of values of the Gaussian hypergeometric function \(F(a,b,c;z)\) \((a,b,c\in\mathbb Q)\) at algebraic points \(z\neq 0\). This could be realized by using a recent deep theorem of G. Wüstholz on the linear independence over \(\overline{\mathbb Q}\) of periods of Abelian varieties over \(\overline{\mathbb Q}\). For almost all choices of \(a, b, c\) this approach works and gives the desired transcendence results. However, as a beautiful bonus, the author discovered a certain set of triples \((a,b,c)\) for which \(F(a,b,c;z)\) assumes algebraic values at a set of algebraic points lying dense in \(\mathbb C\).
I think the significance of these results justifies a short sketch of ideas in this review. Letting \(N\) be the common denominator of \(a, b, c\), the number \(F(a,b,c;z)\) can be written as a quotient \(\oint \omega_ z/\oint \omega_ 0\) where \(\omega_ z\) is a suitable \(l\)-form on the significant factor \(T_ z\) of the Jacobian of the curve \(y^N=x^A(x-1)^B(x-z)^C\). Here \(A, B, C\) depend on \(a, b, c\). Wüstholz's theorem states that if \(T_ z\) and \(T_ 0\) have no factors in common and if \(z\in \overline{\mathbb Q}\), then \(F(a,b,c;z)\) is transcendental over \(\mathbb Q\). The surprise is that there exist \(a, b, c\) such that \(T_ z\) has a factor in common with \(T_ 0\) for infinitely many algebraic \(z\). Families of such \(T_ z\) correspond to certain one-parameter Shimura families of Abelian varieties.
In the present paper all these considerations are worked out quite explicitly and in a very clear style. transcendence of values at algebraic points; Gaussian hypergeometric function; algebraic values; Jacobian; Shimura families Jürgen Wolfart, Werte hypergeometrischer Funktionen, Invent. Math. 92 (1988), no. 1, 187 -- 216 (German, with English summary). Transcendence theory of other special functions, Results involving abelian varieties, Structure of families (Picard-Lefschetz, monodromy, etc.) Values of hypergeometric functions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author makes an attempt to show that the moduli scheme of stable rank-2 vector bundles on \(P_ 3\) with \(c_ 1=0\) and \(h^ 1E(-2)=0\) (i.e., the space of mathematical instanton bundles of rank 2 on \(P_ 3)\) is smooth. One has to prove therefore that \(h^ 2(End(E))=0\). The attempt turns out to be unsuccessful.
In fact one has to prove using the method of the paper that if E is a mathematical instanton bundle then there is a point x in \(P_ 3\) with the properties:
(i) there is a line \(P_ 1\) through x such that \(E_{P_ 1}\) is trivial,
(ii) for any jumping line \(P_ 1\) through x the splitting is \(E_{P_ 1}={\mathcal O}_{P_ 1}(-1)\oplus {\mathcal O}_{P_ 1}(1).\)
Having such data there is still a problem to calculate \(h^ 2(End(E))\) via the properties of the induced theta-characteristic \(R^ 1g_*f^*E(-1,-1)=\theta\). Here f and g are respectively the morphism of blowing up such a point as in (i) and (ii) and the projection of the blow up \(P_ 3\) to the plane parametrizing lines through it. Both steps lack a complete proof. There is a serious mistake in the proof of the fact that the divisor of jumping lines is reduced. The calculation of \(c_ 1(R^ 1\pi_*\tilde E(h-\tau))\) is not correct because one cannot use this resolution of \(\theta\) in the general case not knowing the homological dimension of \(\theta\). Moreover, the correct exact sequence for the sheaf \(R^ 1\lambda_*(E\oplus E(-\tau)| S)\) is the following one
\[
0\to {\mathcal O}_ C\oplus {\mathcal O}_ C(-1)\to R^ 1\lambda_*(E\otimes E(-\tau)| S)\to {\mathcal O}_ C(n)\oplus {\mathcal O}_ C(n-1)\to 0.
\]
So the support of \(R^ 1\lambda_*(E\otimes E(- \tau)| S)\) is not reduced and the method of the paper gives only the tautological equality \(\chi (R^ 1\lambda_*E\oplus E(-\tau)| S)=4n\). There are also some mistakes in {\S} 1. Hence the statement of the vanishing of \(h^ 2(End(E))\) in case \(h^ 1E(-2)=0\) cannot be deduced from this paper. moduli scheme of stable rank-2 vector bundles; mathematical instanton bundles Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Algebraic moduli problems, moduli of vector bundles, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials Smoothness of the moduli scheme of instanton vector bundles with \(c_ 1=0\), \(c_ 2=n\) on \(P^ 3\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The authors give a short course introducing the notion of monodromy. They start with the geometric monodromy of a germ \(f:(\mathbb C^{n+1},0)\to (\mathbb C,0)\) of a holomorphic function at the origin. This is followed by explaining the relation of deformation of a singularity and the monodromy operator, leading to vanishing cycles and the Picard-Lefschetz transformation. An alternative approach to study the monodromy operator is given in the third section, based on resolution of singularities (the authors follow the work of \textit{H. Clemens} [Trans. Am. Math. Soc. 136, 93--108 (1969; Zbl 0185.51302)] and \textit{N. A'Campo} [Comment. Math. Helv. 50, 233--248 (1975; Zbl 0333.14008)]). This leads to a proof of the monodromy theorem. In the final section, it is shown how the zeta function of the monodromy of a hypersurface with a good \(C^*\)-action can be computed. The authors derive their recent result on a relation between the zeta function of the monodromy and the Poincaré series of the coordinate alebra of the singularity [Abh. Math. Semin. Univ. Hamb. 74, 175--179 (2004; Zbl 1070.14004)]. monodromy; Picard-Lefschetz transformation; zeta function; Poincaré series Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Global theory and resolution of singularities (algebro-geometric aspects), Deformations of complex singularities; vanishing cycles, Modifications; resolution of singularities (complex-analytic aspects) Lectures on monodromy | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The paper under review is concerned with motivic zeta functions for Calabi-Yau varieties, in particular, for the toric degeneration of the quartic \(K3\) surface and its mirror dual degeneration. The method used here is the Gross-Siebert program. Let \(K\) be the field of complex Laurent series \({\mathbb{C}}((t)\) and let \(R\) be its valuation ring \({\mathbb{C}}[[t]]\). Denote by \(K_0(\text{{Vac}}_{\mathbb{C}})\) the Grothendieck ring of complex variables, by \({\mathbb{L}}=[{\mathbb{A}}^1_{\mathbb{C}}]\in K_0(\text{{Vac}}_{\mathbb{C}})\) the class of affine line, and by \({\mathcal{M}}_{\mathbb{C}}=K_0(\text{{Vac}}_{\mathbb{C}})[\mathbb{L}^{-1}]\) the localized Grothendieck ring.
Let \(X\) be a smooth, proper, geometrically connected variety over \(K\) with trivial canonical line bundle, and assume that \(X(K)\) is non-empty. The motivic zeta function \(Z_X(T)\) is a formal power series in \(T\) with coefficients in the localized Grothendieck ring \({\mathcal{M}}_{\mathbb{C}}\). \textit{E. Bultot} [C. R., Math., Acad. Sci. Paris 353, No. 3, 261--264 (2015; Zbl 1375.14088)] gave a formula for \(Z_X(T)\) in terms of a log-smooth \(R\)-model of \(X\). Also there is a formula by \textit{A. J. Stewart} and \textit{V. Vologodsky} [Adv. Math. 228, No. 5, 2688--2730 (2011; Zbl 1297.14025)] for the motivic zeta function for \(K3\) surfaces over \(K\) of type III.
The paper computes the motivic zeta functions for the quadrtic and its mirror dual as well as the motivic volume or motivic nearby fiber of \(X\) by taking the limit of \(-Z_X(T)\) for \(T\to +\infty\).
Results: If \(X\) is a \(K3\) surface over \(K\) of type III, then the motivic volume of \(X\) is equal to
\[
2+20{\mathbb{L}}+2{\mathbb{L}}^2\in{\mathcal{M}}_{\mathbb{C}}.
\]
Also applying the formula of Boltot and the combinatorial analysis, the motivic zeta functions for the quartic \(K3\) surface and for the mirror quadratic are computed. motivic zeta function; mirror symmetry; toric degeneration; Gross-Siebert program; quartic \(K3\) surface \(K3\) surfaces and Enriques surfaces, Mirror symmetry (algebro-geometric aspects), Structure of families (Picard-Lefschetz, monodromy, etc.), Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Arithmetic mirror symmetry, Toric varieties, Newton polyhedra, Okounkov bodies, Arcs and motivic integration Motivic zeta functions of the quartic and its mirror dual | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In [Inst. Hautes Étud. Sci., Publ. Math. 12, 5--68 (1962; Zbl 0173.48601)] \textit{B. Dwork} developed a \(p\)-adic cohomology theory for smooth projective hypersurfaces over finite fields. Given \(f\in \mathbb{F}_q [x_0,x_1,\dots, x_N]\), a form of degree \(d\) defined over the field of \(q=p^a\) elements, Dwork constructed a complex \(K.^{\text{Dw}} (f)\) of \(p\)-adic Banach spaces. When the hypersurface \(V(f)\) defined by the vanishing of \(f\) in \(\mathbb{P}^N\) is nonsingular and has nonsingular intersection with every coordinate variety \(H_A= \bigcap_{i\in A}\{x_i=0\}\), where \(A\subset S=\{0,\dots, N\}\), \(A\neq S\), then the complex \(K.^{\text{Dw}}(f)\) is acyclic except in degree 0. The characteristic polynomial of Frobenius acting on \(H_0\) gives the primitive part of the middle-dimensional factor of the zeta function of \(V(f)\). From this vantage point, there remained the problems of extending this work to varieties other than hypersurfaces, as well as to treat even in the hypersurface case open or singular varieties. Of course, the development of crystalline cohomology and rigid cohomology provided an excellent basis for these generalizations.
Our goal in the present paper is to use the approach of exponential modules or twisted de Rham theory pioneered by Dwork in the hypersurface case to treat complete intersections. In this we are continuing the early work of \textit{K. Ireland} [Am. J. Math. 89, 643--660 (1967; Zbl 0197.47201)] and \textit{J. Barshay} [Trans. Am. Math. Soc. 135, 447--458 (1969; Zbl 0174.24302)], who studied projective and multiprojective complete intersections from this point of view also. In their work, they constructed a complex of \(p\)-adic Banach spaces \(K.^{\text{Dw}}\) (related to the complex \(K.(S,S)\) of section 6 of the paper under review), proved the acyclicity except in degree 0 of this complex in the smooth case, and related the characteristic polynomial of Frobenius acting on \(H_0\) to the zeta function of the complete intersection defined by the simultaneous vanishing of forms \(f_1,\dots, f_r\in \mathbb{F}_q [x_0,\dots, x_N]\) in \(\mathbb{P}^N\). Specifically, they showed that this characteristic polynomial equals a product of certain factors (which the Weil conjectures imply are polynomials), from which they concluded this polynomial has the correct degree. They were unable to show the factors themselves are polynomials. In particular, they were unable to construct a finite-dimensional \(p\)-adic vector space with action of Frobenius whose characteristic polynomial is the interesting factor of the zeta function of a smooth projective complete intersection. We construct such a theory here.
The main application of our work that we give here is a proof of the ``Katz conjecture'' (i.e., the assertion that the Newton polygon lies above the Hodge polygon: [see \textit{B. Mazur}, Bull. Am. Math. Soc. 78, 653--667 (1972; Zbl 0258.14006)]) for general smooth complete intersections in an affine space or a torus (as well as another proof in the projective case). Previously, such results were known only in the proper case. We note also that our approach eliminates the need to treat separately the case of hypersurfaces of degree divisible by \(p\). In a future article, we plan to describe the relation between the theory developed here and classical de Rham cohomology. In particular, we believe that the description we give here of middle-dimensional cohomology and of a procedure for finding a basis for it should be useful in calculations involving the Gauss-Manin connection. \(p\)-adic Dwork cohomology; Katz conjecture; exponential modules; twisted de Rham theory; zeta function of the complete intersection; Weil conjectures; characteristic polynomial; Newton polygon; Hodge polygon; hypersurfaces; middle-dimensional cohomology Adolphson, A.; Sperber, S., On the zeta function of a complete intersection, Ann. Sci École Norm. Sup., 4, 287-328, (1996) Varieties over finite and local fields, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Finite ground fields in algebraic geometry, Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry On the zeta function of a complete intersection | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems (Siehe auch JFM 10.0111.01, JFM 10.0111.02, JFM 10.0111.03) Die dritte Notiz der erst genannten Broschüre: Généralisation d'un théorème de M. Smith ist ein Auszug aus den Ann. Soc. Sc. Brux. 2, 211--224 und ist die Abhandlung des englischen Gelehrten im Messenger (2) 7, 81--82 (s. JFM 08.0074.03) entstanden. Der Hauptsatz heisst: Eine Determinante
\[
\sum \pm a_{11}a_{22}\dots a_{nn},
\]
wo
\[
a_{ik}= a_{ki}=a_{i-k, k} \quad \text{oder} \quad a_{i,k-i}
\]
ist eine einfache Function ihrer Diagonalen. Corollar 1). Jedes Product \(x_1 x_2\dots x_n\) kann in die Form einer Determinante dieser Art gebracht werden, wo
\[
a_{11}=x_1 +x_d+ \cdots +x_{d'} +x_i
\]
und \(1, d,\dots d',i\) alle Theiler von \(i\) sind. 2) Jede Determinante dieser Form wird mit \(+1, -1\) oder 0 multiplicirt, wenn man die Elemente einer Linie durch die entsprechenden Elemente der zweiten Diagonale der Determinante ersetzt. Die Arbeit von Catalan enthält einen etwas veränderten Beweis des Hauptsatzes und Corollars 1); die von Le Paige einen directen Beweis des Corollars 1), nur mit Hülfe des Princips der Addition der Reihen. Die zweite Notiz von Mansion enthält den Beweis desselben Corollars durch Multiplication zweier Determinanten, die nach dem oben bezeichnten Princip gebildet sind, so dass alle Elemente oberhalb der Hauptdiagonale Null sind. Theorems of Mansion and Smith; determinant; function of the diagonals; product; divisors; line; proof of a corollary; principle of addition of rows; elements above the principal diagonal Research exposition (monographs, survey articles) pertaining to linear algebra, Determinants, permanents, traces, other special matrix functions, Products, amalgamated products, and other kinds of limits and colimits, Divisors, linear systems, invertible sheaves, Proof theory in general (including proof-theoretic semantics) On a theorem of M. Mansion. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be an \(n\)-dimensional nonsingular real algebraic variety (possibly incomplete). In this case, a local-global spectral sequence is defined,
\[
E_2^{p,q}=\text{H}^p(X,{\mathcal H}^q) \Rightarrow\text{H}_{\text{et}}^{p+q} (X,\mathbb{F}_2),\tag{1}
\]
where \({\mathcal H}^q\) is the sheaf on \(X_{\text{zar}}\) corresponding to the presheaf \(U\mapsto\text{H}^q_{\text{et}}(U, \mathbb{F}_2)\). The spectral sequence (1) was studied by \textit{S. Bloch} and \textit{A. Ogus} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 7, 181-201 (1974; Zbl 0307.14008)], where it was shown that the terms \(E_2^{p,q}\) vanish for \(p>q\). \textit{J.-L. Colliot-Thélène} and \textit{R. Parimala} [Invent. Math. 101, No. 1, 81-99 (1990; Zbl 0756.14013)] proved that, for \(q>n\), there is a canonical isomorphism
\[
E_2^{0,q}=\text{H}^0\bigl(X (\mathbb{R}),\mathbb{F}_2 \bigr),\tag{2}
\]
where \(X (\mathbb{R})\) is the set of real points endowed with the Euclidean topology. \textit{V. A. Krasnov} [Izv. Math. 62, No. 5, 1013-1034 (1998); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 62, No. 5, 165-186 (1988; Zbl 0948.14043)] established that the differentials \(d^{0,q}_r\) of the spectral sequence (1) are zero for \(q>n\). He proved this fact by using equivariant cohomology and it led to a simple proof of the relation (2). In this note we continue the investigation of the spectral sequence (1) by means of equivariant cohomology. Here we prove the following assertion.
Theorem. For \(q>n\), there is a canonical isomorphism \(E_2^{p,q}=\text{H}^p(X (\mathbb{R}), \mathbb{F}_2)\), and the differentials \(d_r^{p,q}\) are zero. Bloch-Ogus spectral sequence; real algebraic variety; local-global spectral sequence; equivariant cohomology Topology of real algebraic varieties, Generalized cohomology and spectral sequences in algebraic topology, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) The Bloch-Ogus spectral sequence of a real algebraic variety | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(k\) be a finite field of characteristic \(p>0\) and \(K\) be an algebraic function field of one variable over \(k\). Consider an elliptic curve \(E\) over \(K\) with \(j\)-invariant transcendental over \(\mathbb{F}_p\). In this paper the author proves that there exists a finite set of primes \(S\), with \(p\in S\), such that for every positive integer \(n\) not divisible by any of the primes of \(S\), the pure \(n\)-torsion points of the fibers of the Néron model of \(E\) describe after completion a smooth irreducible projective curve \(C_n\). The Selmer group \(S_n(E/K)\) embeds canonically into \(_n\text{Pic}^0 (C_n)\), and a necessary and sufficient condition is given for an element of \(_n\text{Pic}^0 (C_n)\) to belong to \(S_n(E/K)\). Moreover, the induced embedding \(E(K)/nE(K) \hookrightarrow {_n\text{Pic}^0} (C_n)\) is described explicitly. Shafarevich-Tate group; Picard groups; transcendental \(j\)-invariant; finite field; algebraic function field; elliptic curve; fibers of the Néron model; irreducible projective curve; Selmer group; embedding Elliptic curves over global fields, Global ground fields in algebraic geometry, Elliptic curves, Arithmetic ground fields for curves Selmer groups and Picard groups | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(\mathbb F\) be the algebraic closure of the field \({\mathbb F}_{p}\) of \(p\) elements and \({\mathbb Q}_{\ell}\) the field of \(\ell\)-adic numbers, \(\ell\) is a prime, \(\ell\not=p.\) The bounded derived category of \(\ell\)-adic sheaves on a scheme \(X\) over \(\mathbb F\) is denoted by \(D^{b}(X, {\mathbb Q}_{\ell}).\) Let \(B\) be a quasicompact scheme over \(\mathbb F\) and \(h: E\to B\) a vector bundle of dimension \(d\) over \(\mathbb F .\) It is shown that the functor \(\overline{R}h h^{\ast}: D^{b}(B, {\mathbb Q}_{\ell})\to D^{b}(B, {\mathbb Q}_{\ell})\) is isomorphic to the (twisted) shift functor \([-2d](-d)\). functorial isomorphism; derived category; vector bundle; \(\ell\)-adic sheaves; quasicompact scheme Derived functors and satellites, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Étale and other Grothendieck topologies and (co)homologies, Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) On functorial isomorphism in the derived category of \(\ell\)-adic sheaves | 0 |
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