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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 The authors review and explore recently discovered, interesting new connections between the three topics mentioned in the title of the paper. In J. Algebra 174, No. 3, 1080-1090 (1995; Zbl 0842.14002), \textit{J. Emsalem} and \textit{A. Iarrobino} observed that there is a relationship between fat points in \(\mathbb{P}^n\) (i.e. zero-dimensional subschemes \(\mathbb{X}\) defined by ideals of the form \(I_{\mathbb{X}}= {\mathfrak p}_1^{\alpha_1} \cap\cdots\cap {\mathfrak p}_s^{\alpha_s}\)) and ideals generated by powers of linear forms. More precisely, for \(j\geq \max\{\alpha_1,\dots, \alpha_s\}\), the \(j\)-th graded piece of the Macaulay inverse system \((I_{\mathbb{X}})^{-1}\) equals the corresponding graded piece of a certain ideal generated by \(s\) powers of linear forms.
In the case of linearly independent linear forms \(\{L_1,\dots, L_s\}\) in two indeterminates, the authors compute explicit formulas for the Hilbert function of \(J= (L_1^{\alpha_1},\dots, L_s^{\alpha_s})\), for the socle degree of \(k[y_0,y_1]/J\), and thus for the minimal graded free resolution of \(J\).
In the paper: Compos. Math. 108, No. 3, 319-356 (1997; Zbl 0899.13016), \textit{A. Iarrobino} also observed that there is a relationship between splines (i.e. piecewise polynomial functions satisfying certain smoothness conditions) on a \(d\)-dimensional simplicial complex \(\Delta\) and the ideals generated by powers of the linear forms defining hyperplanes incident to the interior faces of \(\Delta\). The authors recall a certain chain complex whose top homology module is precisely the module \(C^\alpha (\widehat{\Delta})\) of mixed splines on the cone \(\widehat{\Delta}\) over \(\Delta\) which are smooth of order \(\alpha\). In the case of planar splines \(\Delta\subset \mathbb{R}^2\), they are then able to provide formulas for the number of splines in \(C^\alpha (\widehat{\Delta})_k\) of sufficiently large degrees \(k\gg 0\).
The paper ends with a discussion of the obstacles which have to be overcome in order to get higher-dimensional versions of those results. zero-dimensional scheme; Macaulay inverse system; mixed spline; ideal of linear forms; fat points; Hilbert function; number of splines Geramita, A. V.; Schenck, H., Fat points, inverse systems, and piecewise polynomial functions, J. Algebra, 204, 1, 116-128, (1998) Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Numerical computation using splines Fat points, inverse systems, and piecewise polynomial 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 content of this article, which has very important applications in analytical number theory and other fields, covers the following topics: the logarithm function, multiple zeta values, polylogarithms, partial orders, iterated integrals, the differentials operator $(dt)/t)$ and $(dt)/(1-t)$, the Beta function, the gamma function. The author provides iterated integrals on products of one variable multiple polylogarithms in the algebra of multiple zeta values with a convergent condition. In the divergent case, the author also defines the regularized iterated. Using the same method, the author gives also regularized iterated integrals in the algebra of multiple zeta values. The author provides some applications involving series representations for multiple zeta values. He also gives some remarks and examples involving iterated integrals and multiple zeta values. logarithm function; multiple zeta values; polylogarithms; partial orders; iterated integrals; differentials operator Multiple Dirichlet series and zeta functions and multizeta values, Polylogarithms and relations with \(K\)-theory, Other Dirichlet series and zeta functions, Motivic cohomology; motivic homotopy theory Iterated integrals on products of one variable multiple polylogarithms | 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 These pages contain a short overview on the state of the art of efficient numerical analysis methods that solve systems of multivariate polynomial equations. We focus on the work of Steve Smale who initiated this research framework, and on the collaboration between Stephen Smale and Michael Shub, which set the foundations of this approach to polynomial system--solving, culminating in the more recent advances of Carlos Beltran, Luis Miguel Pardo, Peter Buergisser and Felipe Cucker. Beltrán C and Shub M 2013 The complexity and geometry of numerically solving polynomial systems Recent Advances in Real Complexity and Computation(Contemporary Mathematics vol 604) ed J L Montaña and L Pardo (Providence, RI: American Mathematical Society) pp 71--104 Research exposition (monographs, survey articles) pertaining to numerical analysis, Numerical computation of solutions to systems of equations, Effectivity, complexity and computational aspects of algebraic geometry, Complexity and performance of numerical algorithms, Analysis of algorithms and problem complexity The complexity and geometry of numerically solving polynomial 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 For an ample line bundle on an abelian or \(K3\) surface, minimal with respect to the polarization, the relative Hilbert scheme of points on the complete linear system is known to be smooth. We give an explicit expression in quasi-Jacobi forms for the \(\mathcal X_{-y}\) genus of the restriction of the Hilbert scheme to a general linear subsystem. This generalizes a result of \textit{T. Kawai} and \textit{K. Yoshioka} [Adv. Theor. Math. Phys. 4, No. 2, 397--485 (2000; Zbl 1013.81043)] for the complete linear system on the \(K3\) surface, a result of Maulik, Pandharipande, and Thomas [\textit{D. Maulik} et al., J. Topol. 3, No. 4, 937--996 (2010; Zbl 1207.14058)] on the Euler characteristics of linear subsystems on the \(K3\) surface, and a conjecture of the authors. Göttsche, L.; Shende, V., The \(\chi _{-y}\)-genera of relative Hilbert schemes for linear systems on Abelian and K3 surfaces, Algebr. Geom., 2, 405-421, (2015) Parametrization (Chow and Hilbert schemes), \(K3\) surfaces and Enriques surfaces, Algebraic theory of abelian varieties The \(\mathcal X_{-y}\)-genera of relative Hilbert schemes for linear systems on abelian and \(K3\) 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 Computation of cohomology of local systems is a difficult task. In this paper it is proved that for all rank one local systems \(L\) on an 1-formal smooth complex algebraic variety \(M\), except finitely many local systems in any irreducible component \(W\) of the first characteristic variety, \(H^1(M,L)\) can be computed from the cohomology algebras \(H^*(M_W,\mathbb{C})\), where \(M_W=M\) if \(W\) is a non-translated component but could be a smaller open subset of \(M\) otherwise. This is applied to obtain a simple topological proof of the fact that if the dimension of \(H^1(M,L')\), as \(L'\) varies within \(W\) with \(\dim W>0\), jumps at \(L\in W\), then \(L\) is of finite order.
This result also follows by applying the more difficult results of \textit{A. Dimca, S. Papadima} and \textit{A. I. Suciu} [Int. Math. Res. Not. 2008, Article ID rnm119 (2008; Zbl 1156.32018)], \textit{C. Simpson} [Ann. Sci. Éc. Norm. Supér. (4) 26, No. 3, 361--401 (1993; Zbl 0798.14005)], and \textit{N. Budur} [Adv. Math. 221, No. 1, 217--250 (2009; Zbl 1187.14024)]. As a corollary, if \(M\) is quasi-projective, then \(H^1(M,L)=H^1(M,L^{-1})\). local system; constructible sheaf; twisted cohomology; characteristic variety; resonance variety A. Dimca: On admissible rank one local systems, J. Algebra (2008), doi:10.1016/j.jalgebra.2008.01.039. Homotopy theory and fundamental groups in algebraic geometry On admissible rank one local 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 We show that the Quot scheme \(\mathrm{Quot}_{\mathbb{A}^3}(\mathscr{O}^r,n)\) admits a symmetric obstruction theory, and we compute its virtual Euler characteristic. We extend the calculation to locally free sheaves on smooth 3-folds, thus refining a special case of a recent Euler characteristic calculation of Gholampour-Kool. We then extend Toda's higher rank DT/PT correspondence on Calabi-Yau 3-folds to a local version centered at a fixed slope stable sheaf. This generalises (and refines) the local DT/PT correspondence around the cycle of a Cohen-Macaulay curve. Our approach clarifies the relation between Gholampour-Kool's functional equation for \(\mathrm{Quot}\) schemes, and Toda's higher rank DT/PT correspondence. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Parametrization (Chow and Hilbert schemes) Virtual counts on Quot schemes and the higher rank local DT/PT correspondence | 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 recall in Theorems 2.1 and 2.2 their results with \textit{S. Gitler} in [Bol. Soc. Mat. Mex. II Ser. 30, No. 1, 1--11 (1985; Zbl 0639.57013)] considering generalized higher dimensional versions of the Riemann-Hurwitz formula. In the first part of the paper they apply those results to rational maps of projective varieties. Then, in the second part, they apply them to the study of determinantal varieties realized as the degenerate loci of morphisms of complex vector bundles over a complex projective variety \(X\). They highlight two examples, the general symmetric bundle maps and the flagged bundles. Riemann-Hurwitz formula; rational maps; iterated maps; degeneracy locus; determinantal variety Determinantal varieties, Low-dimensional topology of special (e.g., branched) coverings, Algebraic topology on manifolds and differential topology, Rational and birational maps, Meromorphic mappings in several complex variables, Iteration of holomorphic maps, fixed points of holomorphic maps and related problems for several complex variables Rational and iterated maps, degeneracy loci, and the generalized Riemann-Hurwitz formula | 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 \(Z\) denote a finite collection of reduced points in projective \(n\)-space and let \(I\) denote the homogeneous ideal of \(Z\). The points in \(Z\) are said to be in (\(i,j\))-uniform position if every cardinality \(i\) subset of \(Z\) imposes the same number of conditions on forms of degree \(j\). The points are in uniform position if they are in (\(i,j\))-uniform position for all values of \(i\) and \(j\). We present a symbolic algorithm that, given \(I\), can be used to determine whether the points in \(Z\) are in (\(i,j\))-uniform position. In addition it can be used to determine whether the points in \(Z\) are in uniform position, in linearly general position and in general position. The algorithm uses the Chow form of various \(d\)-uple embeddings of \(Z\) and derivatives of these forms. The existence of the algorithm provides an answer to a question of Kreuzer. uniform position; Chow variety; Chow form; general position; zero-dimensional scheme; points Migliore, J.; Peterson, C.: A symbolic test for (i,j)-uniformity in reduced zero-dimensional schemes, J. symbolic comput. 37, 403-413 (2004) Projective techniques in algebraic geometry A symbolic test for \((i,j)\)-uniformity in reduced zero-dimensional 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 The multiplicity structure of a polynomial system at an isolated zero is identified with the dual space consisting of differential functionals vanishing on the entire polynomial ideal. Algorithms have been developed for computing dual spaces as the kernels of Macaulay matrices. These previous algorithms face a formidable obstacle in handling Macaulay matrices whose dimensions grow rapidly when the problem size and the order of the differential functionals increase.
This paper presents a new algorithm based on the closedness subspace strategy that substantially reduces the matrix sizes in computing the dual spaces, enabling the computation of multiplicity structures for large problems. Comparisons of timings and memory demonstrate a substantial improvement in computational efficiency. numerical examples; closedness subspace method; multiplicity structure; polynomial system; algorithms; Macaulay matrices; computational efficiency Zhonggang Zeng, The closedness subspace method for computing the multiplicity structure of a polynomial system, Interactions of classical and numerical algebraic geometry, Contemp. Math., vol. 496, Amer. Math. Soc., Providence, RI, 2009, pp. 347 -- 362. Numerical computation of solutions to systems of equations, Approximation algorithms, Symbolic computation and algebraic computation, Computational aspects in algebraic geometry, Multiplicity of solutions of equilibrium problems in solid mechanics, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) The closedness subspace method for computing the multiplicity structure of a polynomial system | 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 \(U_i\) be a k-vector space of dimension \( s_i +1 \geq 2\) for \( i = 1, \ldots, k\) and \( k \geq 2\) and \(F\) a vector space of dimension \( n= \sum s_i\) and \( \varphi: F \rightarrow \otimes _{i=1}^k U_i \) a linear transformation and denote by \( \varphi_j: F \otimes k[ \bigoplus_{ i \neq j} U_i ](1,1,1,\dots 1) \) be the corresponding map of free modules. Let \(P_j = \prod_{i \neq j} P^{s_i}\) be the subscheme defined by the ideals of \((s_j +1) \times (s_j +1) \) minor determinants \( I_{s_j +1}( \varphi_j) \subset k[ \bigoplus_{i \neq j}U_i ]\). For any \((k-1)\)-tuple of positive integers \(a\) denote by \( \sigma_a( P_j)\) the Segre embedding given by divisors of type \(a\). The author's main theorem, proposition 3.1, states and proves the equivalence of the following conditions:
(i) for all \( 1 \leq i \leq k\), \(\Gamma_i\) is \(0\)-dimensional and locally Gorenstein, \(\text{codim} (I_{s_i +1}( \varphi)) = n-s_i \) and \(\text{codim}( I_{s_i}( \varphi)) = n-s_i +1 \).
(ii) \(\Gamma_1\) , \(\Gamma_2\) are \(0\)-dimensional.
(iii) \(\text{codim}( I_{s_i + 1}( \varphi)) = n-s_1, \text{codim}( I_{s_1}( \varphi_1)) = n-s_1 + 1 \).
(iv) \( \Gamma_1 \) is \(0\)-dimensional and locally Gorenstein.
Moreover they prove in theorem 3.2 that if one of the equivalent statements of the above is true then all \(\Gamma_ i \) are isomorphic to each other and defining \( a_i = n -\sum_{l=i}^{k-1}(s_l -1) \) for \(s_k \geq 2 \) and \( b_i = \sum_{l = i+1}^{k-1} s_l \) for \( 1\leq i \leq k-2 \) and \( b_k = n -s_k -1 \). Define \( a= ( a_1, \ldots, a_{k-1}), b= ( b_1, \ldots, b_{k-2}, b_k)\) for \( 1 \leq i \leq k-2\) then the Gale transform of \( \sigma_a( \Gamma_k)\) is \( \sigma_b( \Gamma_{k-1})\). The authors' main motivation is to generalize the results of \textit{D. Eisenbud} and \textit{S. Popescu} [J. Algebra 230, No. 1, 127--173 (2000; Zbl 1060.14528)] to cover linear transformations from a vector space \(F\) to a tensor product of three or more vector spaces, replacing Veronese embeddings by Segre embeddings. The paper is organized as follows. In section one they recall the notion of Gale transform for a linear series stating the result of [loc. cit.] for \(k=2\). In section two for \(\Gamma \) a finite Gorenstein scheme they compute in proposition 2.2 the Gale transform of the linear series cut out as \(\Gamma\) by \( H^0( P, O_P(a))\). In proposition 2.4 by focusing on \(\Gamma_j\) they compute a locally free resolution of \( I_F\) from the Eagon-Northcott complex. For the case of \(\dim(\Gamma_j) = 0 \) the authors compute in proposition 2.6 \(\deg( \Gamma_j)\) using a special case of Porteous formula in proposition 2.5. In section three, they state and prove the main theorem given above and theorem 3.2. The proofs of theorems 3.1 and 3.2 are similar to those of [loc. cit] except for the proof of part b) of theorem 3.2 for which they use proposition 2.4 and proposition 2.6 of this paper. determinantal varieties; projective techniques; varieties defined by ring conditions S. M. Cooper and S. P. Diaz, \textit{The Gale transform and multi-graded determinantal schemes}, J. Algebra, 319 (2008), pp. 3120--3127. Determinantal varieties, Projective techniques in algebraic geometry, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) The Gale transform and multi-graded determinantal 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 It is shown that the knowledge of a surjective morphism \(X\rightarrow Y\) of complex curves can be effectively used to make explicit calculations. The method is demonstrated by the calculation of \(j(n\tau)\) (for some small \(n\)) in terms of \(j(\tau)\) for the elliptic curve with period lattice \((1,\tau)\), the period matrix for the Jacobian of a family of genus-\(2\) curves complementing the classic calculations of Bolza and explicit general formulae for branched covers of an elliptic curve with exactly one ramification point. algebraic curves; branched covers; elliptic curves Computational aspects of algebraic curves, Coverings of curves, fundamental group, Special algebraic curves and curves of low genus, Elliptic curves A descent method for explicit computations on 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 \(\mathfrak{g}\) be a reductive Lie algebra over \(\mathbb C\) and \(\mathfrak{h}\) a Cartan subalgebra. The diagonal commutator scheme is \(D=\{(X,Y)\in\mathfrak{g}\oplus\mathfrak{g}\;|\;[X,Y]\in\mathfrak{h}\}.\) It is shown that \(D\) is a reduced complete intersection, having the commuting variety as one of its components. Moreover it is conjectured by the authors that there is only one other component. Another result is that the diagonal commutator scheme has a flat degeneration to the scheme \(\{(X,Y)\mid XY\) is lower triangular, \(YX\) is upper triangular\(\}\) which is also shown to be a reduced complete intersection. \beginbarticle \bauthor\binitsA. \bsnmKnutson, \batitleSome schemes related to the commuting variety, \bjtitleJ. Algebraic Geom. \bvolume14 (\byear2005), no. \bissue2, page 283-\blpage294. \endbarticle \OrigBibText \bibknutsonarticle author=Knutson, Allen, title=Some schemes related to the commuting variety, date=2005, ISSN=1056-3911, journal=J. Algebraic Geom., volume=14, number=2, pages=283\ndash294, url=http://dx.doi.org/10.1090/S1056-3911-04-00389-3, review=, \endOrigBibText \bptokstructpyb \endbibitem Mathematical Reviews (MathSciNet): URL: Link to item Complete intersections, Homogeneous spaces and generalizations Some schemes related to the commuting 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 We construct a virtual fundamental class on the Quot scheme parametrizing quotients of a trivial bundle on a smooth projective curve. We use the virtual localization formula to calculate virtual intersection numbers on Quot. As a consequence, we reprove the Vafa-Intriligator formula; our answer is valid even when the Quot scheme is badly behaved. More intersections of Vafa-Intriligator type are computed by the same method. Finally, we present an application to the non-vanishing of the Pontryagin ring of the moduli space of bundles. Quot scheme; virtual fundamental class; Vafa-Intriligator formula; localization \beginbarticle \bauthor\binitsA. \bsnmMarian and \bauthor\binitsD. \bsnmOprea, \batitleVirtual intersections on the Quot scheme and Vafa-Intriligator formulas, \bjtitleDuke Math. J. \bvolume136 (\byear2007), no. \bissue1, page 81-\blpage113. \endbarticle \OrigBibText A. Marian and D. Oprea, Virtual intersections on the Quot scheme and Vafa-Intriligator formulas , Duke Math. J. 136 (2007), no. 1, 81-113 \endOrigBibText \bptokstructpyb \endbibitem Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Vector bundles on curves and their moduli Virtual intersections on the Quot scheme and Vafa-Intriligator formulas | 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 a new proof of Miyanishi's theorem on the classification of the additive group scheme actions on the affine plane. locally finite iterative higher derivation; additive group scheme action K. Kojima: On approximations for nonlinear Cauchy problems in locally convex spaces (to appear). Group actions on affine varieties, Actions of groups on commutative rings; invariant theory, Derivations and commutative rings Locally finite iterative higher derivations on \(k[x,y]\) | 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 \textit{J. Tits} [in: Centre Belge Rech. math. Colloque d'algèbre supérieure, Bruxelles du 19 au 22 déc. 1956, 261--289 (1957; Zbl 0084.15902)] wondered if there would exist a ``field of one element'' \(\mathbb F_1\) such that for a Chevalley group \(G\) one has \(G (\mathbb F_1)=W\), the Weyl group of \(G\). Recall the Weyl group is defined as \(W=N(T)/Z(T)\) where \(T\) is a maximal torus, \(N(T)\) and \(Z(T)\) are the normalizer and the centralizer of \(T\) in \(G\). He then showed that one would be able to define a finite geometry with \(W\) as automorphism group.
In this paper we will extend the approach of \textit{N. Kurokawa}, \textit{H. Ochiai}, and \textit{M. Wakayama} [Doc. Math., J. DMV Extra Vol. 565--584 (2003; Zbl 1101.11325)] to ``absolute mathematics'' to define schemes over the field of one element. The basic idea of the approach of [loc. cit.] is that objects over \(\mathbb Z\) have a notion of \(\mathbb Z\)-linearity, i.e., additivity, and that the forgetful functor to \(\mathbb F_1\)-objects therefore must forget about additive structures. A ring \(R\) for example is mapped to the multiplicative monoid \((R,\times)\). On the other hand, the theory also requires a ``going up'' or base extension functor from objects over \(\mathbb F_1\) to objects over \(\mathbb Z\). Using the analogy of the finite extensions \(\mathbb F_{1^n}\) as in \textit{C. Soulé} [Mosc. Math. J. 4, 217--244 (2004; Zbl 1103.14003)], we are led to define the base extension of a monoid \(A\) as
\[
A\otimes_{\mathbb F_1}\mathbb Z:=\mathbb Z[A],
\]
where \(\mathbb Z [A]\) is the monoidal ring which is defined in the same way as a group ring. Based on these two constructions, here we lay the foundations of a theory of schemes over \(\mathbb F_1\). field of one element Deitmar, A., Schemes over \(\mathbb{F}_1\), (Number Fields and Function Fields--Two Parallel Worlds. Number Fields and Function Fields--Two Parallel Worlds, Progr. Math., vol. 239, (2005), Birkhäuser Boston: Birkhäuser Boston Boston, MA), 87-100 Schemes and morphisms, Varieties over finite and local fields, Linear algebraic groups over finite fields Schemes over \(\mathbb F_1\) | 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 we introduce new local symbols, which we call 4-function local symbols. We formulate reciprocity laws for them. These reciprocity laws are proven using a new method -- multidimensional iterated integrals. Besides providing reciprocity laws for the new 4-function local symbols, the same method works for proving reciprocity laws for the Parshin symbol. Both the new 4-function local symbols and the Parshin symbol can be expressed as a finite product of newly defined bi-local symbols, each of which satisfies a reciprocity law. The \(K\)-theoretic variant of the first 4-function local symbol is defined in the Appendix. It differs by a sign from the one defined via iterated integrals. Both the sign and the \(K\)-theoretic variant of the 4-function local symbol satisfy reciprocity laws, whose proof is based on Milnor \(K\)-theory (see the Appendix). The relation of the 4-function local symbols to the double free loop space of the surface is given by iterated integrals over membranes. reciprocity laws; complex algebraic surfaces; iterated integrals Horozov, I., Reciprocity laws on algebraic surfaces via iterated integrals, with an appendix by matt Kerr, J. K-Theory, 14, 2, 273-312, (2014) Symbols and arithmetic (\(K\)-theoretic aspects), Surfaces and higher-dimensional varieties, Higher symbols, Milnor \(K\)-theory, Integration on analytic sets and spaces, currents Reciprocity laws on algebraic surfaces via iterated integrals | 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 One can try to study the fundamental group of a smooth, complex projective variety \(S\) by looking at spaces of representations of \(\pi_ 1 (S)\). If the space of representations is a union of isolated points, then this structure reduces to the discrete structure of the set of representations. So a natural problem is to try to find examples of varieties \(S\) where there exist nontrivial continuous families of representations (or, equivalently, local systems on \(S) \). It is not too hard to see that if \(S\) is a projective algebraic curve of genus \(g \geq 2\), then the moduli space of representations of rank \(r \geq 1\) has dimension \((2g - 2) r^ 2 + 2\); or, for example, if \(S\) is a variety with \(\dim H^ 1(S, \mathbb{C}) = a\), then the space of representations of rank 1 has dimension \(a\). Taking tensor products of pullbacks of families of local systems arising in these ways, we obtain some more families. To pursue the problem we ask: Do families other than these exist?
The idea in this article is to use the next natural construction, taking higher direct images of local systems. Suppose that \(f:X \to S\) is a smooth projective morphism and \(\{W_ t\}\) is a family of local systems on \(X\), chosen in a simple way. Let \(V_ t = R^ i f_ * (W_ t)\). This is a collection of local systems, and if the ranks are constant, then it is a continuous family. We can hope that \(\{V_ t\}\) will be an interesting family of local systems on \(S\). The principal question that needs to be addressed is whether, if the family \(\{W_ t\}\) varies nontrivially, the family of direct images \(\{V_ t\}\) varies nontrivially. We are also interested in constructing examples where we can show that the family \(\{V_ t\}\) does not, by some miracle, arise from a simpler construction such as the one described before.
This article consists essentially of two parts. The first (sections 1-5) is devoted to answering our principal question in a fairly general situation. For this we develop the technique of taking the direct image of a harmonic bundle and its associated Higgs bundle. We give a way to calculate the spectral varieties of the Higgs bundles associated to \(V_ t\), as a way of verifying that the \(V_ t\) vary nontrivially. -- The second part (sections 6-8) is concerned with the construction of a particular class of examples and the verification of some additional properties about monodromy groups and possible factorization through morphisms to algebraic curves; these properties serve to show that our examples do not come from tensor products of pullbacks. The methods used in this part are all fairly well known. representations of fundamental group; local systems on complex projective variety; moduli space of algebraic curve; families of representations; direct image of a harmonic bundle; Higgs bundle; spectral varieties of the Higgs bundles Simpson C.: Some families of local systems over smooth projective varieties. Ann. Math. 138, 337--425 (1993) Homotopy theory and fundamental groups in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Families, moduli of curves (algebraic), Variation of Hodge structures (algebro-geometric aspects) Some families of local systems over smooth 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 \(S,T \subset \mathbb{P}^3\) be surfaces. Suppose that \(S \cap T\) is set- theoretically a smooth curve \(C\) of degree \(d\) and genus \(g\). There are several results:
If \(\text{Sing} (S) \cap \text{Sing} (T) = \emptyset\) and \(C\) is not a complete intersection, then \(\deg (S)\), \(\deg (T) < 2d^4\) (and one can improve this given specific values for \(d, g)\).
If \(S\) and \(T\) have only rational singularities, with none in common, and the ground field has characteristic zero, then \(d \leq g + 3\).
If \(S\) is normal, \(d > \deg (S)\), and the ground field has characteristic zero, then \(C\) is linearly normal. In particular, it follows by Riemann- Roch that \(d \leq g + 3\).
Assume that \(S\) is a quartic surface having only rational singularities, and that the ground field has characteristic zero. Then \(C\) is linearly normal. curve blow-up; set theoretic complete intersections of surfaces in \(\mathbb{P}^ 3\); curves on surfaces; bound for surface degree; quartic surface 4. D. B. Jaffe, Applications of iterated blow-up to set theoretic complete intersections in \mathbb{P}3, J. Reine Angew. Math.464 (1995) 1-45. genRefLink(128, 'S0129167X15501049BIB4', 'A1995RP96900001'); Special surfaces, Plane and space curves, Enumerative problems (combinatorial problems) in algebraic geometry, Complete intersections, Singularities of surfaces or higher-dimensional varieties, Projective techniques in algebraic geometry Applications of iterated curve blow-up to set theoretic complete intersections in \(\mathbb{P}^ 3\) | 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 We develop a power structure over the Grothendieck ring of varieties relative to an abelian monoid, which provides a systematic method to compute the class of the generalized Kummer scheme in the Grothendieck ring of Hodge structures. We obtain a generalized version of Cheah's formula for the Hilbert scheme of points, which specializes to Gulbrandsen's conjecture for Euler characteristics. Moreover, in the surface case we prove a conjecture of Göttsche for geometrically ruled surfaces. power structure; Hodge polynomial; Donaldson-Thomas invariant; generalized Kummer scheme Parametrization (Chow and Hilbert schemes), Algebraic theory of abelian varieties, Derived categories of sheaves, dg categories, and related constructions in algebraic geometry Hodge numbers of generalized Kummer schemes via relative power structures | 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 review in this paper with great care 3 different approaches to \(\mathbb F_1\)-geometry, which are Toen-Vaquie's relative schemes, Lorscheid's blue schemes and Connes-Consani's \(\mathbb F_1\)-schemes. They explain certain adjunctions between monoids, rings and blueprints in detail and extend them to the corresponding categories of sheaves.
These adjunctions allows them to merge blue schemes with Connes and Consani's notion of \(\mathbb F_1\)-schemes, which results in the category of so-called \(\mathbb F_1\)-schemes with relations as a common generalization. \(\mathbb{F}_1\)-schemes; blueprints; relative algebraic geometry Geometry over the field with one element, Monoidal categories, symmetric monoidal categories Some remarks on blueprints and \(\mathbb{F}_1\)-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 In this paper, the authors apply Fourier-Mukai transforms to study coherent systems on elliptic curves. Let \(X\) be an elliptic curve. A coherent system \((E,V)\) of type \((r,d,k)\) on \(X\) is a pair consisting of a vector bundle \(E\) of rank \(r\) and degree \(d\) and a \(k\)-dimensional subspace \(V\) of \(H^0(X,E)\). Semistability of such objects depends on a positive rational parameter \(\alpha\) and was defined in \textit{J. Le Potier} [Systèmes cohérents et structures de niveau, Astérisque 214, (1993; Zbl 0881.14008)]. In the same paper, moduli spaces of semsitable coherent systems were constructed. They are usually denoted by \(G ({\alpha}; r,d,k)\).
In order to study this moduli spaces, the authors consider a Fourier-Mukai transform \(\Phi: D(X) \to D(X)\) whose kernel is a sheaf on \(X \times X\). Since such a functor acts naturally on sheaf maps, it is enough to determine which transforms leave invariant the category of coherent systems. To this aim, the authors start by studying the behaviour under Fourier-Mukai transform of the category whose objects are maps \(\phi: V \otimes O_X \to F\), where \(V\) is a finite dimensional vector space and \(F\) a coherent sheaf. The category of coherent system is then a subcategory of the latter. Once the problem solved for the bigger category, one can use the spectral sequence \(F^{p,q}_2 = H^p (X, \Phi^q(F)) \to H^{p+q} (Y, \Phi(F))\) to understand how the Fourier-Mukai acts on coherent systems.
In general, the behaviour of (semistable) coherent systems under Fourier-Mukai is not easy to determine. However, there are two moduli spaces \(G_0(r,d,k)\) and \(G_L(r,d,k)\), with \(r < k\), for which the authors establish the preservation of stability. Finally, they are able to describe Fourier-Mukai transforms \(\Phi_a\), with \(a\) positive integer, such that any \(\Phi_a\) induces isomorphisms of moduli spaces
\[
\begin{aligned} \Phi_a^0: G_0 (r,d,k) &\to G_0(r+ad,d,k),\\ \Phi_a^L: G_L (r,d,k) &\to G_L(r+ad,d,k) \end{aligned}
\]
for \(k<0\) in the latter.
As an application, the authors study birational types of moduli spaces \(G(\alpha;r,d,k)\). In particular, for a given degree \(d\) of the vector bundle and a given dimension \(k\) of the vector subspace, there are at most \(d\) possible different birational types for the moduli space. Fourier-Mukai functors; coherent systems; moduli spaces; elliptic curves Ruipérez, D. Hernández; Prieto, C. Tejero: Fourier-Mukai transforms for coherent systems on elliptic curves, J. lond. Math. soc. 77, 15-32 (2008) Algebraic moduli problems, moduli of vector bundles, Vector bundles on curves and their moduli, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Elliptic curves Fourier-Mukai transforms for coherent systems 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 Let \(X\) be an integral variety over a perfect field \(k\), \(\mathcal L_m(X)\) its jet scheme of level \(m\in \mathbb N\) and \(\mathcal L_{\infty}(X)\) its arc scheme. The general component \(\mathcal G_m(X)\) of \(\mathcal L_m(X)\) is the Zariski closure of \(\mathcal L_m(\mathrm{Reg}(X))\).
If \(X\) is smooth on \(k\) then the geometry and topology of \(\mathcal L_m(X)\) are well understood. In this paper the authors consider the case \(X\) is not smooth and study some properties of the general component \(\mathcal G_m(X)\) by means of a smooth birational model of \(X\).
Indeed, under the further hypothesis that \(X\) is affine embedded in \(\mathbb A^N_k\), the authors prove that a birational model of \(X\) provides a description of \(\mathcal G_m(X)\) that gives rice to an algorithm which computes a Groebner basis of the defining ideal of \(\mathcal G_m(X)\) in \(\mathbb A^N_k\) as a subscheme of \(\mathcal L_m(X)\) (Algorithm~2). The authors also extend to arbitrary integral varieties over perfect fields over arbitrary characteristic another algorithm ''already introduced in the Ph.D. Thesis of Kpognon'' (see also [\textit{K. Kpognon} and \textit{J. Sebag}, Commun. Algebra 45, No. 5, 2195--2221 (2017; Zbl 1376.14018)]) ``for the study of arc scheme associated with integral affine plane curves in characteristic zero'' (Algorithm~1). Several examples and comments to the implementation of the algorithms, which is available in SageMath, are provided in Sections~6 and~7.
The given results are applied for further studies of plane curves, concerning differential operators logarithmic along an affine plane curve and the rationality of a motivic power series that is introduced by the authors and ``which encodes the geometry of all \(\mathcal G_m(X)\)'' (Sections 8 and 9). computational aspects of algebraic geometry; derivation module; jet and arc scheme; singularities in algebraic geometry Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Singularities in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), Arcs and motivic integration, Computational aspects of algebraic curves, Computational aspects of higher-dimensional varieties, Effectivity, complexity and computational aspects of algebraic geometry, Local complex singularities Two algorithms for computing the general component of jet scheme 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 The notions of quasi-prime submodules and developed Zariski topology was introduced by the present authors in [Algebra Colloq. 19, 1089--1108 (2012; Zbl 1294.13011)]. In this paper we use these notions to define a scheme. For an \(R\)-module \(M\), let \(X:= \{Q\in q\operatorname{Spec}(M)\mid (Q:_R M)\in\operatorname{Spec}(R)\}\). It is proved that \((X, \mathcal{O}_{X})\) is a locally ringed space. We study the morphism of locally ringed spaces induced by \(R\)-homomorphism \(M \to N\), and also by ring homomorphism \(R\to S\). Among other results, we show that \((X,\mathcal{O}_X)\) is a scheme by putting some suitable conditions on \(M\). quasi-prime submodule; quasi-primeful module; quasi-prime-embedding module; developed Zariski topology Other special types of modules and ideals in commutative rings, Theory of modules and ideals in commutative rings, Commutative Noetherian rings and modules, Relevant commutative algebra A scheme over quasi-prime spectrum of modules | 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 construct a quantum statistical mechanical system \((A,\sigma_t)\) analogous to the Connes-Marcolli system in the case of Shimura varieties. Along the way, we define a new Bost-Connes system for number fields which has the ``correct'' symmetries and the ``correct'' partition function. We give a formalism that applies to general Shimura data \((G,X)\). The object of this series of papers is to show that these systems have phase transitions and
spontaneous symmetry breaking, and to classify their KMS states, at least for low temperature. E. Ha and F. Paugam, ''Bost-Connes-Marcolli systems for Shimura varieties. I. Definitions and formal analytic properties,'' IMRP Int. Math. Res. Pap. 5, 237--286 (2005). Quantum equilibrium statistical mechanics (general), Many-body theory; quantum Hall effect, Arithmetic aspects of modular and Shimura varieties, Modular and Shimura varieties, Zeta functions and \(L\)-functions of number fields, Classifications of \(C^*\)-algebras, Phase transitions (general) in equilibrium statistical mechanics Bost-Connes-Marcolli systems for Shimura varieties. I: Definitions and formal analytic properties | 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 authors' previous construction of toric manifolds [\textit{A. Bahri} et al., Homology Homotopy Appl. 17, No. 2, 137--160 (2015; Zbl 1342.13029)], an infinite family of toric manifolds was obtained using the simplicial wedge \(J\)-construction and the notion of polyhedral products. In the present paper the authors relate this construction to the Davis-Januskiewicz construction of toric manifolds. Applications related to the ``composed complexes'' as introduced by \textit{A. A. Ayzenberg} [Trans. Mosc. Math. Soc. 2013, 175--202 (2013; Zbl 1302.05214); translation from Tr. Mosk. Mat. O.-va 74, No. 2, 211--245 (2013)] are discussed. toric manifold; quasitoric manifold; \(J\)-construction; Davis-Januszkiewicz construction Compact groups of homeomorphisms, \(n\)-dimensional polytopes, Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes, Toric varieties, Newton polyhedra, Okounkov bodies A generalization of the Davis-Januszkiewicz construction and applications to toric manifolds and iterated polyhedral products | 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 a definition of full level structure on group schemes of the form \(G\times G\), where \(G\) is a finite flat commutative group scheme of rank \(p\) over a \(\mathbb{Z}_p\)-scheme \(S\) or, more generally, a truncated \(p\)-divisible group of height 1. We show that there is no natural notion of full level structure over the stack of all finite flat commutative group schemes. Group schemes, Drinfel'd modules; higher-dimensional motives, etc., Arithmetic aspects of modular and Shimura varieties Full level structure on some 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 The multivariate spline is a piecewise polynomial with certain smoothness, and the piecewise algebraic hypersurfaces with certain smoothness (i.e. the set of all common zeros of multivariate splines) have become useful tools for representing or approximating surfaces. Studying an effective method for the construction of real piecewise algebraic hypersurfaces of a given degree with certain smoothness and prescribed topology is one of the important ways to solve the problem of how to represent or approximate geometric objects with certain topological structure (especially the complex topology structure), and also a new and important topic of computational geometry and algebraic geometry. Parametric piecewise polynomial systems are not only closely related to a series of research, such as intersection of surfaces, blending curves and surfaces, generation of transition surfaces, but also are essential generalizations of the parametric semi-algebraic systems. The purpose of this paper is to introduce some recent research progress on the construction of real piecewise algebraic hypersurfaces, and parametric piecewise polynomial systems. piecewise algebraic hypersurfaces; parametric polynomial systems; Viro method; number of real zeros Numerical computation using splines, Numerical smoothing, curve fitting, Spline approximation, Approximation by arbitrary nonlinear expressions; widths and entropy, Computational aspects of algebraic surfaces Progress in construction of piecewise algebraic hypersurfaces and solving parametric piecewise polynomial 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 Hecke algebras of affine Weyl groups with equal parameters are important to representation theory of p-adic reductive groups because their simple modules give those representations that admit an Iwahori fixed vector. But also Hecke algebras with unequal parameters occur in representation theory, i.e. as endomorphism algebras of induced representations.
The paper is concerned with the graded version of H of such an algebra with respect to a natural filtration. Similar to [\textit{D. Kazhdan}, \textit{G. Lusztig}, Invent. Math. 87, 153-215 (1987; Zbl 0613.22004)] where affine Hecke algebras are treated, the algebra H is realized in equivariant homology of a generalized flag variety of G, where G is the complex Lie group attached to the root data defining H. Thus the simple H-modules are classified in a way similar to [loc. cit.]. Hecke algebras; affine Weyl groups; p-adic reductive groups; simple modules; equivariant homology; generalized flag variety; root data Lusztig, G, Cuspidal local systems and graded Hecke algebras, I, Publ. Math. IHÉ,S, 67, 145-202, (1988) Representations of Lie and linear algebraic groups over local fields, Group actions on varieties or schemes (quotients), Group rings of infinite groups and their modules (group-theoretic aspects), Applications of methods of algebraic \(K\)-theory in algebraic geometry Cuspidal local systems and graded Hecke algebras. 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 This paper describes the relative Hilbert Chow morphism for a flat projective family \(\pi: X \to B\) of generically nonsingular curves which are at worst nodal over an arbitrary irreducible base \(B\). The relative Hilbert Chow morphism is the cycle map \(c_m: X^{[m]}_B \to X^{(m)}_B\), where \(X^{[m]}_B\) is the relative Hilbert scheme of \(m\) points and \(X^{(m)}_B\) is the relative symmetric product. The main theorem states that \(c_m\) is equivalent to the blowing up of the discriminant locus \(D^m \subset X^{(m)}_B\). The author works over the complex numbers and uses Serre's GAGA principle, constructing a local analytic model \(H\) for \(X^{[m]}_B\) and reverse engineering an ideal sheaf \(G\) in \(X^{(m)}_B\) to have syzygies so that the blow up at \(G\) maps to the pullback \(OH\) of \(H\) over the Cartesian product via a map \(\gamma\). A local analysis shows that \(\gamma\) is an isomorphism and that \(G\) defines the ordered diagonal, hence descends to the isomorphism claimed. This provides the details of a proof sketched in the author's earlier paper [in: Projective varieties with unexpected properties. A volume in memory of Giuseppe Veronese. Proceedings of the international conference ``Varieties with unexpected properties'', Siena, Italy, June 8--13, 2004. Berlin: Walter de Gruyter. 361--378 (2005; Zbl 1186.14027)].
In the second half of the paper the author uses the local model \(H\) to glean information about the singularity stratification of \(X^{[m]}_B\), specifically the structure of certain node polyscrolls he used earlier [Asian J. Math. 17, No. 2, 193--264 (2013; Zbl 1282.14097)] to develop an intersection calculus for the Hilbert scheme. This extends the intersection theory and enumerative geometry of a single smooth curve developed by \textit{I. G. Macdonald} [Topology 1, 319--343 (1962; Zbl 0121.38003)] to families with at worst nodal singularities, extending work of \textit{E. Cotterill} [Math. Z. 267, No. 3--4, 549--582 (2011; Zbl 1213.14064)]. nodal curves; relative Hilbert-Chow morphism; enumerative geometry; node scrolls Parametrization (Chow and Hilbert schemes), Families, moduli of curves (algebraic), Singularities of curves, local rings Structure of the cycle map for Hilbert schemes of families of nodal 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 \(H^N\) be the Hilbert scheme of \(N\) points in the projective plane. \textit{G. Gotzmann} [Math. Z. 199, 539--547 (1988; Zbl 0637.14003)] observed that \(H^N\) is naturally stratified by the Hilbert function, and he showed the irreducibility and dimension of the strata. The current paper, instead, studies the nested Hilbert scheme \(H_{N-i,N}\) parameterizing pairs \((Z,W)\) such that \(Z \in H^{N-i}, W \in H^{N}\) and \(Z \subset W\). In the case \(i=1\) this is known to be smooth and irreducible. In this paper, \(H_{N-1,N}\) is shown to again be stratified by irreducible subvarieties of the type \(H_{\phi, \psi} = \{ (Z,W) \in H_{N-1,N} | Z \in H_\phi, W \in H_\psi \}\), where \(H_\phi\) is the locally closed subscheme of the Hilbert scheme parameterizing zero-dimensional schemes with Hilbert function \(\phi\). The dimension of these strata is also computed.
On the other hand, when \(i>1\), it is shown that \(H_{N-i,N}\) is never smooth, and the corresponding strata may be reducible. However, if the Hilbert functions \(\phi\) and \(\psi\) are very close to each other (in a sense made precise in the paper), then the strata are irreducible, and the dimensions are computed. The authors also show that \(H_{N-2,N}\) is irreducible.
An important ingredient in the proof is the classical notion that algebraic liaison can be used (for codimension two arithmetically Cohen-Macaulay subschemes) to pass from any scheme to a simpler one. Here the authors reduce a statement about a specific nested scheme \(H_{\phi,\psi}\) to a corresponding one about a ``simpler'' nested scheme \(H_{\psi^*,\phi^*}\), via liaison. As a byproduct they find a new proof of Gotzmann's results. The results of this paper are motivated by the authors' study of globally generated and very ample Hilbert functions [Math. Z. 245, 155--181 (2003; Zbl 1079.14057)], and has applications to the classification of globally generated line bundles on the blow-up of \(\mathbb P^2\) at points, and to the study of Cayley-Bacharach schemes in \(\mathbb P^2\). globally generated line bundles; Cayley-Bacharach schemes; Hilbert function Parametrization (Chow and Hilbert schemes), Linkage, Projective techniques in algebraic geometry On the stratification of nested 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 Let \(k\) be a field, \(R = k[X_1, \dots, X_n]\) a polynomial ring, \(\mathfrak m = (X_1, \dots, X_n)\) and \(M_2\), \dots, \(M_{d-1}\) graded \(R\)-modules of finite length where \(2 < d \leq n-2\). The authors give a homogeneous ideal \(I\) of \(R\) such that \(\dim R/I = d\) and H\(_{\mathfrak m}^i(R/I) \cong M_i\) (up to shift) for \(2 \leq i \leq d-1\). If \(n-d = 2\), then Evans and Griffith already gave such a ring by using Bourbaki sequence. However the authors use the theory of orientable modules and give the ring when \(n - d \geq 3\). local cohomology; prescribed cohomology; graded modules; orientable modules Migliore, J.; Nagel, U.; Peterson, C., Constructing schemes with prescribed cohomology in arbitrary codimension, J. Pure Appl. Algebr., 152, 1-3, 245-251, (2000) Local cohomology and commutative rings, Local cohomology and algebraic geometry, Syzygies, resolutions, complexes and commutative rings, (Co)homology theory in algebraic geometry, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Varieties and morphisms Constructing schemes with prescribed cohomology in arbitrary codimension | 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 Motivated by the minimal model program of the moduli spaces of Bridgeland-semistable sheaves on surfaces, this paper computes the nef cones of the nested Hilbert scheme \(X^{[n+1,n]} \subset X^{[n+1]} \times X^{[n]}\) and the universal family \(Z=X^{[n,1]}\) of the Hilbert scheme \(X^{[n]}\) of \(n\)-points, where \(X\) is the projective plane, a Hirzebruch surface, or a \(K3\) surface of Picard rank \(1\). Several interesting questions are also concluded in the last section. nested Hilbert schemes; nef cones Parametrization (Chow and Hilbert schemes), Rational and birational maps, Rational and ruled surfaces, \(K3\) surfaces and Enriques surfaces Nef cones of nested 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 In this note the author gives a direct description (without proof) of the Hodge filtration of the mixed Hodge structure on the vanishing cohomology of an isolated singularity. Gauss Manin system; Hodge filtration of the mixed Hodge structure; vanishing cohomology of an isolated singularity Transcendental methods, Hodge theory (algebro-geometric aspects), Local complex singularities, Singularities in algebraic geometry, Transcendental methods of algebraic geometry (complex-analytic aspects) Systèmes de Gauss-Manin et structure de Hodge mixte sur la cohomologie évanescente | 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 discusses techniques relevant in the theory of resolution of singularities of algebraic varieties. It is largely an exposition of previous results of the authors and other researchers (Hironaka, Encinas, Benito, García-Escamilla, etc.), but also contains some new results. Let us explain its content more carefully.
Trying to resolve the singularities of an algebraic variety \(X\) over a field \(k\), an approach (due to Hironaka) is to associate to \(X\) an upper semicontinuous function \(F_X\) into a (fixed) totally ordered set \(\Lambda\), which in some sense measures how bad the singularities of \(X\) are.
The set \(C\) of points of \(X\) where \(F_X\) reaches its maximum value is closed. For suitable \(F_X\), blowing up \(C\) we should get a variety whose ``worst'' singularities are ``better '' than those of \(X\). In this paper, the cases where \(F_X=HS_X\) or \(F_X=e_X\), the Hilbert-Samuel or the multiplicity function respectively, are considered.
To control this process, an important concept is that of \textit{representation}. In general, an \textit{idealistic pair} (or just a pair) on a smooth variety \(W\) is an ordered pair \((J,b)\) where \(J\) is a coherent sheaf of ideals on \(W\) and \(b \geq 0\) an integer. The singular locus \(\mathrm{Sing}(J,b)\) of the pair is the set of points of \(W\) where \(\mu_x(J)\), the order of the stalk \(J_x\) in \({\mathcal O}_{W,x}\), is \(\geq b\). There is a natural notion of \textit{resolution of pairs} (eventually, the singular locus is empty). For an upper semicontinuous function \(F_X\) as above, if \(X\) is embedded as a closed subvariety of a regular \(W\), a \textit{representation of \(F_X\)} is a pair \((J,b)\) on \(W\) such that \(\mathrm{Sing}(J,b)=\mathrm{Max}(F_X)\). If this can be done in such a way that this equality is preserved after taking suitable blowing-ups (``permissible blowing-ups'') then from a resolution of the pair \((J,b)\) one reaches a situation where the maximum value of the function \(F\) has dropped.
For \(F_X=HS_X\) or \(F_X=e_X\) representation is available, locally in the étale topology. Pairs can be resolved in characteristic zero. In the usual proofs there is an essential inductive step (on the dimension of \(W\), in the previous notation), which uses the local existence of certain smooth subvarieties of \(W\), called of \textit{maximal contact}. These are not available over fields of positive characteristic, which prevents the argument to generalize. Villamayor proposed to replace these subvarieties of maximal contact by suitable projections onto smooth varieties of smaller dimension. To do induction successfully, it seems convenient not to use pairs as above, but essentially equivalent more algebraic objects, called \textit{Rees algebras}. These are certain graded subalgebras of \({\mathcal O}_W[T]\) (with \(T\) an indeterminate). To find the necessary suitable projections, a numerical invariant \(\tau\), also introduced by Hironaka, plays an important role. This provides an alternative presentation of the proof of resolution in characteristic zero. For instance, certain fundamental results on Rees algebras obtained by the authors allow them to greatly simplify the proof that the local process of representation mentioned before globalizes. But, more importantly, this approach might work also in positive characteristic. Some partial results were obtained by the authors.
All these developments, including many of the technicalities involved, as well as other topics, are discussed in the reviewed paper. A good part of the article is on the necessary commutative algebra. One of the new results presented is the theory of \textit{identifiable pairs}, useful when one must compare, in a suitable way, Rees algebras defined over different ambient spaces.
Several relevant examples are examined. Overall this article is a good introduction to the subject. resolution of singularities; blowing up; multiplicity; integral closure; elimination theory; Rees algebra; equivalence invariant Bravo, A., Villamayor, O.E.: On the behavior of the multiplicity on schemes: stratification and blow ups. In: Ellwood, D., Hauser, H., Mori, S., Schicho, J. (eds.) The Resolution of Singular Algebraic Varieties. Clay Institute Mathematics Proceedings, vol. 20, pp. 81-207 (340pp.). AMS/CMI, Providence (2014) Global theory and resolution of singularities (algebro-geometric aspects), Local structure of morphisms in algebraic geometry: étale, flat, etc., Singularities of surfaces or higher-dimensional varieties, Birational geometry, Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics, Integral closure of commutative rings and ideals, Multiplicity theory and related topics On the behavior of the multiplicity on schemes: stratification and blow ups | 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 \(Y\) be a curve over an algebraically closed field \(k\) of characteristic \(p>0\). Let \(W\) be a complete local ring with residue field \(k\) and \(G\) a finite flat group scheme over \(W\) which acts faithfully on \(Y\). Let \(\text{Def}(Y,G)\) be the functor which associates to a local Artinian \(W\)-algebra \(R\) with residue field \(k\) the set of isomorphism classes of \(G\)--equivariant deformations of \(Y\) to \(R\). The functor \(\text{Def}(Y,G)\) is studied using cohomological methods. The first 3 chapters of the paper give an exposition of certain cohomological methods for studying equivariant deformations of curves with group scheme actions. This theory is applied to the case of three point covers with bad reduction. There are three appendices including Picard stacks, the cohomology of affine group schemes and some spectral sequences. equivariant deformation; group schemes; cotangent complex [11] S. Wewers, `` Formal deformation of curves with group scheme action {'', \(Ann. Inst. Fourier (Grenoble)\)55 (2005), no. 4, p. 1105-1165. Cedram | &MR 21 | &Zbl 1079.} Local deformation theory, Artin approximation, etc., Coverings of curves, fundamental group, Deformations and infinitesimal methods in commutative ring theory Formal deformation of curves with group scheme action | 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 assigns to an abelian category \(\mathcal A\) a ringed space \({\mathbf X}_{\mathcal A}\) such that if \(\mathcal A\) and \(\mathcal A'\) are equivalent categories, then \({\mathbf X}_{\mathcal A}\) is naturally isomorphic to \({\mathbf X}_{{\mathcal A}'}\). If \(\mathcal A\) is the category of quasi-coherent sheaves on a scheme \(\mathbf X\), then it is shown that \({\mathbf X}_{\mathcal A}\), which is called the reconstruction of \(\mathbf X\), is canonically isomorphic to \(\mathbf X\). The result generalises a result by \textit{P. Gabriel} where \(\mathbf X\) is a noetherian scheme using an assignment of a spectrum to an abelian category different than the Gabriel spectrum. abelian categories; reconstruction of schemes; quasi-coherent sheaves; noetherian; spectrum Rosenberg, A. L., The spectrum of abelian categories and reconstruction of schemes, (Rings, Hopf Algebras, and Brauer Groups, Lect. Notes Pure Appl. Math., vol. 197, (1998), Marcel Dekker), 257-274 Abelian categories, Grothendieck categories, Schemes and morphisms, Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] The spectrum of abelian categories and reconstruction of 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 consider two examples of a fully decodable combinatorial encoding of Bernoulli schemes: the encoding via Weyl simplices and the much more complicated encoding via the RSK (Robinson-Schensted-Knuth) correspondence. In the first case, decodability is quite a simple fact, while in the second case, this is a nontrivial result obtained by \textit{D. Romik} and \textit{P. Śniady} [Ann. Probab. 43, No. 2, 682--737 (2015; Zbl 1360.60028)] and based on [\textit{S. V. Kerov} and the author, SIAM J. Algebraic Discrete Methods 7, 116--124 (1986; Zbl 0584.05004)], [the author and \textit{S. V. Kerov}, Sov. Math., Dokl. 18, 527--531 (1977; Zbl 0406.05008); translation from Dokl. Akad. Nauk SSSR 233, 1024--1027 (1977)], and other papers. We comment on the proofs from the viewpoint of the theory of measurable partitions; another proof, using representation theory and generalized Schur-Weyl duality, will be presented elsewhere. We also study a new dynamics of Bernoulli variables on P-tableaux and find the limit 3D-shape of these tableaux. coding; RSK-correspondence; filtration; limit shape Combinatorial aspects of representation theory, Exact enumeration problems, generating functions, Grassmannians, Schubert varieties, flag manifolds, Representations of infinite symmetric groups, Signal theory (characterization, reconstruction, filtering, etc.), Theory of error-correcting codes and error-detecting codes Combinatorial encoding of Bernoulli schemes and the asymptotic behavior of Young tableaux | 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. The existence of the moduli space \(H_{N,p}\) parametrizing the embedded curve singularities in \((\mathbb C^N,0)\) with admissible Hilbert polynomial \(p\) is proved. \(H_{N,p}\) is not locally of finite type. It is a projective limit of schemes of finite type. The tangent space ot \(H_{N,p}\) at a closed point is computed. moduli space; space curves; embedded curve singularities Singularities of curves, local rings, Deformations of singularities, Fine and coarse moduli spaces A moduli scheme of embedded curve 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 Here we look at the postulation of general unions of zero-dimensional schemes more general than fat points w.r.t. a very positive linear system. Projective techniques in algebraic geometry, Divisors, linear systems, invertible sheaves Zero-dimensional schemes, blowing-up and asymptotic postulation | 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 Hilbert scheme \(\mathrm{Hilb}^{p(t)} (\mathbb P^n)\) parametrizing closed subschemes of \(\mathbb P^n\) with Hilbert polynomial \(p(t)\) has been of great interest every since Grothendieck constructed it in the early 1960s. Early results include the connectedness theorem of \textit{R. Hartshorne} [Publ. Math., Inst. Hautes Étud. Sci. 29, 5--48 (1966; Zbl 0171.41502)] and smoothness of \(\mathrm{Hilb}^{p(t)} (\mathbb P^2)\) due to \textit{J. Fogarty} [Am. J. Math. 90, 511--521 (1968; Zbl 0176.18401)]. \textit{A. Reeves} and \textit{M. Stillman} showed that every non-empty Hilbert scheme contains a smooth Borel-fixed point [J. Algebr. Geom. 6, No. 2, 235--246 (1997; Zbl 0924.14004)] and \textit{A. P. Staal} classified those with exactly one such fixed point, which are necessarily smooth and irreducible [Math. Z. 296, No. 3--4, 1593--1611 (2020; Zbl 1451.14010)].
The main result classifies Hilbert schemes with two Borel-fixed points over a field \(k\) of characteristic zero. To describe the result, express the Hilbert polynomial \(p(t)\) in the form used by \textit{Gotzmann}, namely
\[
p(t) = \sum_{i=1}^m \binom{t+\lambda_i-i}{\lambda_i-1}
\]
where \(\lambda_1 \geq \lambda_2 \geq \dots \geq \lambda_m \geq 1\) [\textit{G. Gotzmann}, Math. Z. 158, 61--70 (1978; Zbl 0352.13009)]. Writing \(\mathbf{\lambda} = (\lambda_1,\dots,\lambda_m)\), the theorem lists for exactly which \(\mathbf{\lambda}\) the Hilbert scheme \(\mathrm{Hilb}^{p(t)} (\mathbb P^n)\) has two Borel-fixed points and further determines when it is (a) smooth, (b) irreducible and singular or (c) a union of two components. In each case the irreducible components are normal and Cohen-Macaulay and the singularities of the Hilbert scheme appear as cones over certain Segre embeddings of \(\mathbb P^a \times \mathbb P^b\). Since the writing of his paper, (a) \textit{A. P. Staal} [``Hilbert schemes with two Borel-fixed points in arbitrary characteristic'', Preprint, \url{arXiv:2107.02204}] has shown that the theorem is valid in all characteristics with a small modification when char \(k=2\) and (b) \textit{R. Skjelnes} and \textit{G. G. Smith} [J. Reine Angew. Math. 794, 281--305 (2023; Zbl 07640144)] have classified the smooth Hilbert schemes are described their geometry.
Despite the difficulty of the content, the paper is readably written. Section 1 gives preliminaries on Borel-fixed (strongly stable) ideals and the resolution of \textit{S. Eliahou} and \textit{M. Kervaire} [J. Algebra 129, No. 1, 1--25 (1990; Zbl 0701.13006)], while Section 2 identifies the tuples \(\mathbf{\lambda} = (\lambda_1,\dots,\lambda_m)\) corresponding to Hilbert schemes with two components. Section 3 uses the comparison theorem of \textit{R. Piene} and \textit{M. Schlessinger} [Am. J. Math. 107, 761--774 (1985; Zbl 0589.14009)] to compute the tangent space of the non-lexicographic Borel-fixed ideal \(I(\mathbf{\lambda})\) and give a partial basis for the second cohomology group of \(k[x_0,\dots,x_n]/I(\mathbf{\lambda})\). These are used in Section 4 where the main theorem is proved to describe the universal deformation space of \(I(\mathbf{\lambda})\) and hence the nature of singularities of the Hilbert schemes. Finally in Section 5 the author gives examples of Hilbert schemes with three Borel-fixed points. The last three examples relate to Hilbert schemes studied in the literature [\textit{S. Katz}, in: Zero-dimensional schemes. Proceedings of the international conference held in Ravello, Italy, June 8-13, 1992. Berlin: de Gruyter. 231--242 (1994; Zbl 0839.14001); \textit{D. Chen} and \textit{S. Nollet}, Algebra Number Theory 6, No. 4, 731--756 (2012; Zbl 1250.14004); \textit{D. Chen} et al., Commun. Algebra 39, No. 8, 3021--3043 (2011; Zbl 1238.14012)]. Hilbert scheme; singularities; Borel-fixed points; deformations of ideals Syzygies, resolutions, complexes and commutative rings, Parametrization (Chow and Hilbert schemes), Fine and coarse moduli spaces, Singularities of surfaces or higher-dimensional varieties Hilbert schemes with two Borel-fixed 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 Each elliptic curve can be embedded uniquely in the projective plane, up to projective equivalence. The hessian curve of the embedding is generically a new elliptic curve, whose isomorphism type depends only on that of the initial elliptic curve. One gets like this a rational map from the moduli space of elliptic curves to itself. We call it the hessian dynamical system. We compute it in terms of the \(j\)-invariant of elliptic curves. We deduce that, seen as a map from a projective line to itself, it has 3 critical values, which correspond to the point at infinity of the moduli space and to the two elliptic curves with special symmetries. Moreover, it sends the set of critical values into itself, which shows that all its iterates have the same set of critical values. One gets like this a sequence of dessins d'enfants. We describe an algorithm allowing to construct this sequence. hession; elliptic curves; modular curves; dessins d'enfants Popescu-Pampu, P.: Iterating the Hessian: a dynamical system on the moduli space of elliptic curves and dessins d'enfants. In: Noncommutativity and Singularities, Adv. Stud. Pure Math., vol. 55, pp. 83-98. Math. Soc. Japan, Tokyo (2009) Families, moduli of curves (algebraic), Singularities in algebraic geometry, Complex surface and hypersurface singularities, Modifications; resolution of singularities (complex-analytic aspects), Dessins d'enfants theory Iterating the hessian: a dynamical system on the moduli space of elliptic curves and 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 We compute the Euler characteristics of the generalized Kummer schemes associated to \(A\times Y\), where \(A\) is an abelian variety and \(Y\) is a smooth quasi-projective variety. When \(Y\) is a point, our results prove a formula conjectured by Gulbrandsen. Shen, J.: The Euler characteristics of generalized Kummer schemes. arXiv:1502.03973 Parametrization (Chow and Hilbert schemes), Algebraic theory of abelian varieties The Euler characteristics of generalized Kummer 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 For every scheme \(X\) over a field \(k\), the \textit{arc scheme} of \(X\) is a \(k\)-scheme that parameterizes formal arcs on \(X\): for every field extension \(k'\) of \(k\), the \(k'\)-points of the arc scheme correspond canonically to points on \(X\) with values in the ring of formal power series \(k'[[t]]\). This construction can be generalized to a relative setting, replacing \(X\) by a morphism of schemes. In this chapter, we present three constructions of the arc scheme: one in terms of Weil restrictions, and the two others in terms of differential algebra. Each construction highlights some particular features of the arc scheme. Further results on arc schemes can be found in [\textit{A. Chambert-Loir} et al., Motivic integration. New York, NY: Birkhäuser (2018; Zbl 06862764)]. Arcs and motivic integration, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Classical problems, Schubert calculus Arc schemes in geometry and differential algebra | 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 (even not integral). A coherent system on \(X\) is a pair \((E,V)\), where \(E\) is a depth \(1\) sheaf on \(X\) and \(V\) is a linear subspace of \(H^0(E)\). The discrete invariants of \((X,V)\) are the rank and degree of \(E\) and the integer \(k:= \dim V\). There is the notion of stability for the pair \((E,V)\) depending a the real paramenter \(\alpha\) and moduli spaces for \(\alpha\)-stable/semistable coherent system [\textit{A. D. King} and \textit{P. E. Newstead}, Int. J. Math. 6, No. 5, 733--748 (1995; Zbl 0861.14028)]. In the paper under review the authors start the study of coherent system when \(X\) is reducible nodal curve of compact type. They fix a polarization \(w\) on \(X\), i.e. on \(X\), i.e. a positive integer for each irreducible component, and consider \((w,\alpha)\) stability, mainly for \(\alpha\) large. If \(k>r\) they prove the existence of \(\beta\in \mathbb{R}\) such that for all \(\alpha >\beta\) \(V\) generically spans \(E\) for each \((w,\alpha)\)-stable. Then they get smoothness and the expected dimension for the moduli space if the restrictions of \(E\) to each component \(X_i\) has degree \(\ge \max \{2p_a(X_i)+1,p_a(X_i)+r\}\). Many strong results are obtained in the case \(k=r+1\). coherent system; stability; curves of compact type; nodal curves Vector bundles on curves and their moduli, Algebraic moduli problems, moduli of vector bundles Coherent systems on curves of compact type | 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 Hausel and Rodriguez-Villegas (2015, \textit{Astérisque} 370, 113-156) recently observed that work of Göttsche, combined with a classical result of Erdös and Lehner on integer partitions, implies that the limiting Betti distribution for the Hilbert schemes \((\mathbb{C}^2)^{[n]}\) on \(n\) points, as \(n\rightarrow +\infty\), is a \textit{Gumbel distribution}. In view of this example, they ask for further such Betti distributions. We answer this question for the quasihomogeneous Hilbert schemes \(((\mathbb{C}^2)^{[n]})^{T_{\alpha ,\beta}}\) that are cut out by torus actions. We prove that their limiting distributions are also of Gumbel type. To obtain this result, we combine work of Buryak, Feigin, and Nakajima on these Hilbert schemes with our generalization of the result of Erdös and Lehner, which gives the distribution of the number of parts in partitions that are multiples of a fixed integer \(A\geq 2\). Furthermore, if \(p_k(A;n)\) denotes the number of partitions of \(n\) with exactly \(k\) parts that are multiples of \(A\), then we obtain the asymptotic
\[
p_k(A,n)\sim \frac{24^{\frac k2-\frac14}(n-Ak)^{\frac k2-\frac34}}{\sqrt2\left(1-\frac1A\right)^{\frac k2-\frac14}k!A^{k+\frac12}(2\pi)^k}e^{2\pi\sqrt{\frac1{6}\left(1-\frac1A\right)(n-Ak)}},
\]
a result which is of independent interest. Betti numbers; Hilbert schemes; partitions Parametrization (Chow and Hilbert schemes), (Co)homology theory in algebraic geometry, Analytic theory of partitions Limiting Betti distributions of Hilbert schemes on \(n\) 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 settle a number of open questions in unstable motivic homotopy theory. In particular, they prove that the sheaf of \(\mathbb{A}^1\)-connected components of a smooth variety is in general not homotopy invariant, not a birational invariant, and does not (in general) coincide with the sheaf of \(\mathbb{A}^1\)-chain connected components. Additionally they give examples of smooth varieties \(X\) such that \(\text{Sing}_*(X)\) fails to be \(\mathbb{A}^1\)-local. These are all counterexamples to conjectures of Morel and Asok-Morel.
In order to prove their results, the authors study carefully the sheaf \(\pi_0^{\mathbb{A}^1}(X)\). Their techniques can sometimes be used to establish positive results; to this end they prove that the conjectures of Morel and Asok-Morel hold for non-uniruled surfaces. \(\mathbb A^1\)-homotopy theory; \(\mathbb A^1\)-connected components C.~Balwe, A.~Hogadi, A.~Sawant, \(\mathbb{A}^1\)-connected components of schemes, Adv. Math. 282 (2015), 335--361. DOI 10.1016/j.aim.2015.07.003; zbl 1332.14025; MR3374529; arxiv 1312.6388 Motivic cohomology; motivic homotopy theory, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] \(\mathbb A^1\)-connected components of 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 [For the entire collection see Zbl 0527.00002.]
The following result is proved: A noetherian subalgebra A of a \({\mathbb{C}}\)- algebra of finite type is finitely generated if and only if the Gel'fand topology on Spec max A is locally compact. The Gel'fand topology is the weakest topology such that for every \(f\in A\) the function Spec max \(A\to {\mathbb{C}}\), \(m\mapsto residue\quad of\quad f\quad in\quad A/{\mathfrak m}={\mathbb{C}},\) is continuous. Actually there is a slightly more general result in terms of Spec A. spectrum; finite generation of subalgebra; algebra of finite type Relevant commutative algebra, Commutative rings and modules of finite generation or presentation; number of generators On the algebraization of some complex 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 \(C\) be a smooth projective curve of genus \(g\) and let \(E\) be a vector bundle over \(C\). The Segre invariant, semistability and cohomological semistability for \(E\) are closely related and have been studied from different points of view for many years. The author consider \(S:=\mathbb{P}(E)\) the associated projective bundle and he describes the inflectional loci of projective models \(\psi: S\dashrightarrow \mathbb{P}^n\) in terms of Quot Scheme of \(E\). The author gives a characterization of the Segre invariant \(s_1(E)\) in terms of the osculating space and the inflectional locus. This allows to give an equivalence between the semistability of bundle \(E\) with slope \(\mu<1-2g \) and the osculating space. In the same direction, the author gives a natural characterization of cohomological stability of \(E\) via inflectional loci. Finally, he proved that for general \(S\), the inflectional loci are the expected dimension. curve; scroll; quot scheme; inflectional locus Vector bundles on curves and their moduli, Projective techniques in algebraic geometry Quot schemes, Segre invariants, and inflectional loci of scrolls over 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 A scheme \(X\subset \mathbb P^n\) of codimension \(c\) is called \textit{standard determinantal} if its homogeneous saturated ideal can be generated by the maximal minors of a homogeneous \(t\times (t+c-1)\) matrix. It is called \textit{good determinantal} if in addition it is a generic complete intersection. Given integers \(a_0\leq\dotsb\leq a_{t+c-2}\) and \(b_1\leq\dotsb\leq b_t\), denote by \(W(\underline{b};\underline{a})\) the set of good determinantal schemes as above, given as the maximal minors of matrices whose entries are homogeneous of degree \(a_j-b_i\) (where \(0\leq j\leq t+c-2\) and \(1\leq i\leq t\)).
In the paper under review, the authors are interested in some natural problems concerning \(W(\underline{b};\underline{a})\):
(1) to determine the dimension of \(W(\underline{b};\underline{a})\) in terms of \(a_j\) and \(b_i\);
(2) is the closure of \(W(\underline{b};\underline{a})\) an irreducible component of the Hilbert scheme?
(3) Is the Hilbert scheme generically smooth along \(W(\underline{b};\underline{a})\)?
If \(a_{t+3}>a_{t-2}\) they almost solve the first problem, while for the other two problems they generalize their previous results of [Trans. Am. Math. Soc. 357, No. 7, 2871--2907 (2005; Zbl 1073.14063)] substantially. Good determinantal schemes; Hilbert schemes Kleppe, J.O.; Miró-Roig, R.M., Families of determinantal schemes, Proc. am. math. soc., 139, 3831-3843, (2011) Determinantal varieties, Parametrization (Chow and Hilbert schemes), Families, moduli of curves (algebraic), Families, moduli, classification: algebraic theory Families of determinantal 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 \(R\) be a local Artin ring with maximal ideal \({\mathfrak m}\) and residue class field of characteristic \(p>0\). We show that every finite flat group scheme over \(R\) is annihilated by its rank, whenever \({\mathfrak m}^p= p{\mathfrak m}=0\). This implies that any finite flat group scheme over an Artin ring, the square of whose maximal ideal is zero, is annihilated by its rank. characteristic \(p\); local Artin ring; finite flat group scheme Group schemes, Commutative Artinian rings and modules, finite-dimensional algebras, Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Characteristic \(p\) methods (Frobenius endomorphism) and reduction to characteristic \(p\); tight closure Finite flat group schemes over local Artin 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 A (contravariant) pseudofunctor over a category \(\mathcal C\) is assigning an object \(X\) of \(\mathcal C\) to a category \(\mathcal D(X)\) and a morphism \(f \: X \to Y\) to a functor \(\mathcal D(Y) \to \mathcal D(X)\). For example, the twisted inverse image \((-)^!\) is a pseudofunctor on the category of schemes. In the present paper, the authors construct a new pseudofunctor \((-)^\sharp\) on the category of formal schemes with codimension function. They assign such a scheme to a category of Cousin complexes. However, their definition of Cousin complexes is different from Hartshorne's one. Roughly speaking, they call \(M^\bullet\) a Cousin complex on a formal scheme~\(X\) if it has a similar structure as a Cousin complex of a sheaf on~\(X\) with Hartshorne's definition. Cousin complex; twisted inverse image; codimension function; formal scheme Lipman, J., Nayak, J., Sastry, S.: Pseudofunctorial behavior of Cousin complexes on formal schemes. Variance and duality for Cousin complexes on formal schemes. Contemporary Mathematics, vol. 375, pp. 3--133. American Mathematical Society, Providence (2005) Local cohomology and algebraic geometry, Foundations of algebraic geometry, (Co)homology theory in algebraic geometry, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Categorical algebra, Radical theory Pseudofunctorial behavior of Cousin complexes on formal 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 \textit{E. Bombieri} and \textit{J. Pila} [Duke Math. J. 59, No. 2, 337--357 (1989; Zbl 0718.11048)] introduced a method for bounding the number of integral lattice points that belong to a given arc under several assumptions. We generalize the Bombieri-Pila method to the case of function fields of genus 0 in one variable. We then apply the result to counting the number of elliptic curves contained in an isomorphism class and with coefficients in a box. integral points; function fields; Diophantine equations Lattice points in specified regions, Algebraic functions and function fields in algebraic geometry On the Bombieri-Pila method 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 The paper studies the Weil-étale topology which is to the usual étale topology what is the cyclic group \(\mathbb{Z}\) to its profinite completion. Namely for a scheme over a finite field \(k\) a Weil-étale sheaf on \(X\) is an étale sheaf on the base-extension to the algebraic closure \(k\), together with an isomorphism to its Frobenius-pullback. (Note that the total Frobenius is the product of the geometric and the arithmetic Frobenius, and it induces a natural equivalence on sheaves.) The properties of the corresponding cohomology are studied, also for non-torsion coefficients like the integers \(\mathbb{Z}\). This relies heavily on previous work of \textit{J. Milne} [``Étale cohomology'', Princeton Univ. Press (1980; Zbl 0433.14012); Am. J. Math. 108, 297--360 (1986; Zbl 0611.14020)]. Finally, the author states, and proves in some cases, a conjecture relating Euler-characteristics and zeta-values at \(s= 0\). étale topology; Weil-group Lichtenbaum, S., The Weil-étale topology on schemes over finite fields, Compositio Math., 141, 3, 689-702, (2005) Étale and other Grothendieck topologies and (co)homologies, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) The Weil-étale topology on schemes over finite 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 The stratification associated with the number of generators of the ideals of the punctual Hilbert scheme of points on the affine plane has been studied since the 1970s. In this paper, we present an elegant formula for the E-polynomials of these strata. Symmetric functions and generalizations, Polynomial rings and ideals; rings of integer-valued polynomials, Parametrization (Chow and Hilbert schemes) A note on the E-polynomials of a stratification of the Hilbert scheme 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 nonnegative Grassmannian is a cell complex with rich geometric, algebraic, and combinatorial structures. Its study involves interesting combinatorial objects, such as positroids and plabic graphs. Remarkably, the same combinatorial structures appeared in many other areas of mathematics and physics, e.g., in the study of cluster algebras, scattering amplitudes, and solitons. We discuss new ways to think about these structures. In particular, we identify plabic graphs and more general Grassmannian graphs with polyhedral subdivisions induced by 2-dimensional projections of hypersimplices. This implies a close relationship between the positive Grassmannian and the theory of fiber polytopes and the generalized Baues problem. This suggests natural extensions of objects related to the positive Grassmannian. total positivity; positive Grassmannian; hypersimplex; matroids; positroids; cyclic shifts; Grassmannian graphs; plabic graphs; polyhedral subdivisions; triangulations; zonotopal tilings; associahedron; fiber polytopes; Baues poset; generalized Baues problem; flips; cluster algebras; weakly separated collections; scattering amplitudes; amplituhedron; membranes Combinatorial aspects of algebraic geometry, Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.), Cluster algebras, Grassmannians, Schubert varieties, flag manifolds Positive Grassmannian and polyhedral subdivisions | 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 finite group \(G \subset \mathrm{GL}(n, \mathbb{C})\), the \(G\)-Hilbert scheme is a fine moduli space of \(G\)-clusters, which are 0-dimensional \(G\)-invariant subschemes \(Z\) with \(H^0(\mathcal{O}_Z)\) isomorphic to \(\mathbb{C}[G]\). In many cases, the \(G\)-Hilbert scheme provides a good resolution of the quotient singularity \(\mathbb{C}^n/G\), but in general it can be very singular. In this note, we prove that for a cyclic group \(A \subset \mathrm{GL}(n, \mathbb{C})\) of type \(\frac{1}{r}(1, \dots, 1, a)\) with \(r\) coprime to \(a\), \(A\)-Hilbert Scheme is smooth and irreducible. \(A\)-Hilbert schemes; cyclic quotient singularities Parametrization (Chow and Hilbert schemes), Vanishing theorems in algebraic geometry, Arcs and motivic integration, Complex surface and hypersurface singularities, Singularities of surfaces or higher-dimensional varieties, Variation of Hodge structures (algebro-geometric aspects) A-Hilbert schemes for \(\displaystyle\frac{1}{r}(1^{n-1},a)\) | 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 set of the common zeros of the multivariate splines, the piecewise algebraic variety is a kind of generalization of the classical algebraic variety. In this paper, we discuss the computation problem of the piecewise algebraic variety. The approach presented here is the interval iterative algorithm by introducing the concept of \(\epsilon \)-deviation solutions. Meanwhile, we present a simple method to evaluate the bound on the value of the derivative of the function on a given region. piecewise algebraic variety; interval iterative algorithm; Krawczyk algorithm; \(\epsilon \)-deviation solutions; derivative matrix R. H. Wang and X. L. Zhang, ``Interval iterative algorithm for computing the piecewise algebraic variety,'' Computers & Mathematics with Applications, vol. 56, no. 2, pp. 565-571, 2008. General methods in interval analysis, Computational aspects in algebraic geometry Interval iterative algorithm for computing the piecewise 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 [For the entire collection see Zbl 0635.00006.]
Let \(m_ s\) denote the rational equivalence class of the s-fold points of a morphism \(f:\quad X\to Y,\) where X,Y are smooth algebraic varieties defined over an algebraically closed field. The author shows the existence of certain polynomials \(P_ i\) with rational coefficients, appearing in an iterative formula for the classes \(m_ s\). Moreover, he gives a computatonal algorithm for the \(P_ i's.\)
These results are then applied to two enumerative problems: finding the number of space conics meeting a given curve 8 times, and - by a degeneration method - finding the number of conics contained in a general quintic threefold in \(P^ 4\). The latter number had previously been determined by the author [Compos. Math. 60, 151-162 (1986; Zbl 0606.14039)], but he gives good reasons for hoping that the present method can also be applied to find the number of twisted cubics on a general quintic threefold. incidence; rational equivalence class; s-fold points; enumerative problems; number of conics; number of twisted cubics Enumerative problems (combinatorial problems) in algebraic geometry, Singularities in algebraic geometry Iteration of multiple point formulas and applications to conics | 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 goal of this article under review is to extend results on the Cayley-Bacharach properties of \(K\)-rational points in \(\mathbb P^n_k\) to arbitrary 0-dimensional subschemes over an arbitrary field \(K\) using the technique of liaison theory.
The Cayley-Bacharach property of a set of \(K\)-points \(Z\) in \(\mathbb P^n_k\) is well-studied.
Among others cited in the paper, [\textit{P. Griffiths} and \textit{J. Harris}, Ann. Math. (2) 108, 461--505 (1978; Zbl 0423.14001); \textit{D. Eisenbud} et al., Bull. Am. Math. Soc., New Ser. 33, No. 3, 295--324 (1996; Zbl 0871.14024)] are important references.
We say \(Z \subset \mathbb{P}^n\) satisfies the Cayley-Bacharach property with respect to the linear system \(|\mathcal O(l)|\) (abbreviated \(CBP(l)\)) if whenever a divisor \(D\) in \(|\mathcal O(l)|\) contains a co-length one subscheme of \(Z\), it must contain all of \(Z\).
This definition is subtly extended for arbitrary \(0\)-dimensional subschemes \(Z\) over an arbitrary field \(K\) using the notion of ``maximal \(p_j\) subschemes''.
The regularity index \(r_Z\) of \(Z\) is the minimal degree after which the Hilbert function and the Hilbert polynomial of \(Z\) coincide.
The scheme \(Z\) is said to be a Cayley-Bacharach scheme if it satisfies \(CBP(r_Z-1)\), which is the highest possible degree \(d\) such that \(Z\) can satisfy \(CBP(d)\).
It is known that for a set of \(K\)-points \(Z\), being a Cayley-Bacharach scheme is equivalent to the condition b) where a generic element of the least degree in the ideal of its link in a complete intersection does not vanish anywhere on \(Z\).
The main result of this paper is Theorem 3.5 which characterizes the property of being a Cayley-Bacharach scheme in terms of different conditions on its link.
The authors show that condition b) is only sufficient but necessary for \(Z\) to be a Cayley-Bacharach scheme in the general setting.
Last but not least, the authors characterize the Cayley-Bacharach property
of degree \(d\) using the canonical module of \(Z\), and use this result to give an equivalent condition for \(Z\) to be arithmetic Gorenstein in terms of the Hilbert function of the Dedekind different of \(Z\). zero-dimensional scheme; Cayley-Bacharach property; Hilbert function; liaison theory; Dedekind different Linkage, complete intersections and determinantal ideals, Linkage, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Projective techniques in algebraic geometry An application of liaison theory to zero-dimensional 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 The purpose of this paper is to define a reasonable topological cyclic homology (\(\text{TC}\)) for schemes. Many interesting results are proven in this paper, and the authors have taken the opportunity to spell out yet another foundation for the theory using the symmetric spectra of \textit{M. Hovey, B. Shipley} and \textit{J. Smith} [J. Am. Math. Soc. 13, No.~1, 149-208 (2000; Zbl 0931.55006)].
Topological cyclic homology was introduced by \textit{M. Bökstedt, W. C. Hsiang} and \textit{I. Madsen} [Invent. Math. 111, No.~3, 465-539 (1993; Zbl 0804.55004)], and has proved to be a good invariant for studying algebraic \(K\)-theory. \textit{R. McCarthy} [Acta Math. 179, No.~2, 197-222 (1997; Zbl 0913.19001)] and the reviewer [\textit{B. Dundas}, Acta Math. 179, No. 2, 223-242 (1997; Zbl 0913.19002)] have proven that the difference between algebraic \(K\)-theory and topological cyclic homology is in some sense ``locally constant''. Several authors, including Bökstedt, Hesselholt, Madsen and Rognes has used this property to calculate algebraic \(K\)-theory in important examples by analyzing topological cyclic homology. In particular, the last author and Madsen have demonstrated that topological cyclic homology is well behaved for algebras over perfect fields of positive characteristic and deduced many interesting \(K\)-theoretic results, see e.g. \textit{L. Hesselhold} [Acta Math. 177, No.~1, 1-53 (1996; Zbl 0892.19003)].
In the paper under review, the authors extend the definition of topological Hochschild homology to schemes using Thomason's hypercohomology construction \(\mathbb{H}^{\dot{}}(X,-)\). The standard approach is then applied to construct topological cyclic homology from topological Hochschild homology. This definition is reasonable in that it agrees with the old definition for affine schemes, and it does not depend on the topology coarser than the étale topology. For smooth schemes over perfect fields of positive characteristic \(p\) the authors identify the topological cyclic homology sheaf for the Zariski and étale topology. In fact, if \(X\) is such a scheme, the equivalence of Thomason and Trobaugh
\[
K(X)^{\widehat{}}_p\simeq\mathbb{H}^{\dot{}}(X_{\text{Zar}},K(-)^{\widehat{}}_p)
\]
and the equivalence proved in the paper
\[
\text{TC}(X,p)\simeq\mathbb{H}^{\dot{}}(X_{\text{ét}},K(-)^{\widehat{}}_p)
\]
allows one to identify the cyclotomic trace with the change of topology map
\[
\mathbb{H}^{\dot{}}(X_{\text{Zar}},K(-)^{\widehat{}}_p)\to \mathbb{H}^{\dot{}}(X_{\text{ét}},K(-)^{\widehat{}}_p).
\]
Using the last equivalence above they give a formula calculating the topological cyclic homology groups of a field of positive characteristic in terms of its \(K\)-groups. Another interesting result is that \(\text{TC}(X,p)\) vanishes above the dimension of \(X\).
The appendix contains a proof of the fact that the cyclotomic trace respects the multiplicative structure for commutative rings. More precisely, if \(\mathcal C\) is an exact category with a strict symmetric monoidal bi-exact structure, then the cyclotomic trace
\[
K(\mathcal C)\to \text{TC}(\mathcal C,p)
\]
is a map of symmetric spectra. topological cyclic homology; hypercohomology; algebraic K-theory of schemes; cyclotomic trace Geisser, Thomas; Hesselholt, Lars, Topological cyclic homology of schemes.Algebraic \(K\)-theory, Seattle, WA, 1997, Proc. Sympos. Pure Math. 67, 41-87, (1999), Amer. Math. Soc., Providence, RI \(K\)-theory and homology; cyclic homology and cohomology, Applications of methods of algebraic \(K\)-theory in algebraic geometry, \(K\)-theory of schemes Topological cyclic homology of 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 This paper is devoted to the study of the group \(\pi_1 (X)/N\), where \(X\) is a projective smooth variety, \(Y \subset X\) is a closed subvariety, \(f : Z \to Y\) is surjective, all components of \(Z\) are complete and normal, and \(N \subset \pi_1 (X)\) is the normal subgroup generated by the images of the components of \(Z\). The result is the following theorem:
Assume \(\pi_1 (Y,y) \to \pi_1 (X,y)\) is surjective, or that \(\pi_1^{\text{alg}} (Y,y) \to \pi^{\text{alg}}_1 (X,y)\) is surjective; then for any integer \(n\), there are only finitely many conjugacy classes of representations
\[
\pi_1 (X)/N \to \text{GL} (n, \mathbb{C}).
\]
The proof uses Weil's construction of the set of representations of \(\pi_1 (X)\) modulo conjugacy as an algebraic variety, and the description of its tangent space as an \(H^1\)-group of a local system on \(X\). It also uses Simpson's result on the existence of local systems \(V\) underlying a variation of Hodge structure in each component of this variety, and the fact that the group \(H^1 (\text{End} V)\) for such a local system has a functorial mixed Hodge structure. The proof of the infinitesimal version of the theorem, namely the injectivity of the map \(H^1 (X, \text{End} V) @>f^*>> H^1 (Z, \text{End} V)\), then follows from a weight argument at such a point \(V\). complex local systems; fundamental group; mixed Hodge structure Lasell, B. : Complex local systems and morphisms of varieties. Dissertation , University of Chicago (1994). Coverings in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry Complex local systems and morphisms of 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 \(S\) be a smooth minimal surface of general type over the field of complex numbers and denote by \({\mathcal M} (S)\) the coarse moduli space of surface of general type homeomorphic to \(S\), \({\mathcal M} (S)\) is a quasi-projective variety by Gieseker's theorem. Since \(K^2_S>0\), the divisibility \(r(S)\) of the canonical class \(k_S= c_1(K_S) \in H^2 (S,\mathbb{Z})\) is well defined, i.e. \(r(S)= \max \{r\in\mathbb{N} \mid r^{-1} c_1(S) \in H^2 (S,\mathbb{Z})\}\). \(r(S)\) is a positive integer which is invariant under deformation and the set
\[
{\mathcal M}_d (S)=\bigl\{[S']\in {\mathcal M} (S) \mid r(S') =r(S)\bigr\}
\]
is a subvariety of \({\mathcal M} (S)\) and the number of connected components of \({\mathcal M}_d(S)\) is bounded by a function \(\delta\) of the numerical invariants \(K^2_S\), \(\chi ({\mathcal O}_S)\). It is known that \(\delta\) is not bounded [see \textit{M. Manetti}, Compos. Math. 92, N. 3, 285-297 (1994; Zbl 0849.14016)]. Here we prove that ``in general'' \(\delta\) takes quite large values, more precisely we have
Theorem A. For every real number \(4\leq \beta \leq 8\) there exists a sequence \(S_n\) of simply connected surfaces of general type such that:
(a) \(y_n= K^2_{S_n}\), \(x_n= \chi ({\mathcal O}_{S_n}) \to\infty\) as \(n\to \infty\).
(b) \(\lim_{n\to \infty} (y_n/x_n) =\beta\).
(c) \(\delta (S_n) \geq y_n^{(1/5) \log y_n}\) (here \(\delta (S_n)\) is the number of connected components of \({\mathcal M}_d (S_n))\).
Theorem A relies on the explicit description of the connected components in the moduli space of a wide class of surfaces of general type whose Chern numbers spread in all the region \({1\over 2} c_2 \leq c^2_1 \leq 2c_2 \).
Definition B. A finite map between normal algebraic surfaces \(p: X\to Y\) is called a simple iterated double cover associated to a sequence of line bundles \(L_1, \dots, L_n \in\text{Pic} (Y)\) if the following conditions hold:
(1) There exist \(n+1\) normal surfaces \(X=X_0, \dots, X_n=Y\) and \(n\) flat double covers \(\pi_i: X_{i-1} \to X_i\) such that \(p=\pi_n \circ \cdots \circ \pi_1\).
(2) If \(p_i: X_i\to Y\) is the composition of \(\pi_j\)'s \(j>i\) then we have for every \(i=1, \dots, n\) the eigensheaves decomposition \(\pi_{i*} {\mathcal O}_{X_{i-1}}= {\mathcal O}_{X_i} \oplus p^*_i (-L_i)\).
For any sequence \(L_1, \dots, L_n\in \text{Pic} (\mathbb{P}^1 \times \mathbb{P}^1)\) define \((N(L_1, \dots, L_n)\) as the image in the moduli space of the set of surfaces of general type whose canonical model is a simple iterated double cover of \(\mathbb{P}^1 \times \mathbb{P}^1\) associated to \(L_1,\dots,L_n\). The main theme of this paper is to determine sufficient conditions on the sequence \(L_1,\dots,L_n\) in such a way that the set \(N(L_1, \dots, L_n)\) has ``good'' properties; the conditions we find are summarized in the following definition:
Definition C. A sequence \(L_1, \dots, L_n\), \(L_i= {\mathcal O}_{\mathbb{P}^1 \times \mathbb{P}^1} (a_i,b_i)\), \(n\geq 2\) of line bundles on \(\mathbb{P}^1 \times\mathbb{P}^1\) is called a good sequence if it satisfies the following conditions.
(C1) \(a_i\), \(b_i\geq 3\) for every \(i=1, \dots, n\).
(C2) \(\max_{j<i} \min (2a_i-a_j\), \(2b_i-b_j) <0\).
(C3) \(a_n\geq b_n+2\), \(b_{n-1} \geq a_{n-1}+2\).
(C4) \(a_i\), \(b_i\) are even for \(i=2, \dots, n\).
(C5) For every \(i<n\), \(2a_i -a_{i+1} \geq 2\), \(2b_i- b_{i+1} \geq 2\).
The main result we prove is:
Theorem D. Let \(L_1,\dots,L_n\) be a good sequence in sense of definition C, then:
(a) \(N(L_1, \dots, L_n)\) is a nonempty connected component of the moduli space.
(b) \(N(L_1, \dots, L_n)\) is reduced, irreducible and unirational. (For (a) and (b) the condition C5 is not necessary.)
(c) The generic \([\text{S}] \in N(L_1, \dots, L_n)\) has \(\Aut (S)= \mathbb{Z}/2 \mathbb{Z}\).
(d) If \(M_1, \dots, M_m\) is another good sequence and \(N (L_1, \dots, L_n)= N(M_1, \dots, M_m)\) then \(n=m\) and \(L_i=M_i\) for every \(i=1, \dots, n\).
Theorem D gives us some new interesting examples of homeomorphic but not deformation equivalent surfaces of general type. divisibility of canonical class; Picard group; number of connected components of coarse moduli space; Chern class; surface of general type; simple iterated double cover Manetti, M, Iterated double covers and connected components of moduli spaces, Topology, 36, 745-764, (1996) Families, moduli, classification: algebraic theory, Enumerative problems (combinatorial problems) in algebraic geometry, Picard groups, Coverings in algebraic geometry, Moduli, classification: analytic theory; relations with modular forms, Complex-analytic moduli problems Iterated double covers and connected components of moduli 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 It is known that the integrable ellipsoidal billiard can be regarded as the limit of a geodesic flow on an ellipsoid, when one of its semi-axis tends to zero. This phenomenon can be used to construct discrete analogs of other algebraic integrable systems. Trajectories of such discrete systems are defined as limit points of certain asymptotic trajectories of the continuous counterparts. The latter are linearized on generalized Jacobians of singular spectral curves, whereas the former are described as shifts on reductions of such Jacobians. We illustrate this approach by producing a discretization of the classical Euler top problem on the group SO(3) and give its geometric model of motion. generalized Jacobi-Mumford system; ellipsoidal billiard; geodesic flow; asymptotic trajectories; generalized Jacobians; Euler top Fedorov, Yu.; Levi, D. (ed.); Ragnisco, O. (ed.), Discrete versions of some algebraic integrable systems related to generalized Jacobians, 147-160, (2000) , Relationships between algebraic curves and integrable systems, Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) Discrete versions of some algebraic integrable systems related to generalized Jacobians | 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 smooth algebraic variety \(X\) its Chow group \(A*X\) has a ring structure given by the intersection product. However, when \(X\) has singularities the situation is much more complicated. In some cases it is impossible to define a natural intersection product on the group \(A*X\) [see \textit{A. Zobel}, Mathematika 8, 39-44 (1961; Zbl 0099.15902)]. When the singularities are mild, various attempts have been made to define such products. Danilov constructed intersection products for simplicial toric varieties [\textit{V. I. Danilov}, Russ. Math. Surv. 33, No. 2, 97-154 (1978); translation from Usp. Mat. Nauk 33, No. 2(200), 85-134 (1978; Zbl 0425.14013)], \textit{D. Mumford} did this for the moduli space of stable curves [in: Arithmetic and geometry, Pap. dedic I. R. Shafarevich, Vol. II: Geometry, Prog. Math. 36, 271-328 (1983; Zbl 0554.14008)]. \textit{S. Kleiman} and \textit{A. Thorup} defined the category of \(C_\mathbb{Q}\) orthocyclic schemes in: Algebraic geometry, Proc. Summer Res. Inst., Brunswick 1985, part II, Proc. Symp. Pure Math. 46, 321-370 (1987; Zbl 0664.14031). These schemes have been exensively studied by \textit{A. Vistoli} [Invent. Math. 97, No. 3, 613-670 (1989; Zbl 0694.14001) and Compos. Math. 70, No. 3, 199-225 (1989; Zbl 0702.14002)] who called them Alexander schemes.
The category of Alexander schemes is convenient from the point of view of the intersection theory with \(\mathbb{Q}\)-coefficients. In the paper the author answers affirmatively the conjecture of A. Vistoli that the Alexander property is Zariski local. The author constructs a category of bivariant sheaves and shows (theorem 4.6) that the scheme is Alexander if and only if all the first cohomolgoy groups of the bivariant sheaves vanish. Then the author proves that a union of two Alexander subschemes is again Alexander. intersection product; orthocyclic schemes; Alexander schemes Kimura, S.: A cohomological characterization of Alexander schemes. Invent. math. 137, 575-611 (1999) Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Vanishing theorems in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] A cohomological characterization of Alexander 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 \(k\) be an algebraically closed field of characteristic \(p\). A \(k\)-group \(\mathcal G\) is uniserial if \(\mathcal G\) has a unique composition series. Uniserial groups play an important role in determining if an infinitesimal group is representation-finite, that is it admits only finitely many isomorphism classes of finite-dimensional indecomposable modules.
This paper gives a complete classification of isomorphism classes of non-trivial infinitesimal unipotent commutative uniserial \(k\)-groups. It turns out that there are six different types of isomorphism classes, all of which can be described as kernels of Witt vectors (or their duals).
The classification is facilitated by the use of (classical) Dieudonné modules. A Dieudonné module \(M\) corresponds to a uniserial group if and only if either \(M/FM\) or \(M/VM\) is a simple module (over the Dieudonné ring \(\mathbb{D}\)). In this case, \(M\) is also called uniserial. The authors construct a list of Dieudonné modules with \(M/VM\) simple, and then pass to Cartier duality to get the others.
The results of this classification enable an examination of representation-finite infinitesimal groups, and the article concludes with such a study, as well as with a discussion of the classification problem for unipotent groups of complexity 1. uniserial groups; infinitesimal groups; finite representation type; Witt vectors; Dieudonné modules; simple modules; group schemes Rolf Farnsteiner, Gerhard Röhrle, and Detlef Voigt, Infinitesimal unipotent group schemes of complexity 1, Colloq. Math. 89 (2001), no. 2, 179 -- 192. Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers, Modular Lie (super)algebras, Group schemes Infinitesimal unipotent group schemes of complexity 1 | 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 theory of higher-order automorphic forms has come into existence only within the past five years or so, but it already has an interesting history thanks to the work of a number of leading mathematicians. In [Acta Arith. 103, No. 3, 209--223 (2002; Zbl 1020.11025)], \textit{G. Chinta}, \textit{N. Diamantis} and \textit{C. O'Sullivan} introduced holomorphic higher-order forms and the larger space of smooth \((C^\infty)\) higher-order forms, and classified the latter for order \(s= 2\).
In both spaces the forms are ``entire'' in the Hecke sense; that is, they have at most polynomial growth at the parabolic cusps of the relevant Fuchsian group. (The interest in the larger space of smooth forms appears to stem from the possibility of a future study of higher-order Maass wave forms.) The purpose of the article under review is to generalize the classification theorem of the 2002 paper to higher-order automorphic forms of all orders \(\geq 2\).
Let \(\Gamma\) be a finitely generated Fuchsian group of the first kind (i.e., with finite hyperbolic volume), acting on the upper half-plane \(H\) and having parabolic elements. For their classification result (Theorem 4.4) the authors require \(\Gamma\) to be torsion-free as well, let \(k\in 2\mathbb Z^+\). If \(f\) is a function defined on \(H\), consider the operator \((f|_k\gamma)(z)= (cz+ d)^{-k} f(\gamma z)\), for \(\gamma= (\begin{smallmatrix} * & *\\ c & d\end{smallmatrix})\in \Gamma\). Traditionally, an automorphic form of weight \(k\) satisfies the transformation law
\[
f|_k(\gamma- 1)= 0\quad\text{for all }\gamma\in\Gamma.\tag{\(*\)}
\]
In the definition of higher-order automorphic form of weight \(k\) and order \(s\) \((s\in \mathbb Z^+)\), \((*)\) is replaced by the more general transformation property
\[
f|_k(\gamma_1- 1)\cdots(\gamma_s- 1)= 0\quad\text{for }\gamma_j\in \Gamma,\;1\leq j\leq s.\tag{\(**\)}
\]
Theorem 4.4 may be stated as follows. Suppose \(\Gamma\) has \(2g\) hyperbolic generators and \(m\geq 1\) parabolic generators. (For \(\Gamma\) torsion-free there are no elliptic generators.) Let \(M^s_k\) denote the space of smooth, entire automorphic forms of weight \(k\) and order \(s\geq 2\). Then,
\[
M^{s+1}_k\cong M^s_k\oplus M_k',\tag{\(***\)}
\]
where the indicated isomorphism refers to the vector space structure and the index \(i\) runs from 1 to \((2g+ m-1)^2\). Note that \((***)\) can be iterated to obtain \(M^{s+1}_k\) as (an isomorphic image of) a direct sum of
\[
\sum^s_{\ell= 0}(2g+ m-1)^\ell
\]
copies of \(M_k'\). (The space of order 2 forms characterized in the 2002 paper is somewhat smaller than that which appears here, since in the earlier article the authors imposed the further condition, \(f|_k(\pi- 1)= 0\), for all parabolic \(\pi\in\Gamma\).)
\textit{K.-T. Chen's} result on iterated integrals referred to in the title [Trans. Am. Math. Soc. 156, 359--379 (1971; Zbl 0217.47705), Theorem 3.1] is needed here to establish Proposition 4.1, and in the authors' proof of this proposition the application of Chen's Theorem 3.1 requires that \(\pi_1(\Gamma/H^+)= \Gamma\). This fails if \(\Gamma\) has elements of finite order. We note that, in the absence of Proposition 4.1, the result here would be the much weaker statement: \(M^{s+1}_k\cong M^s_k\oplus S\), with \(S\) some subspace of the direct sum of \((2q+ m-1)^2\) copies of \(M_k'\). Since the Acta paper derives the characterization in the case \(s= 2\) without the assumption that \(\Gamma\) is free of torsion, it seems reasonable to suspect that the result of the present work carries over to the more general case as well. higher-order automorphic forms; iterated integrals; filtration Mazur, B., Rubin, K.: The statistical behavior of modular symbols and arithmetic conjectures. Presentation at Toronto, Nov 2016 (2016). http://www.math.harvard.edu/~mazur/papers/heuristics.Toronto.12.pdf Automorphic forms, one variable, Transcendental methods, Hodge theory (algebro-geometric aspects) Iterated integrals and higher order automorphic forms | 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 paper under review, the author studies particular loci of the Hilbert scheme \(\mathcal{H}\mathrm{ilb}^{r}_{n}\) of \(r\) points in the affine space \(\mathbb{A}^n\). In a previous paper [J. Commut. Algebra 3, No. 3, 349--404 (2011; Zbl 1237.14012)], the author introduced the functor \(\mathcal{H}\mathrm{ilb}^{\mathrm{Lex}\Delta}_{n,k}: (k\mathrm{-Alg}) \rightarrow (\mathrm{Sets})\) that associates to any algebra \(B\) over a ring \(k\) the set of reduced Gröbner bases in the ring \(B[x_1,\ldots,x_n]\) with respect to the lexicographic order with a given standard set \(\Delta\) of \(r\) monomials. He proved that this functor is representable and represented by a locally closed subscheme of \(\mathcal{H}\mathrm{ilb}^{r}_{n}\) called Gröbner stratum. In this paper, the author studies the subfunctor \(\mathcal{H}\mathrm{ilb}^{\mathrm{Lex}\Delta,\text{ét}}_{n,k}\) of \(\mathcal{H}\mathrm{ilb}^{\mathrm{Lex}\Delta}_{n,k}\) that considers only the reduced Gröbner bases of ideals defining reduced points. The main result of the paper is that \(\mathcal{H}\mathrm{ilb}^{\mathrm{Lex}\Delta,\text{ét}}_{n,k}\) is representable and, in the case of a ring \(k\) such that \(\mathrm{Spec}\, k\) is irreducible, all the connected components of the representing scheme have the same dimension. Moreover, the number of connected components and their dimension are nicely described in terms of combinatorial properties of the standard set \(\Delta\). Hilbert scheme of points; Gröbner stratum; lexicographic order; reduced points Mathias Lederer (2014). Components of Gröbner strata in the Hilbert scheme of points. \textit{Proc. Lond. Math. Soc}. (3) \textbf{108}(1), 187-224. ISSN 0024-6115. URL http://dx.doi.org/10.1112/plms/pdt018. Parametrization (Chow and Hilbert schemes), Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Polynomials over commutative rings, Enumerative problems (combinatorial problems) in algebraic geometry Components of Gröbner strata in the Hilbert scheme 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 Here we extend the definition and the main properties of separators of a connected component \(mP\) of a zero-dimensional scheme \(Z:= mP\cup Z''\subset\mathbb P^n\) introduced by \textit{E. Guardo, L. Marino} and \textit{A. Van Tuyl} [J. Algebra 324, No. 7, 1492--1512 (2010; Zbl 1216.13010)] if \(Z''\) is a disjoint union of fat point. zero-dimensional scheme; fat point; Hilbert function Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Syzygies, resolutions, complexes and commutative rings, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Projective techniques in algebraic geometry Separations of zero-dimensional schemes in \(\mathbb P^n\) | 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 in this paper a canonical form for the elements in the ring of continuous piecewise polynomial functions. This new representation is based on the use of a particular class of functions
\[
\{C_i(P):P\in\mathbb Q[x],\quad i=0,\dots,\deg(P)\}
\]
defined by
\[
C_i(P)(x)=\begin{cases} 0 & \text{ if }x\leq\alpha \\ P(x) & \text{ if }x\geq\alpha\end{cases}
\]
where \(\alpha\) is the \(i\)th real root of the polynomial \(P\). These functions will allow us to represent and manipulate easily every continuous piecewise polynomial function through the use of the corresponding canonical form.
It will be also shown how to produce a ``rational'' representation of each function \(C_i(P)\) allowing its evaluation by performing only operations in \(\mathbb Q\) and avoiding the use of any real algebraic number. continuous piecewise polynomial functions; Pierce-Birkhoff conjecture; canonical form for functions; conversion algorithms Computer-aided design (modeling of curves and surfaces), Semialgebraic sets and related spaces A canonical form for the continuous piecewise polynomial 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 authors consider a smooth, algebraic curve \(C\) of genus \(g \geq 2\) over \(\mathbb{C}\) and consider the moduli of coherent systems over \(C\). A \textit{coherent system} of type \((n,d,k)\) is a pair \((E,V)\), where \(E \to C\) is a rank-\(n\) vector bundle of degree \(d\) and \(V \subseteq H^0(C;E)\) is a \(k\)-dimensional linear subspace.
The notion of slope stability extends to coherent systems, depending on a parameter \(\alpha \in \mathbb{R}\), via the slope \(\mu_\alpha = (d+\alpha k)/n\). \(G(\alpha;n,d,k)\) denotes the moduli space of \(\alpha\)-stable coherent systems on \(C\), which is related to the \textit{Brill-Noether locus} \(B(n,d,k)\) of \(\mu\)-stable bundles \(E \to C\) of rank \(n\) and degree \(d\) with \(h^0(C; E) \geq k\) (considered inside the moduli \(M(n,d)\) of all \(\mu\)-stable rank-\(n\) bundles of degree \(d\)).
The authors had previously studied spaces of coherent systems [J. Reine Angew. Math. 551, 123--143 (2002; Zbl 1014.14012); Int. J. Math. 18, No. 4, 411--453 (2007; Zbl 1117.14034); Int. J. Math. 14, No. 7, 683--733 (2003; Zbl 1057.14041)] in the case \(k \leq n\). The present paper presents a generalization to arbitrary \(k\), subject to the constraint \(d\leq2n\); the final Section~7 illustrates with an example why the case \(d>2n\) is more complicated.
The main results are these: \(G(\alpha;n,d,k)\) is irreducible whenever it is non-empty. With the exception of the case where \(C\) is hyperelliptic and \(d=2n\) and \(k>n\), the conditions for non-emptiness of \(G(\alpha;n,d,k)\) are the same as for non-emptiness of \(B(n,d,k)\). With the exception of the case where \(C\) is hyperelliptic and \((n,d,k) = (n,2n,n+1)\) and \(n<g-1\), the dimension of \(G(\alpha;n,d,k)\) is the \textit{Brill-Noether number}
\[
\beta(n,d,k) {{\mathop:}=} n^2(g-1)+1-k(k-d+n(g-1)) \text{ .}
\]
We always require \(\alpha>0\), \(d>0\) and \(\alpha(n-k)<d\) for non-emptiness. The authors prove sharp inequalities on \(n\), \(d\) and \(k\) for the non-emptiness of \(G(\alpha;n,d,k)\). For example, when \(C\) is not hyperelliptic, then \(G(\alpha;n,d,k) \neq \varnothing\) if and only if \(k\leq n+(d-n)/g\) and \((n,d,k)\neq(n,n,n)\), or if \((n,d,k) = (g-1,2g-2,g)\); otherwise \(G(\alpha;n,d,k) = \varnothing = B(n,d,k)\) for all \(\alpha\). Moreover, a coherent system \((E,V)\) may be presented as an exact sequence
\[
0 \longrightarrow D^* \longrightarrow V \otimes \mathcal{O}_C \longrightarrow E \longrightarrow F \oplus T \longrightarrow 0 \text{ ,}
\]
where \(D\) and \(F\) are vector bundles, \(T\) is a torsion sheaf and \(h^0(C;D^*)=0\). The authors prove that if \((E,V) \in G(\alpha;n,d,k)\) is a generic element then: \(T=0=D\) if \(k<n\), \(F=0=D\) if \(k=n\), and \(F=0=T\) if \(k>n\). The case when \(C\) is hyperelliptic is treated separately with different methods.
Section~2 of the paper is a nice presentation of the basic properties of coherent systems, and Section~3 proves existence results for \(\alpha\)-stable coherent systems, in analogy to the existence of \(\mu\)-stable vector bundles on curves; these sections may be of interest in their own right. Sections~4, 5 and 6 contain the technical work of the paper, proving respectively irreducibility of \(G(\alpha;n,d,k)\), non-emptiness for non-hyperelliptic \(C\) and for hyperelliptic \(C\). algebraic curves; Brill-Noether loci; coherent systems; moduli of vector bundles Bradlow, B.; García-Prada, O.; Mercat, V.; Muñoz, V.; Newstead, P., Moduli spaces of coherent systems of small slope on algebraic curves, Commun. algebra, 37, 8, 2649-2678, (2009) Vector bundles on curves and their moduli, Algebraic moduli problems, moduli of vector bundles, Special divisors on curves (gonality, Brill-Noether theory) Moduli spaces of coherent systems of small slope on algebraic 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 paper, we study the tangent spaces of the smooth nested Hilbert scheme \(\mathrm{Hilb}^{n,n-1}(\mathbb A^2)\) of points in the plane, and give a general formula for computing the Euler characteristic of a \(\mathbb T^2\)-equivariant locally free sheaf on \(\mathrm{Hilb}^{n,n-1}(\mathbb A^2)\). Applying our result to a particular sheaf, we conjecture that the result is a polynomial in the variables \(q\) and \(t\) with non-negative integer coefficients. We call this conjecturally positive polynomial as the ``nested \(q,t\)-Catalan series'', for it has many conjectural properties similar to that of the \(q,t\)-Catalan series. Atiyah-Bott Lefschetz formula; (nested) Hilbert scheme of points; tangent spaces; diagonal coinvariants Can, M.: Nested Hilbert schemes and the nested q,t-Catalan series Parametrization (Chow and Hilbert schemes), Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Nested Hilbert schemes and the nested \(q,t\)-Catalan 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 For homogeneous Gorenstein ideals \(I\subseteq R= K[x_0,\dots, x_n]\) of height \(\geq 4\), there exist only very few constructions: intersections of \(G\)-linked aCM-varieties introduced by \textit{C. Peskine} and \textit{L. Szpiro} [Invent. Math 26, 271--304 (1974; Zbl 0298.14022)], certain divisors on aCM-schemes studied by \textit{J. O. Kleppe}, \textit{J. C. Migliore}, \textit{R. M. Miró-Roig}, \textit{U. Nagel} and \textit{C. Peterson} [in: ``Gorenstein liaison, complete intersection liaison invariants and unobstructedness'', Mem. Am. Math Soc. 732 (2001; Zbl 1006.14018)], and the top-dimensional components of zeroschemes of regular sections of Buchsbaum-Rim sheaves found by \textit{J. C. Migliore} and \textit{C. Peterson} [Trans. Am. Math. Soc. 349, No. 9, 3803--3824 (1997; Zbl 0885.14022)] and extended by \textit{J. C. Migliore, U. Nagel} and \textit{C. Peterson} [J. Algebra 219, No. 1, 378-420 (1999; Zbl 0961.14031)]. Since Gorenstein schemes are essential for performing \(G\)-liaison, the authors' new entry on this short list is most welcome.
The proposed construction works as follows: Let \(V\) be an arithmetically Gorenstein scheme containing a complete intersection \(W\) of the same codimension. Under suitable hypotheses we can then replace \(W\) with a complete intersection \(W'\) such that \((V\setminus W)\cup W'\) is still arithmetically Gorenstein. This construction can be applied iteratively. The proof of its correctness is obtained by producing the free resolution of the new Gorenstein ideal.
The authors demonstrate the usefulness of their construction in several ways. They show that every arithmetically Gorenstein zero-dimensional scheme of degree \(n + 2\) in \(\mathbb{P}^n\) can be constructed in this way, and they give examples of Gorenstein schemes constructed using their method which cannot be obtained using the other methods. Furthermore, they analyze the effect of their construction in codimension three by comparing the Buchsbaum-Eisenbud matrices of the input and output Gorenstein ideals. homogeneous Gorenstein ideal; \(G\)-liaison; complete intersection; free resolution of Gorenstein ideal; arithmetically Gorenstein scheme Bocci, C; Dalzotto, G; Notari, R; Spreafico, ML, An iterative construction of Gorenstein ideals, Trans. Am. Math. Soc., 354, 1417-1444, (2004) Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Linkage, complete intersections and determinantal ideals, Syzygies, resolutions, complexes and commutative rings, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) An iterative construction of Gorenstein 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 \textit{A. Iarrobino} and \textit{V. Kanev} introduced in [Power sums, Gorenstein algebras, and determinantal loci. With an appendix `The Gotzmann theorems and the Hilbert scheme' by Anthony Iarrobino and Steven L. Kleiman. Berlin: Springer (1999; Zbl 0942.14026)] the catalecticant varieties and schemes in relation to the problem of representing a homogeneous form as a sum of powers of linear forms as well as related questions. Of special interest are the irreducible components and smoothness properties of such schemes. For zero-dimensional complete intersections with homogeneous ideal generators of equal degrees over an algebraically closed field of characteristic zero, the author gives a combinatorial proof of the smoothness of the corresponding catalecticant schemes along an open subset of a particular irreducible component. catalecticant schemes and varieties Geometric invariant theory, Actions of groups on commutative rings; invariant theory, Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes, Complete intersections A combinatorial proof of the smoothness of catalecticant schemes associated to 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 In this article the authors study the limits of Weierstrass points in a family of curves degenerating to some stable curve, under the assumption that the limit stable curve has exactly two irreducible components. They also give a concrete approach for the general situation, but not the complete solution. limits of Weierstrass points; degeneration to stable curves; family of curves E. Esteves and N. Medeiros, Limit canonical systems on curves with two components, Inventiones Mathematicae 149 (2002), 267--338. Riemann surfaces; Weierstrass points; gap sequences, Families, moduli of curves (algebraic) Limit canonical systems on curves with two components | 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 gives the necessary and sufficient conditions for the existence of unitary local systems with prescribed local monodromies on \(\mathbb{P}^1-S\) where \(S\) is a finite set. This is used to give an algorithm to decide if a rigid local system on \(\mathbb{P}^1-S\) has finite global monodromy, thereby answering the following question of \textit{N. M. Katz} [Rigid local systems, Ann. Math. Stud. 139 (1969; Zbl 0864.14013)]. Let \(L\) be a rigid local system on \(\mathbb{P}^1-\{p_1, \dots,p_s\}\) with finite monodromies at the punctures. When does the local system \(L\) have finite global monodromy (in terms of the Jordan canonical forms of the local monodromies around the punctures)? Clearly the local monodromies should be of finite order. But of course there are more conditions. The above problem is related to the following problem concerning \(\text{SU} (n)\): Let \(\overline A_1,\overline A_2,\dots,\overline A_s\) be conjugacy classes in \(\text{SU}(n)\). When can we lift to matrices \(A_i\) in \(\text{SU}(n)\) with the conjugacy class of \(A_i=\overline A_i\) and so that \(A_1 A_2\dots A_s=I\)?
We see that an affirmative answer for all \(\text{Gal} (\overline\mathbb{Q}/ \mathbb{Q})\) conjugates of the local monodromies for the second problem is what is needed for the first problem. The \(\text{SU} (n)\) problem is related to quantum cohomology. I. Biswas had previously considered and solved this problem for \(\text{SU}(2)\).
Firstly, by the theorem of \textit{V. Mehta} and \textit{C. Seshadri} [Math. Ann. 248, 205--239 (1980; Zbl 0454.14006)] (modified to the \(\text{SU}(n)\) setting in the appendix to this paper) the existence problem for lifting is the same as the existence of a semistable parabolic vector bundle with prescribed local weights on \(\mathbb{P}^1\). Using the openness of semi-stability, this is reduced to checking if the trivial vector bundle with generic flags and the prescribed weights is semistable. This means that no subbundle should contradict semi-stability. The subbundles of a given rank and degree of the trivial vector bundle of rank \(n\) form an open subset of a Quot scheme and also of the moduli space of maps from \(\mathbb{P}^1\) to an appropriate Grassmann variety. Now the question of existence of a subbundle is translated into existence of a map from \(\mathbb{P}^1\) to a Grassmann variety, such that the prescribed points on \(\mathbb{P}^1\) go to appropriate generic Schubert cycles. The existence is (with a little bit more work) realized as the nonvanishing of certain Gromov-Witten numbers. The Harder-Narasimhan filtration is used to conclude that only inequalities corresponding to intersections which are numerically one need to be considered.
The above problem is related to one considered by \textit{A. A. Klyachko} [Sel. Math., New Ser. 4, 419--445 (1998; Zbl 0915.14010)]. Using the Harder-Narasimhan filtration in that context the author concludes that only intersections which are numerically one need to be considered. rigid local systems; nonvanishing of Gromov-Witten numbers; finite global monodromy; quantum cohomology; Harder-Narasimhan filtration P. Belkale, ''Local systems on % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuqr1ngBPrgarmWu51MyVXguY9 % gCGievaerbd9wDYLwzYbWexLMBbXgBcf2CPn2qVrwzqf2zLnharyav % P1wzZbItLDhis9wBH5garqqtubsr4rNCHbGeaGqiVu0Je9sqqrpepC % 0xbbL8F4rqqrFfpeea0xe9Lq-Jc9vqaqpepm0xbba9pwe9Q8fs0-yq % aqpepae9pg0FirpepeKkFr0xfr-xfr-xb9adbaqaaeGaciGaaiaabe % qaamaaeaqbaaGcbaWefv3ySLgznfgDOjdarCqr1ngBPrginfgDObcv % 39gaiyaacqWFzecudaahaaWcbeqaaiabigdaXaaaaaa!47A8! $ \mathbb{P}^1 $ - S for S a finite set,'' Compositio Math. 129(1), 67--86 (2001).
Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Local systems on \({\mathbb{P}}^1-S\) for \(S\) a finite 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 A class of autonomous systems, \(\dot y = V(y)\), \(y(0) = x \in R^ n\), where \(V\) is a \(C^ 1\)-vector field on \(R^ n\), is that having polynomial flow, i.e., the solution depends polynomially upon the initial conditions. A well known result of \textit{B. Coomes} and \textit{V. Zurkowski} [J. Dyn. Differ. Equations 3, No. 1, 29-66 (1991; Zbl 0728.34001)] shows that an autonomous system has polynomial flow if and only if the derivation induced by the vector field that defines the system on the \(n\)-variable polynomial ring is locally finite.
In this paper the author studies the notion of locally finite and locally nilpotent derivations on a commutative algebra obtaining simple results on them and then applies these results to prove that the Lorenz system and the Maxwell-Bloch equations have no polynomial flows. These results were proved before with some restrictions and in a more complicated way.
Moreover, the author provides an algorithm that decides if a derivation on a polynomial ring in two variables is locally finite. This algorithm is easily programmable with any language of symbolic computation. Using this algorithm the author obtains a new proof of the classification theorem of Bass and Meisters concerning vector fields on \(R^ 2\).
A section about the automorphism group of polynomial rings is included. In this section the author gives an equivalent formulation of the Jacobian conjecture in terms of locally nilpotent derivations. A good survey on this conjecture is one of the same author in [Computational aspects of Lie group representations and related topics, Proc. Comput. Algebra Semin., Amsterdam/Neth. 1990, CWI Tracts 84, 29-44 (1991; Zbl 0742.14008)]. polynomial flow; autonomous systems; derivations on polynomial rings; Lorenz equations; Maxwell-Bloch equations Den Essen, A. Van: Locally finite and locally nilpotent derivations with applications to polynomial flows and polynomial morphisms. Proc. amer. Math. soc. 116, No. 3, 861-871 (1992) Dynamics induced by flows and semiflows, Determinants, permanents, traces, other special matrix functions, Morphisms of commutative rings, Rational and birational maps, Birational automorphisms, Cremona group and generalizations, Polynomial rings and ideals; rings of integer-valued polynomials Locally finite and locally nilpotent derivations with applications to polynomial flows and polynomial morphisms | 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 of characteristic \(0\) and denote by \(A_{n+1}(K)\) the Weyl algebra attached to the polynomial ring \(K[t,x_1,\dots,x_n]\).
The author gives algorithms for detecting whether a finitely generated module over \(A_{n+1}(K)\) is specializable along \(t=0\) and for computing the corresponding \(b\)-functions. He also obtains algorithms for computing restrictions and algebraic local cohomology with respect to \(t=0\). The main ingredient is the technique of Gröbner bases for rings of differential operators, as introduced in [\textit{J. Briançon} and \textit{Ph. Maisonobe}, Enseign. Math. 30, 7-38 (1984; Zbl 0542.14008)] and [\textit{F. Castro}, C. R. Acad. Sci., Paris, Sér. I 302, No. 14, 487-490 (1986; Zbl 0606.32007)]. In order to compute Gröbner bases with respect to the Malgrange-Kashiwara filtration, the author uses a homogenization process for elements in \(A_{n+1}(K)\), which is analogous to the algorithm for the tangent cone in local algebra.
A similar idea has been used in [\textit{A. Assi, F. J. Castro-Jiménez} and \textit{J. M. Granger}, C. R. Acad. Sci., Paris, Sér. I 320, No. 2, 193-198 (1995; Zbl 0849.13019) and Compos. Math. 104, No. 2, 107-123 (1996; Zbl 0862.32005)] [see also \textit{F. J. Castro-Jiménez} and the reviewer, Homogenising differential operators, Prep. 36, Fac. Matemáticas, Univ. Sevilla, June (1997)]. Full details have appeared in the author's paper [Duke Math. J. 87, No. 1, 115-132 (1997; see the paper above)]. Weyl algebra; Gröbner basis; \(b\)-function; D-module Oaku, T.: Algorithms for b-functions, induced systems and algebraic local cohomology of D-modules. Proc. Japan acad. 72, 173-178 (1996) Sheaves of differential operators and their modules, \(D\)-modules, Rings of differential operators (associative algebraic aspects), Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs Algorithms for \(b\)-functions, induced systems, and algebraic local cohomology of \(D\)-modules | 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 non-commutative scheme. A Grothendieck category of quasi-coherent sheaves on \(X\) is called a quasi-scheme and in a certain sense it can be identified with \(X\). Suppose that there is a regularly embedded hypersurface \(Y\subset X\). The authors investigate curves on \(X\) which are in general position with respect to \(Y\). In particular it is shown that the category of quasi-coherent sheaves on such a curve \(C\) is isomorphic to a certain quotient category of graded modules over the \(n\)-dimensional commutative polynomial ring graded by \(\mathbb{Z}^n\), where \(n\) is the number of points of the intersection \(C\cap Y\). The results obtained are applied to some interesting examples of non-commutative schemes such as the enveloping algebra of the two-dimensional non-Abelian Lie algebra, the quantum affine and projective planes, the quantum projective space of \textit{M. Vancliff} [J. Algebra 165, No. 1, 63-90 (1994; Zbl 0837.16023)], and others. non-commutative schemes; quasi-schemes; quasi-coherent sheaves; quantum algebras; quantum planes; curves; Grothendieck categories; graded modules; enveloping algebras S. P. Smith and J. J. Zhang,Curves on quasi-schemes, Algebras and Representation Theory1 (1998), 311--351. Graded rings and modules (associative rings and algebras), Homological dimension in associative algebras, Special algebraic curves and curves of low genus, Relationships between algebraic curves and physics, Grothendieck categories, \(K_0\) of other rings, Rings arising from noncommutative algebraic geometry, Quantum groups (quantized enveloping algebras) and related deformations Curves on quasi-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 Denote by \(\mathfrak{g}\simeq \mathfrak {gl}_n^{[F_v^+:\mathbf{Q}_p]}\) (respectively, \(\mathfrak{b}\)) the \(L\)-Lie algebra of \(G:=({\text{Res}}_{F^+_v/\mathbf{Q}_p}\text{GL}_{n/F_v^+})_L\) (respectively, of the Borel subgroup \(B\) of upper triangular matrices).
The authors describe the completed local rings of the trianguline variety at certain points of integral weights in terms of completed local rings of algebraic varieties related to Grothendieck's simultaneous resolution \(\widetilde{\mathfrak{g}}\rightarrow \mathfrak{g}\) of singularities, where \(\widetilde{\mathfrak{g}}:=
\{(gB,\psi)\in G/B\times \mathfrak{g}: \text{Ad}(g^{-1})\psi\in \mathfrak{b}\}\subseteq G/B\times \mathfrak{g}\).
The authors derive several local consequences at these points for the trianguline variety: local irreducibility, description of all local companion points in the crystalline case, and combinatorial description of the completed local rings of the fiber over the weight map. Taylor-Wiles assumptions are as follows: \(p>2\), the field \(F\) is unramified over \(F^+\), \(F\) does not contain a nontrivial root \(\sqrt[p]{1}\) of unity, and \(G\) is quasi-split at all finite places of \(F^+\), \(U_v\) is hyperspecial when the finite place \(v\) of \(F^+\) is inert in \(F\), and
\(\overline{\rho}(\text{Gal}(\overline F/F(\sqrt[p]{1}))\) is adequate. Combined with the patched Hecke eigenvariety under the usual Taylor-Wiles assumptions, these results have global consequences, such as classicality of crystalline strictly dominant points on global Hecke eigenvarieties, existence of all expected companion constituents in the completed cohomology, and existence of singularities on global Hecke eigenvarieties. simultaneous resolution of singularities; trianguline variety; Taylor-Wiles hypothesis; Hecke eigenvariety; global Hecke eigenvarieties; Galois representations Homogeneous spaces and generalizations, Structure theory for Lie algebras and superalgebras, Lie algebras and Lie superalgebras, Group schemes, Linear algebraic groups and related topics A local model for the trianguline variety 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 Minimal algebraic surfaces of general type with the smallest possible invariants have geometric genus zero and \( K^2=1\) and are usually called numerical Godeaux surfaces. Although they have been studied by several authors, their complete classification is not known. In this paper we classify numerical Godeaux surfaces with an involution, i.e. an automorphism of order 2. We prove that they are birationally equivalent either to double covers of Enriques surfaces or to double planes of two different types: the branch curve either has degree 10 and suitable singularities, originally suggested by Campedelli, or is the union of two lines and a curve of degree 12 with certain singularities. The latter type of double planes are degenerations of examples described by Du Val, and their existence was previously unknown; we show some examples of this new type, also computing their torsion group. Godeaux surface; involution; torsion group Calabri, A.; Ciliberto, C.; Mendes Lopes, M., Numerical godeaux surfaces with an involution, \textit{Trans. Amer. Math. Soc.}, 359, 4, 1605-1632, (2007) Surfaces of general type, Special surfaces, Automorphisms of surfaces and higher-dimensional varieties Numerical Godeaux surfaces with an involution | 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 commutative ring. For a \(k\)-algebra \(A\) one has Hochschild homology \(HH_* (A)\) and cyclic homology \(HC_* (A)\) which are related by Connes' SBI-sequence. For a scheme \(X\) over \(k\) one may sheafify the Hochschild complex of \(A = {\mathcal O}_X (U)\), \(U \subset X\) open, to obtain a complex of sheaves \({\mathcal C}^h_*\). One defines the Hochschild homology of \(X\), \({\mathcal H} H_n (X) : = {\mathcal H}^{-n} (X, {\mathcal C}^h_*)\). \textit{S. C. Geller} and the author [cf. ``Étale descent for Hochschild and cyclic homology'', Comment. Math. Helv. 66, No. 3, 368-388 (1991; Zbl 0741.19007)] showed that \({\mathcal H} H_* (X)\) has the properties of a cyclic homology theory, i.e. (i) functoriality, (ii) Mayer-Vietoris, and (iii) the property that for an affine scheme \(X = \text{Spec} (A)\), one has \({\mathcal H} H_n (X) \cong {\mathcal H} H_n (A)\). In the underlying paper, the same result is established for cyclic homology of schemes. One sheafifies Connes' double complex \((B_{**}, b, B)\) to obtain a double complex of sheaves \({\mathcal B}_{**}\). The cyclic homology of \(X\) is defined by \({\mathcal H} C_n (X) : = {\mathcal H}^{-n} (X, \text{Tot} {\mathcal B}_{**})\). The definition goes back to \textit{J. Loday}, but the key property (iii) was still lacking. Here it is proved by a truncation procedure on the columns of the double complex \({\mathcal B}_{**}\). The hypercohomology \({\mathcal H}C_*\) becomes the limit of the hypercohomology of the truncated double complexes \(\tau_{q < r} {\mathcal B}_{**}\). The subtleties of hypercohomology (of not necessarily bounded below complexes) is explained in an appendix. Hochschild homology; cyclic homology; schemes; hypercohomology Weibel, C., \textit{cyclic homology for schemes}, Proc. Amer. Math. Soc., 124, 1655-1662, (1996) \(K\)-theory and homology; cyclic homology and cohomology, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Other (co)homology theories (cyclic, dihedral, etc.) [See also 19D55, 46L80, 58B30, 58G12] Cyclic homology for 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 Here we study the postulation with respect to a linear system of the general curvilinear zero-dimensional subscheme whose support is a general point of a prescribed irreducible subvariety. Projective techniques in algebraic geometry General curvilinear zero-dimensional subschemes with prescribed support and their postulation | 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 an arrangement algorithm for plane curves. The inputs are (1) continuous, compact, \(x\)-monotone curves and (2) a module that computes approximate crossing points of these curves. There are no general position requirements. We assume that the crossing module output is \(\epsilon\) accurate, but allow it to be inconsistent, meaning that three curves are in cyclic \(y\) order over an \(x\) interval. The curves are swept with a vertical line using the crossing module to compute and process sweep events. When the sweep detects an inconsistency, the algorithm breaks the cycle to obtain a linear order. We prove correctness in a realistic computational model of the crossing module. The number of vertices in the output is \(V=2n+N+\min(3kn, n^2/2)\) and the running time is \(O(\log n)\) for \(n\) curves with \(N\) crossings and \(k\) inconsistencies. The output arrangement is realizable by curves that are \(O(\epsilon+kn\epsilon)\) close to the input curves, except in \(kn\epsilon\) neighborhoods of the curve tails. The accuracy can be guaranteed everywhere by adding tiny horizontal extensions to the segment tails, but without the running time bound. An implementation is described for semi-algebraic curves based on a numerical equation solver. Experiments show that the extensions only slightly increase the running time and have little effect on the error. On challenging data sets, the number of inconsistencies is at most \(3N\), the output accuracy is close to \(\epsilon\), and the running time is close to that of the standard, non-robust floating point sweep. V. Milenkovic, E. Sacks, An approximate arrangement algorithm for semi-algebraic curves, in: SCG '06: Proceedings of the Twenty-Second Annual ACM Symposium on Computational Geometry, 2006, pp. 237 -- 246 Semialgebraic sets and related spaces, Numerical aspects of computer graphics, image analysis, and computational geometry, Computer graphics; computational geometry (digital and algorithmic aspects) An approximate arrangement algorithm for semi-algebraic 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 aim of this paper is to continue the work done by the author and \textit{J. Migliore} [Commun. Algebra 18, No. 9, 3015-3040 (1990; Zbl 0711.14031)] on the following problem: Let \(H\subseteq\mathbb{P}^ N\) be a hyperplane and \(B\subseteq H\) a 0-dimensional scheme (possibly not reduced) spanning \(H\). Find a ``nice'' curve \(C\subseteq\mathbb{P}^ N\) such that \(B=C\cap H\), as schemes. -- The main result of the paper is the following:
Let \(d,k,t\in\mathbb{Z}\), \(d>t\geq 2\) and \(t\leq d+1\). Let \(H\subseteq\mathbb{P}^{k+1}\) be a hyperplane which meets transversally a smooth connected rational curve \(D\subseteq\mathbb{P}^{k+1}\) of degree \(d\), and \(D\cap H=A\) (as schemes). -- Let \(B\) be a double structure on \(A\) (i.e. \(A=B_{\text{red}}\) and, at each point \(x\in A\), length\((B)=2)\). Then there is a double structure \(N\subseteq\mathbb{P}^{k+1}\) on \(D\) such that \(N\cap H=B\) and \(p_ a(N)=t-1\).
The main tool for the proof is the well known ``Ferrand construction'' (which allows to give doublings of a locally complete intersection curve when a rank one quotient of their conormal bundle is given).
In the last part of the paper the case of the possible genera (geometric or arithmetic) of curves with a given hyperplane section is treated, and the main result there (proposition 2.1) gives, for a certain range of the values \(t,d,\Delta\), that a general subset \(Z\subseteq E\) of cardinality \(d\) of an integral, nondegenerate rational curve \(E\subseteq\mathbb{P}^{k+1}\) is the hyperplane section of an integral curve \(C\subseteq\mathbb{P}^{k+1}\), with \(p_ a(C)-\Delta\leq p_ g(C)\leq p_ a(C)\). Ferrand construction; genus; complete intersection curve; hyperplane section of an integral curve Curves in algebraic geometry, Surfaces and higher-dimensional varieties, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Curves whose hyperplane section is a given scheme | 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 Motivated by applications in computer vision, the authors study multiview varieties. These are subvarieties of \((\mathbb P^2)^n\) that contain the images of \(n\) linear projections from \(\mathbb P^3\) to \(\mathbb P^2\) (``cameras''). Knowledge about multiview varieties and their ideals is important for the numeric reconstruction of points from their images. The authors' findings go well-beyond these immediate applications and reveal a fascinating algebraic and combinatorial geometry of the varieties under scrutiny.
A first important result is that a known generating system of generic multiview ideals is in fact a universal Gröbner basis. This observation is followed by a thorough investigation of multiview ideals and, in particular, a distinguished monomial subideal. An explicit description is given and interesting results on its Hilbert function in the \(\mathbb Z^n\)-grading are provided. The article's main result states that the Hilbert scheme that parametrizes the \(\mathbb Z^n\)-homogeneous ideals with this Hilbert function contains the space of camera positions as a special component.
The above statements are true in generic cases. Other results pertain to collinear and infinitesimally neighbouring cameras or less than five cameras with a toric multiview variety. multigraded Hilbert scheme; computer vision; monomial ideal; Groebner basis; generic initial ideal C. Aholt, B. Sturmfels, and R. Thomas, \textit{A Hilbert scheme in computer vision}, Canad. J. Math., 65 (2013), pp. 961--988, . Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Toric varieties, Newton polyhedra, Okounkov bodies, Projective techniques in algebraic geometry, Projective analytic geometry, Computational issues in computer and robotic vision A Hilbert scheme in computer vision | 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 book attempts to provide a coherent account of the current state of étale homotopy theory. Since its introduction by \textit{M. Artin} and \textit{B. Mazur} [Étale homotopy, Lect. Notes Math. 100 (1969; Zbl 0182.260)] étale homotopy theory has been refined - and thereby rendered prohibitively technical - in the course of the pursuit and development of the ideas of Quillen and Sullivan. With a view to making the subject accessible the author gives a fairly thorough, basic introduction to the étale homotopy type and cohomology related to the étale site of a simplicial scheme. Applications to such topics as the Adams conjecture and self-maps of classifying spaces are given. The generalised cohomology properties of the étale homotopy type are developed a little - treating such topics as Poincaré duality, tubular neighbourhoods, fibrations and function spaces.
The successes of étale homotopy theory have been in its application of results from algebraic geometry to solve problems in topology. Regrettably, the use of topological methods in conjunction with the étale topology to solve problems in K-theory are not touched on in this book. This is a shame. It came too early, I suppose, to include the major advances of R. Thomason and A. Suslin in algebraic K-theory. However, this volume would have been a suitable place for a discussion, for example, of characteristic classes in the generalised cohomology of the étale homotopy types of schemes - an inclusion which K-theorists would have welcomed. étale homotopy of simplicial schemes; étale cohomology; Adams conjecture; self-maps of classifying spaces; Poincaré duality; tubular neighbourhoods; fibrations; function spaces Friedlander, Eric M., Étale homotopy of simplicial schemes, Annals of Mathematics Studies 104, vii+190 pp., (1982), Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo Research exposition (monographs, survey articles) pertaining to algebraic topology, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Homotopy theory, Classical real and complex (co)homology in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry, Stable classes of vector space bundles in algebraic topology and relations to \(K\)-theory, Topological \(K\)-theory Etale homotopy of simplicial 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 \(R = {\mathbb C}[x_1,\ldots, x_n]\) be the polynomial ring graded by a non-negative \(d\times n\)-matrix \(A = (a_1, \ldots, a_n)\) of non-negative integers such that \(\deg x_i = a_i \in {\mathbb N}^d\) is given by a vector. This defines a decomposition \(R = \bigoplus_{b \in {\mathbb N}A} R_b,\) where \({\mathbb N}A\) denotes the subsemigroup of \({\mathbb N}^d\) generated by the vectors \(a_1, \ldots, a_n\) and \(R_b\) is the \({\mathbb C}\)-span of an element \(b\) of the subsemigroup. The toric Hilbert scheme parametrizes all the \(A\)-homogeneous ideals \(I \subset R\) with the property that the graded component \((R/I)_b\) is a 1-dimensional \({\mathbb C}\)-vector space. This concept was summarized by \textit{B. Sturmfels} in the first chapter of his book ``Gröbner bases and convex polytopes'', Univ. Lect. Ser. 8 (1996; Zbl 0856.13020)]. In the paper under review, the authors illustrate the use of Macaulay 2 for exploring the structure of toric Hilbert schemes. It is known that all the components of the scheme are toric varieties. Among them there is a fairly well understood component, the coherent component. The results contribute to the open problem whether toric Hilbert schemes are always connected. In their investigations the authors encounter algorithms from commutative algebra, polyhedral theory and geometric combinatorics. Macaulay2; toric Hilbert scheme; semigroup algebra Stillman, M., Sturmfels, B., Thomas, R.: Algorithms for the toric Hilbert scheme. In: Computations in Algebraic Geometry using Macaulay 2, D. Eisenbud et al. (eds.), Algorithms and Computation in Mathematics Vol 8, Springer, 2002, pp. 179--213 Computational aspects of higher-dimensional varieties, Parametrization (Chow and Hilbert schemes), Software, source code, etc. for problems pertaining to algebraic geometry, Symbolic computation and algebraic computation Algorithms for the toric Hilbert scheme | 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 arrangements of slightly skewed tropical hyperplanes, called blades by A. Ocneanu, on the vertices of a hypersimplex \(\Delta_{k,n}\), and we investigate the resulting induced polytopal subdivisions. We show that placing a blade on a vertex \(e_J\) induces an \(\ell \)-split matroid subdivision of \(\Delta_{k,n}\), where \(\ell\) is the number of cyclic intervals in the \(k\)-element subset \(J\). We prove that a given collection of \(k\)-element subsets is weakly separated, in the sense of the work of Leclerc and Zelevinsky on quasicommuting families of quantum minors, if and only if the arrangement of the blade \(((1, 2, \ldots, n))\) on the corresponding vertices of \(\Delta_{k,n}\) induces a matroid (in fact, a positroid) subdivision. In this way we obtain a compatibility criterion for (planar) multi-splits of a hypersimplex, generalizing the rule known for 2-splits. We study in an extended example a matroidal arrangement of six blades on the vertices \(\Delta_{3,7}\). combinatorial geometry; matroid subdivisions; weakly separated collections Configurations and arrangements of linear subspaces, Combinatorial aspects of tropical varieties, Combinatorial aspects of tessellation and tiling problems, Discrete geometry, Algebraic combinatorics From weakly separated collections to matroid subdivisions | 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 analyze the computational complexity of the problem of interpolating real algebraic functions given by a black box for their evaluations, extending the results of the authors [SIAM J. Comput. 23, No. 1, 1-11 (1994)] on interpolation of sparse rational functions. computational complexity; real algebraic functions; interpolation; sparse rational functions D. Grigoriev, M. Karpinski, M. Singer, Computational complexity of sparse real algebraic function interpolation, Computational Algebraic Geometry, Birkhäuser, Boston, 1993, pp. 91--104. Analysis of algorithms and problem complexity, Numerical interpolation, Real algebraic sets Computational complexity of sparse real algebraic function interpolation | 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 discuss Hilbert functions and graded Betti numbers of arithmetically Gorenstein subschemes of projective space, and the recent results obtained by \textit{J. Migliore} and \textit{U. Nagel} [Adv. Math. 180, 1--63 (2003; Zbl 1053.13006)] for arithmetically Gorenstein subschemes with the subspace property, a generalization of the weak Lefschetz property. Artinian-Hilbert function; Stanley-Iarrobino sequence; SI-sequence Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Syzygies, resolutions, complexes and commutative rings, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Linkage, complete intersections and determinantal ideals Reduced arithmetically Gorenstein schemes with prescribed properties. | 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 If \(X\) is a reduced irreducible variety of codimension \(c\) in \(\mathbb{P}^ r\) over an algebraically closed field \(F\) of characteristic 0, then a general plane of dimension \(c\) meets \(X\) in a set of reduced points in linearly general position; that is, no \(k+2\) of them are contained in a \(k\)-plane for \(k<c\). For this reason, reduced sets of points in linearly general position play a significant role in many arguments of algebraic geometry, perhaps most notably those of Castelnuovo theory, which gives a bound on the genus of a variety in terms of its degree. In certain applications, however, it is desirable to extend the theory to more general subschemes of projective space.
Definition: A finite subscheme \(\Gamma\) of \(\mathbb{P}^ r\) (over some algebraically closed field) is in linearly general position if for every proper linear subspace \(\Lambda \subset \mathbb{P}^ r\) we have \(\deg \Lambda \cap \Gamma \leq 1 + \dim \Lambda\).
Algebraic interpretation: If we let \(W\) be an \((r+1)\)-dimensional vector space over \(F\), and write \(\mathbb{P}^ r = \mathbb{P}(W)\), then a finite subscheme of \(\mathbb{P}^ r\) corresponds (cf. \S1) to a finite-dimensional \(F\)-algebra \(A = {\mathcal O}_ \Gamma (\Gamma)\) and a map from \(W\) to \(A\) whose image includes the identity element. It turns out that the subscheme is in linearly general position iff for every ideal \(I\) of \(A\), the composite map \(W \to A \to A/I\) is either a monomorphism or an epimorphism. -- We will be interested here in whether the lemma of Castelnuovo, which says that a (reduced) set of \(r+3\) points in linearly general position in \(\mathbb{P}^ r\) must lie on a rational normal curve, remains valid in the context of schemes. Elementary examples suggest that this might not be the case! The first infinitesimal neighborhood of a point in \(\mathbb{P}^ r\) is a scheme of degree \(r+1\) in linearly general position, and certainly lies on no smooth curves at all if \(r \geq 2\). We shall see similar examples for all \(r\) later on. However, as soon as the degree of \(\Gamma\) is at least \(r+3\), such examples are impossible, and the scheme-theoretic version of Castelnovo's lemma does hold.
Main result. (Theorem 1): Suppose \(\Gamma\) is a finite subscheme of \(\mathbb{P}^ r\) in linearly general position over an algebraically closed field.
(a) If \(\deg \Gamma \geq r + 3\), then \(\Gamma\) lies on a smooth curve which is unramified at each point in the support of \(\Gamma\).
(b) If \(\deg \Gamma = r + 3\), then \(\Gamma\) lies on a unique (smooth) rational normal curve of degree \(r\).
Corollary. Any two subschemes \(\Gamma\) and \(\Gamma'\) of degree \(r+3\) in linearly general position in \(\mathbb{P}^ r\) are conjugate by an automorphism of \(\mathbb{P}^ r\) provided that they are the same as cycles and their supports contain at most three points.
Corollary. If \(\Gamma\) is a finite subscheme of degree \(\geq r + 3\) in linearly general position in \(\mathbb{P}^ r\) and \(q\) is a general point, then \(\Gamma \cup \{q\}\) is in linearly general position. finite subscheme in linearly general position; Castelnuovo theory Eisenbud, David; Harris, Joe: Curves in projective space, Sémin. math. Supér. 85 (1982) Schemes and morphisms, Projective techniques in algebraic geometry, Cycles and subschemes Finite projective schemes in linearly general position | 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 describes explicitly the maximal torus and maximal unipotent subgroup of the Picard variety of a proper scheme over a perfect field. Picard scheme; torus; unipotent subgroup; semi-normalization; etale cohomology T. Geisser, ''The affine part of the Picard scheme,'' Compos. Math., vol. 145, iss. 2, pp. 415-422, 2009. Picard schemes, higher Jacobians The affine part of the Picard scheme | 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 find generating functions for the Poincaré polynomials of hyperquot schemes for all partial flag varieties. These generating functions give the Betti numbers of hyperquot schemes, and thus give dimension information for the cohomology ring of every hyperquot scheme. This can be viewed as a step towards understanding a presentation and the structure of the cohomology rings.
Let
\({\mathbf F}(n;{\mathbf s})\) denote the partial flag variety corresponding to flags of the form
\[
V_1\subset V_2\subset \dots\subset V_l\subset V=\mathbb{C}^n; \quad \dim V_i=s_i.
\]
It is classically known that its Poincaré polynomial \({\mathcal P}({\mathbf F}(n;{\mathbf s}))=\sum_Mb_{2M}({\mathbf F}(n;{\mathbf s}))z^M\) is equal to the following generating function for the Betti numbers of the partial flag variety:
\[
{\mathcal P}\bigl({\mathbf F}(n:{\mathbf s})\bigr)= \sum_Mb_{2M}({\mathbf F})z^M=\frac {\prod^n_{i=1}(1-z^i)}{\prod^{l+1}_{j=1}\prod^{s_j-s_{j-1}}_{i=1}(1-z^i)}
\]
with \(s_{l+1}:=n\) and \(s_0:=0\). Defining \(f_k^{i,j}:=1-t_i \dots,t_jz^k\), the main result is:
Theorem 1.
\[
\sum_{d_1,\dots,d_l}{\mathcal P}\biggl( {\mathcal H}{\mathcal Q}_{\mathbf d}\bigl({\mathbf F}(n;{\mathbf s})\bigr)\biggr) t_1^{d_1}\dots t_l^{d_l}=\;{\mathcal P} \bigl({\mathbf F}(n;{\mathbf s})\bigr)\cdot\prod_{1\leq i\leq j\leq l}\prod_{s_{i-1}<k\leq s_i} \left( \frac{1}{f^{i,j}_{s_j-k}}\right) \left(\frac{1}{f^{i,j}_{s_{j+1}-k+1}} \right).
\]
Chen, L, Poincaré polynomials of hyperquot schemes, Math Ann, 321, 235-251, (2001) Grassmannians, Schubert varieties, flag manifolds, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Parametrization (Chow and Hilbert schemes) Poincaré polynomials of hyperquot 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 The authors work on the problem of determining which zero dimensional schemes deform to a set of distinct points, i.e. are smoothable. This is a fundamental problem in the theory of Hilbert schemes of points -- and active and exciting area of research in algebraic geometry and commutative algebra.
The authors define a syzygetic invariant which gives insight to the problem mentioned above. Using their invariant the authors are able to deduce several interesting examples including: families which are not smoothable; Hilbert schemes of points which have components intersecting away from the smoothable component; and information about the Hilbert scheme of nine points in five dimensional affine space.
The paper is expertly written and contains necessary background information on Hilbert schemes, inverse systems, and regularity for homogeneous ideals. Several enlightening examples are given including computations of their introduced \(\kappa\)-vector. Theorems establishing necessary, as well as both necessary and sufficient conditions for certain classes of schemes of regularity two to be smoothable are given. Artinian algebras; smoothability; syzygies; deformation theory; punctual Hilbert schemes; Hilbert schemes of points Erman D., Velasco M.: syzygetic approach to the smoothability of 0-schemes of regularity two. Adv. Math. 224(3), 1143--1166 (2010) Parametrization (Chow and Hilbert schemes), Syzygies, resolutions, complexes and commutative rings, Deformations and infinitesimal methods in commutative ring theory, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Commutative Artinian rings and modules, finite-dimensional algebras A syzygetic approach to the smoothability of zero-dimensional 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 ``straight equisingular deformations'', a linear subfunctor of all equisingular deformations and describe their seminuniversal deformation by an ideal containing the fixed Tjurina ideal. Moreover, we show that the base space of the seminuniversal straight equisingular deformation appears as the fibre of a morphism from the \(\mu\)-constant stratum onto a punctual Hilbert scheme parametrizing certain zero-dimensional schemes concentrated in the singular point. Although equisingular deformations of plane curve singularities are very well understood, we believe that this aspect may give a new insight in their inner structure. Singularities in algebraic geometry, Deformations of singularities, Local complex singularities, Equisingularity (topological and analytic) Straight equisingular deformations and 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 The author considers a germ of a nodal curve \(X\), and studies the Hilbert scheme of length \(m\) subschemes of \(X\). The knowledge of this Hilbert scheme Hilb\(_m(X)\) is an important step for attaching some enumerative problems via the degeneration of curves to nodal curves.
The author proves that Hilb\(^0_m(X)\), the Hilbert scheme of length \(m\) subschemes of the node, consists in a union of \(m-1\) rational curves, meeting transversally. Then the author describes Hilb\(_m(X)\) as a formal scheme defined along Hilb\(^0_m\). Let \(\tilde X\to B\) denote the deformation of some smooth germ of curve to a nodal one (formally defined by \(xy-t\)). The author uses the previous result to show that, when the total space of the family is smooth, then the relative Hilbert scheme Hilb\(_m(\tilde X/B)\) is a smooth, \((m+1)\)-dimensional formal variety. Similarly, some relative flag Hilbert schemes for \(\tilde X/B\) are proven to be normal and complete intersection. nodal curves Ran, Z., A note on Hilbert schemes of nodal curves, J. Algebra, 292, 2, 429-446, (2005) Parametrization (Chow and Hilbert schemes), Families, moduli of curves (algebraic) A note on Hilbert schemes of nodal 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 construct a local model for Hilbert-Siegel moduli schemes with \(\Gamma_1(p)\)-level structures, when \(p\) is unramified in the totally real field. Our key tool is a variant of the ring-equivariant Lie complex defined by Illusie. variété de Shimura; modèle local; complexe cotangent Arithmetic aspects of modular and Shimura varieties, Modular and Shimura varieties Local model of Hilbert-Siegel moduli schemes with \(\Gamma_1(p)\)-level structures | 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 classical theory of adeles in algebraic number theory, which works also for curves in algebraic geometry, has been generalized over the past few years, first to smooth proper surfaces over a perfect field, and then to all noetherian schemes \(X\). A construction of adeles in the latter case was given by \textit{A. A. Bejlinson} [in Funct. Anal. Appl. 14, 34-35 (1980; translation from Funkts. Anal. Prilozh. 14, No. 1, 44-45 (1980; Zbl 0509.14018)]. This construction associates to a quasi-coherent sheaf \({\mathcal F}\) on \(X\) a complex (rather than a single ring as in the classical case) of adeles \(\mathbb{A}^*(X,{\mathcal F})\), functorially in \({\mathcal F}\), and it has several applications, principally because of the following theorem: The cohomology of the complex \(\mathbb{A}^*(X,{\mathcal F})\) is isomorphic to the cohomology of \({\mathcal F}\).
As Bejlinson's article is very short and concise, the present paper aims mainly at providing a more accessible exposition of this construction with proofs in the language of commutative algebra. The author also gives a new construction of rational (as opposed to Bejlinson's analytic) adeles and shows that the complex of rational adeles also computes the cohomology of \({\mathcal F}\). adele cohomology; Parshin-Beilinson adeles; complex of rational adeles H A.~Huber, \emph On the Parshin--Beilinson adeles for schemes, Abh. Math. Sem. Univ. Hamburg \textbf 61 (1991), 249--273. Étale and other Grothendieck topologies and (co)homologies, Adèle rings and groups On the Parshin-Beilinson adeles for 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 In this article the author generalizes the notions of smooth, unramified and étale morphisms to the category of fine logarithmic schemes and gives criteria which are analogs of criteria in the category of schemes.
First he recalls the definition of logarithmic schemes, which have been introduced by \textit{K. Kato}: a log structure \((M_X , \alpha)\) on a scheme \(X\) is a sheaf of monoids \(M_X\) on the étale site on \(X\) and a homomorphism \(\alpha : M_X \to {\mathcal O}_X\) such that \(\alpha ^{-1} ({\mathcal O} _X ^*) \buildrel {\scriptstyle \simeq} \over \longrightarrow {\mathcal O} _X ^*\); a chart \((P \to M_X)\) of \(M_X\) is a homomorphism \(P_X \to M_X\), where \(P_X\) is the constant sheaf of monoids of value \(P\), with \(P\) finitely generated and integral.
Then, as in the category of schemes, we can give the following definitions: A morphism \(f : X \to Y\) of fine log schemes is formally smooth (resp. formally unramified, resp. formally étale) if for any strict closed immersion of affine fine log schemes \(Z_0 \buildrel {\scriptstyle i} \over \hookrightarrow Z\) and any morphism \(Z \to Y\) the map \(\text{Hom}_Y (Z,X) \to \text{Hom}_Y (Z_0 ,X)\) induced by \(i\) is surjective (resp. injective, resp. bijective); and we get the same basic properties for these morphisms of fine log schemes when we consider composition, base change, etc... A morphism \(f : X \to Y\) of fine log schemes is called smooth (resp. unramified, resp. étale) if it is formally smooth (resp. formally unramified, resp. formally étale) and the underlying morphism of schemes is locally of finite presentation. We have again the same properties as in the category of schemes, and the author gives the following characterization:
Theorem: Let \(f : X \to Y\) be a morphism of fine log schemes such that the underlying morphism of schemes is of finite presentation; the following properties are equivalent:
(a) \(f\) is unramified.
(b) The diagonal \(\Delta : X \to X \times _Y X\) is étale.
(c) The differential module \(\Omega _{X/Y} ^1\) is zero.
(d) For any \(y \in Y\) the fibre \(X_y = f^{-1} (y)\) provided with the induced log structure is unramified over Spec\(k(y)\).
(e) Étale locally on \(X\) there exists a chart \( (Q \to M_X , P \to M_Y, P \to Q)\) extending a given chart on \(P \to M_Y\) such that: (i) \(P^{gp} \to Q^{gp}\) is injective with finite cokernel of order invertible on \(X\), (ii) the induced morphism of schemes \(X \to Y \times _{\text{Spec}(\mathbb Z [P ])} \text{Spec}(\mathbb Z [Q ])\) is unramified.
In the last part the author gives criteria for a morphism to be smooth, flat or étale. First he shows that the morphism \(f\) is smooth if and only if locally it can be factorized over an étale map into the standard log affine space over \(Y\). Then he recalls the following theorem of Kato:
Theorem: Let \(f : X \to Y\) be a morphism of fine log schemes; the following properties are equivalent:
(a) \(f\) is smooth (resp. étale).
(b) Étale locally on \(X\) there exists a chart \( (Q \to M_X , P \to M_Y, P \to Q)\) of \(f\) extending a given chart on \(P \to M_Y\) such that: (i) \(P^{gp} \to Q^{gp}\) is injective and the torsion part of its cokernel (resp. its cokernel) is finite of order invertible on \(X\), (ii) the induced morphism of schemes \(X \to Y \times _{\text{Spec}({\mathbb Z} [P ])} \text{Spec}(\mathbb Z [Q ])\) is smooth (resp. étale).
The author gives the following definition: A morphism of fine log schemes \(f : X \to Y\) is called ``flat'' if fppf locally on \(X\) and \(Y\), there exist a chart \( (Q \to M_X ,P \to M_Y, P \to Q) \) of \(f\) such that: (i) \(P^{gp} \to Q^{gp}\) is injective, (ii) the induced morphism of schemes \(X \to Y \times _{\text{Spec}(\mathbb Z [P ])} \text{Spec}(\mathbb Z [Q ])\) is flat.
With this definition he can generalize the usual fibre criterion for flatness: Let \(f : X \to Y\) be an \(S\)-morphism of fine log schemes, with \(X/S\) flat, then \(f\) is flat if and only if the induced maps on the fibres \(f_s : X_s \to Y_s\) are flat.
He can also generalize criteria for étale and smooth morphisms: \(f : X \to Y\) is étale if and only if \(f\) is flat and unramified; \(f : X \to Y\) is smooth if and only if \(f\) is flat and the induced maps on the fibres \(f_y : X_y \to y\) are smooth. category of fine logarithmic schemes; log structure; morphism of fine log schemes; fibre criterion for flatness Werner Bauer, On smooth, unramified étale and flat morphisms of fine logarithmic schemes, Math. Nachr. 176 (1995), 5 -- 16. Local structure of morphisms in algebraic geometry: étale, flat, etc., Schemes and morphisms, Étale and other Grothendieck topologies and (co)homologies On smooth, unramified, étale and flat morphisms of fine logarithmic 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 There is a natural class of topological rings which contains both the affinoid rings (from rigid analytic geometry) and the noetherian adic rings. Namely the class of topological rings which have an open adic subring with a finitely generated ideal of definition. We call such a ring \(f\)-adic. For every \(f\)-adic ring \(A\) the topological space \(\text{Cont }A\) of all continuous valuations of \(A\) is a spectral space. If \(A\) has an open noetherian adic subring or if for every \(n \in N\) the ring \(A\langle X_ 1,\dots, X_ n\rangle\) of restricted power series in \(n\) variables over \(A\) is noetherian then there is a natural sheaf \({\mathcal O}_ A\) of topological rings on \(\text{Cont }A\). All stalks of \({\mathcal O}_ A\) are local rings. We call a topologically and locally ringed space which is locally isomorphic to some \(\text{Spa }A := (\text{Cont }A,{\mathcal O}_ A)\) adic. There are natural fully faithful functors from the category of rigid analytic varieties and the category of locally noetherian formal schemes to the category of adic spaces. In the first case one assigns to an affinoid rigid analytic variety \(\text{Sp }A\) the adic space \(\text{Spa }A\) and in the latter case one assigns to an affine noetherian formal scheme \(\text{Spf }A\) the adic space \(\text{Spa }A\). affinoid rings; adic rings; \(f\)-adic ring; continuous valuations; formal schemes; rigid analytic variety R. Huber, A generalization of formal schemes and rigid analytic varieties, Math. Z. 217 (1994), no. 4, 513-551. Local ground fields in algebraic geometry, Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects), Formal groups, \(p\)-divisible groups, Global topological rings A generalization of formal schemes and rigid analytic varieties | 0 |
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