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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 A determination of the fixed components, base points and irregularity is made for arbitrary numerically effective divisors on any smooth projective rational surface having an effective anticanonical divisor. All of the results are proven over an algebraically closed field of arbitrary characteristic. Applications, to be treated in separate papers, include questions involving: points in good position, birational models of rational surfaces in projective space, and resolutions for 0-dimensional subschemes of \(\mathbb{P}^2\) defined by complete ideals. linear systems; fixed components; base points; irregularity; numerically effective divisors; rational surface; anticanonical divisor Brian Harbourne, ``Anticanonical rational surfaces'', Trans. Am. Math. Soc.349 (1997) no. 3, p. 191-1208 Divisors, linear systems, invertible sheaves, Rational and ruled surfaces, Rational and unirational varieties, Projective techniques in algebraic geometry Anticanonical rational surfaces | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In dieser kurzen Note wird mitgeteilt, daß der Autor die algebraische Klassifikation der Differentialkörpererweiterungen L/K vom Transzendenzgrad 1 ohne bewegliche Singularitäten [siehe \textit{M. Matsuda}, First order algebraic differential equations (Lect. Notes Math. 804) (1980; Zbl 0447.12014)] auf Transzendenzgrad 2 und 3 ausgedehnt hat, wobei naturgemäß neben den elliptischen Funktionen in Matsudas Klassifikation abelsche Funktionen auftreten.
Ein kurzer Hinweis auf die Methode, die im Gegensatz zu Matsudas (reine Differential-Algebra), substantiell Gebrauch von der Geometrie (Deformationstheorie) macht, erklärt, weshalb der Autor neben der Beschränkung auf Körper meromorpher Funktionen für Dimension 3 noch eine Hypothese über das Verschwinden der Kodaira-Dimension verwendet (mit der Vermutung, daß sie bei den betrachteten Körpern stets erfüllt ist). differential function fields; differential field; movable singularity; abelian function; differential Galois extension; ADE; algebraic differential equations Differential algebra, Formal methods and deformations in algebraic geometry Fields of meromorphic functions | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In this paper we study special Cremona transformation, i.e. those the base scheme \(Y\) of which is smooth and connected. Our main results classify them when they are quadro-quadric and when \(\operatorname{codim}(Y)=2\). In the first case, they are just the maps given by systems of quadrics through Severi varieties (by a result of Zak there are just four of these). In the second case, either \(Y\) is a quintic elliptic scroll in \(P^4\) or \(Y\) or is defined in \(P^n\) by the vanishing of the \(n\times n\) minors of a \(n\times (n+1)\) matrix with linear entries, and \(n\le 5\). determinantal variety; classification of Cremona transformations; systems of quadrics through Severi varieties; quintic elliptic scroll Ein L. and Shepherd-Barron N., Some special Cremona transformations, Amer. J. Math. 111 (1989), 783-800. Birational automorphisms, Cremona group and generalizations, Rational and birational maps Some special Cremona transformations | 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 some examples of QRT families of biquadratic curves which are of degree 4 (with high degree term \(x^2y^2\)), but such that every curve of such a family is singular, with genus 0, or is reducible (for some finite set of values of the parameter). This contrasts with classical examples of QRT families studied by many authors. We give also examples of QRT families of degree 4 whose every curve is reducible. Then we sketch out to study the dynamical systems associated to such QRT families of genus zero. QRT maps; genus of curves; dynamical systems Completely integrable discrete dynamical systems, Relations of finite-dimensional Hamiltonian and Lagrangian systems with algebraic geometry, complex analysis, special functions, Rational and birational maps, Relationships between algebraic curves and integrable systems Dynamical systems associated with QRT families of degree four biquadratic curves each of them with genus zero | 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 affine group schemes of automorphisms of the multilinear \(r\)-fold cross products on finite-dimensional vectors spaces over fields of characteristic not two are determined. Gradings by abelian groups on these structures, that correspond to morphisms from diagonalizable group schemes into these group schemes of automorphisms, are completely classified, up to isomorphism. cross product; automorphism group scheme; grading Other \(n\)-ary compositions \((n \ge 3)\), Nonassociative algebras satisfying other identities, Automorphisms, derivations, other operators (nonassociative rings and algebras), Infinite automorphism groups, Linear algebraic groups over arbitrary fields, Affine algebraic groups, hyperalgebra constructions, Multilinear algebra, tensor calculus Cross products, automorphisms, and gradings | 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 Gorenstein trigonal curve with \(g:= p_a(Y)\geq 0\). Here we study the theory of special linear systems on \(Y\), extending the classical case of a smooth \(Y\) given by \textit{A. Maroni} [Ann. Mat. Pura Appl., IV. Ser. 25, 343-354 (1946; Zbl 0061.35407)]. As in the classical case, to study it we use the minimal degree surface scroll containing the canonical model of \(Y\). The answer is different if the degree 3 pencil on \(Y\) is associated to a line bundle or not. We also give the easier case of special linear series on hyperelliptic curves. The unique hyperelliptic curve of genus \(g\) which is not Gorenstein has no special spanned line bundle. Gorenstein trigonal curve; linear systems; scroll; canonical model Ballico E.,Trigonal Gorenstein curves and special linear systems, Israel J. Math.,119 (2000), 143--155. Special divisors on curves (gonality, Brill-Noether theory), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Divisors, linear systems, invertible sheaves Trigonal Gorenstein curves and special linear 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 Here the author uses linkage techniques to obtain many curves \(C\subset {\mathbb{P}}^ 3\) with \(H^ 1(N_ C(-2))=0\) (hence with normal bundle \(N_ C\) semistable). The results seem very useful. space curve; Hilbert scheme; deformation theory; liaison; semistable normal bundle; linkage D. PERRIN . - C. R. Acad. Sci. Paris, 300, Série I, N^\circ 2, 1985 , p. 39-42. MR 86h:14026b | Zbl 0586.14025 Special algebraic curves and curves of low genus, Projective techniques in algebraic geometry, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Parametrization (Chow and Hilbert schemes), Formal methods and deformations in algebraic geometry Courbes gauches, fibré normal et liaison. (Space curves, normal bundle and linkage) | 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 survey starts with a leisurely exposition of the notion of integration with respect to the Euler characteristic; with the use of the right cohomology theory the Euler characteristic becomes an additive (bot not non-negative) function on the algebra of constructible sets, and can be used like a measure to integrate constructible functions. The integral is in fact a finite sum, but is has a Fubini theorem. Some examples of this point of view are presented, like the Riemann-Hurwitz formula and A'Campo's formula for the monodromy zeta-function.
Then generalisations are discussed: integration over the infinite-dimensional spaces of arcs and functions, with applications such as the definition and computation of Poincaré series of multi-index filtrations, and motivic integration. Euler charactersitic; zeta-function; Poincaré series; motivic integration Gusein-Zade, S. M., Integration with respect to the Euler characteristic and its applications, Uspekhi Matem. Nauk, 65, 5, (2010) Singularities in algebraic geometry, Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants, Milnor fibration; relations with knot theory, Zeta functions and \(L\)-functions Integration with respect to the Euler characteristic and its 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 Using arithmetic intersection theory, \textit{G. Faltings} [Ann. Math., II. Ser. 133, No. 3, 549-576 (1991; Zbl 0734.14007)] defined a height of cycles in multiprojective space over a number field \(K\). Let \(X\) be a complete scheme over \(K\) and \({\mathcal M}_ 0,\dots, {\mathcal M}_ t\) isomorphism classes of basepoint-free line bundles on \(X\). There is a multiprojective realization of \({\mathcal M}_ 0,\dots, {\mathcal M}_ t\). Using pushforward of cycles, one gets a height of any \(t\)-dimensional effective cycle \(Z\) of \(X\). The main theorem of the paper states that the height does not depend on the choice of the realization up to \(O(\sum d_ i)\), where \(i\) ranges over \(0,\dots,t\) and \(d_ i(Z)\) is the degree of \(Z\) relative to \({\mathcal M}_ 0,\dots, {\mathcal M}_{i-1}\), \({\mathcal M}_{i+1},\dots, {\mathcal M}_ t\). The proof is based on a formula which describes the difference of the heights corresponding to different realizations defined over the ring of integers. As in the classical case of points, one gets the height machine and Néron-Tate heights. For even \({\mathcal M}_ 0= \cdots= {\mathcal M}_ t\), the latter are the same as the canonical heights of \textit{P. Philippon} [Math. Ann. 289, No. 2, 255-283 (1991; Zbl 0704.14017)].
In an appendix a different approach to the above mentioned Néron-Tate height for cycles on abelian varieties (over \(\mathbb{Q})\) is given using arithmetical compactifications. height of cycles; complete scheme; line bundles; height machine; Néron- Tate heights Gubler, Walter, Höhentheorie, Math. Ann., 298, 3, 427-455, (1994) Arithmetic varieties and schemes; Arakelov theory; heights, Arithmetic ground fields for abelian varieties, Algebraic cycles, Arithmetic algebraic geometry (Diophantine geometry) Theory of heights. With an appendix by Jürg Kramer: An alternative foundation of the Néron-Tate height | 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 algebraic set in \(\mathbb{R}^n\). Real-valued functions, defined on subsets of \(X\), that are continuous and admit a rational representation have some remarkable properties and applications. We discuss recently obtained results on such functions, against the backdrop of previously developed theories of arc-symmetric sets, arc-analytic functions, approximation by regular maps, and algebraic vector bundles. real algebraic set; semialgebraic set; regular function; rational function; regulous function; arc-symmetric set; arc-analytic function; approximation; vector bundle Real algebraic sets, Semialgebraic sets and related spaces, Arcs and motivic integration, Real rational functions, Topology of vector bundles and fiber bundles, Real-analytic and Nash manifolds From continuous rational to regulous 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 [For the entire collection see Zbl 0511.00003.]
Let F be a field, and let a finite group G act faithfully by permutations on a finite set of indeterminates \(x_ 1,...,x_ n\). Is the field of invariants \(F(x_ 1,...,x_ n)^ G\) purely transcendental \(over\quad F?\) This question was raised by Emmy Noether, hence is tranditionally termed Noether's problem. The paper under review is a very pleasant survey on the work related to this problem, mainly when G is Abelian. Here is a brief list of the contents: Noether's idea of parametrizing all Galois extensions with a given group and Saltman's approach in terms of generic Galois extensions, Saltman's sufficient condition for the existence of such an extension when G is Abelian, a detailed account of Saltman's nice observation that Wang's counterexample to Grunwald's theorem provides a negative answer to Noether's problem with \(F={\mathbb{Q}}\quad and\quad G={\mathbb{Z}}/8;\) Fischer's classical positive answer when G is Abelian and F contains enough roots of unity, and Masuda's idea of applying Galois descent, which as observed by Voskresenskij realizes the field of invariants as the function field of a torus; a detailed account of the algebraic invariants used to control tori up to stable birational equivalence, as developed by Swan, Voskresenkij, Endo and Miyata, Lenstra jun., Sansuc and the reviewer; a very clear presentation of the author's original negative answer to Noether's problem \((F={\mathbb{Q}},G={\mathbb{Z}}/47);\) a statement of the general answer to the problem for G Abelian (work of Endo and Miyata, Voskresenkij, Lenstra jun. - for the intricate story of whose is what, see \textit{V. E. Voskresenskij}'s book [Algebraic Tori (1977; Zbl 0499.14013)]). The paper closes with a compact description of the main results of \textit{J.-J. Sansuc} and the reviewer on R-equivalence upon the rational points of tori [Ann. Sci. Ec. Norm. Supér., IV. Sér. 10, 175-229 (1977; Zbl 0356.14007)].
Since the writing of the paper, there has been more work on Noether's problem, mostly by \textit{D. J. Saltman} [Isr. J. Math. 47, 165-215 (1984); Invent. Math. 77, 71-84 (1984); Multiplicative field invariants (preprint)]. The last two papers give a negative answer to Noether's problem when F is algebraically closed and G is a suitable non-Abelian group (also a long-standing question). A torus-theoretic approach to Saltman's analysis of the Noether problem in the Abelian case is included in the preprint ''Principal homogeneous spaces under flasque tori; applications'' by Sansuc and the reviewer. invariants of finite group; inverse problem of Galois theory; Noether's problem; function field of a torus; Algebraic Tori; rational points of tori Swan, R. G.: Noether's problems in Galois theory. Symposium ''emmy Noether in bryn mawr'' (1983) Transcendental field extensions, Separable extensions, Galois theory, Rational and unirational varieties, Finite automorphism groups of algebraic, geometric, or combinatorial structures, Linear algebraic groups over arbitrary fields, Integral representations of finite groups Noether's problem in Galois theory | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(G\) be a \(1\)-truncated Barsotti-Tate group scheme (= a \(BT_1\), group scheme) i.e., a finite, locally free group scheme \(G \to S\) for which \(\ker F_G = \operatorname {Im} V_G\) and \(G[V]=F[G]\). \(G\) is indecomposible if it is not isomorphic to a product \(G_1 \times G_2\) of \(BT_1\), group schemes with \(G_1, G_2\) nonzero and is simple if the only \(BT_1\), subgroup schemes it contains are \(O\) and \(G\). It is minimal if it is isomorphic to \(H(\beta) [p]_k\) for some Newton polygon \(\beta\), where \(H(\beta) = \times_i H_{m_i, n_i}\) and the groups \(H_{m,n}\) are as defined by the author and \textit{A. de Jong} in Section 5 of [J. Am. Math. Soc. 13, No. 1, 209--241 (2000; Zbl 0954.14007)]. The paper under review proves two theorems. The first is that for \(BT_1\) group schemes over an algebraically closed field indecomposable plus minimal is equivalent to simple. The second is that if \(G\) is a \(BT_1\) group scheme over an algebraically closed field which is not minimal then there are infinitely many isomorphism classes of groups \(X\) for which \(X[p] \simeq G\). The second result was announced in an earlier paper of the author [Ann. Math. (2) 161, No. 2, 1021--1036 (2005; Zbl 1081.14065)] where it was shown that if \(G\) is minimal then all \(X\) for which \(X[p]\simeq G\) are isomorphic. Barsotti-Tate group; finite group scheme; Dieudonné module. F. Oort, ''Simple \(p\)-kernels of \(p\)-divisible groups,'' Adv. Math., vol. 198, iss. 1, pp. 275-310, 2005. Formal groups, \(p\)-divisible groups Simple \(p\)-kernels of \(p\)-divisible groups | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Wie verändert sich das Geschlecht eines Funktionenkörpers einer Variablen bei Reduktion nach einer nichtarchimedischen Bewertung des Konstantenkörpers ?
Im Falle von diskreten Bewertungen liefern eine Reihe von Arbeiten Antworten auf diese Frage [siehe z.B. \textit{H. Mathieu}, Arch. Math. 20, 597-611 (1969; Zbl 0235.14010)]. Der Autor kündigt hier Erweiterungen dieser Ergebnisse für nichtdiskrete Bewertungen an, die er mit Methoden der rigiden analytischen Geometrie erzielt hat. Beweise dazu erscheinen an anderer Stelle. valued function fields; genus change; algebraic function field; reduction of constants; rigid analytic geometry; non-discrete valuation Arithmetic theory of algebraic function fields, Algebraic functions and function fields in algebraic geometry, Non-Archimedean valued fields, Arithmetic ground fields for surfaces or higher-dimensional varieties Genre des corps de fonctions values après Deuring, Lamprecht et Mathieu. (Genus of valued function fields after Deuring, Lamprecht and Mathieu) | 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 survey the author's result, which describes special values of zeta functions of (singular) varieties over finite fields by using their higher Chow groups. To be precise, we show that the special value at \(s=0\) is equal to the alternating product of the ratio of the cardinalities of kernels and cokernels of maps from higher Chow group to another invariant which is called weight homology. The principal idea of proof is to reduce the problem to the case that the variety is smooth by using weight spectral sequences. special value; zeta function; higher Chow group; weight homology; weight complex Zeta and \(L\)-functions in characteristic \(p\), Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), (Equivariant) Chow groups and rings; motives, Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects) Special values of zeta functions of varieties over finite fields via higher Chow groups: a survey | 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 notion, due to Nakumura, of \(G\)-Hilbert scheme \(\mathrm{Hilb}^G{\mathbb C}^n\) for any finite, abelian subgroup \(G\) of \(\mathrm{GL}(n,{\mathbb C})\). The \(G\)-Hilbert scheme can be described in terms of \(G\)-sets. In the article under review the author describes the \(G\)-Hilbert scheme when \(G\) is a finite cyclic group generated by a \((3\times 3)\) matrix of a particular form. After giving an classification of all possible \(G\)-sets, the author also obtains a formula for the number of different \(G\)-sets that appear for these groups. \(G\)-sets; \(G\)-Hilbert scheme O. Kȩdzierski, The G-Hilbert scheme for \(\frac{1}{r}\)(1,a,r-a), Glasg. Math. J. 53 (2010), 115 -129. McKay correspondence, Parametrization (Chow and Hilbert schemes), Singularities in algebraic geometry The \(G\)-Hilbert scheme for \(\frac 1{r} (1,a, r-a)\) | 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 Generalized algebraic geometry (GAG) codes were proposed by Xing, Neiderreiter and Lam [\textit{C. Xing} et al., IEEE Trans. Inf. Theory 45, No. 7, 2498--2501 (1999; Zbl 0956.94023)] with the purpose of improving the performances of the algebraic geometry (Goppa) codes [\textit{V. D. Goppa}, Sov. Math., Dokl. 24, 170--172 (1981; Zbl 0489.94014); translation from Dokl. Akad. Nauk SSSR 259, 1289--1290 (1981)]. In a previous paper the author of the present work and \textit{A. G. Spera} [Discrete Math. 309, No. 2, 328--340 (2009; Zbl 1166.94013)] studied the automorphisms of certain GAG codes (associated with rational, elliptic and hyperelliptic function fields). Now the author studies automorphisms for other GAG codes.
Section 2 recalls the concept and parameters of GAG codes (while Goppa codes use rational places of a function field, GAG codes make use of places of a (fixed) degree \(n>1\)) and Section 3 gives the definition of \(n\)-automorphism of a GAG code and recalls some necessary facts proved in [Zbl 1166.94013].
Section 4 shows results for the group of \(n\)-automorphisms of GAG codes associated with the so-called \textit{admissible} function fields, whose better well-known example is the Hermitian function field. The main result (Theorem 4.5) proves that, under suitable conditions, that group of \(n\)-automorphisms is isomorphic to a subgroup of the automorphisms of the corresponding function field. generalized algebraic geometry codes; n-automorphisms; admissible function fields; Hermitian function fields A. Picone, Automorphisms of generalized algebraic geometry codes, Ph.D. Thesis, Università degli Studi di Palermo, 2007 Algebraic functions and function fields in algebraic geometry, Applications to coding theory and cryptography of arithmetic geometry, Geometric methods (including applications of algebraic geometry) applied to coding theory Automorphisms of Hermitian generalized algebraic geometry codes | 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 an Archimedean real closed field and \(K\) be the function field of an integral, projective, smooth \(R\)-curve. We show, by generalizing some demonstrations of an earlier paper [\textit{A. Ducros}, J. Reine Angew. Math. 504, 73-114 (1998; Zbl 0934.14012)] which traited the case of fibrations in conics, that the obstruction to the reciprocity of the Hasse principle is the only one for \(K\)-varieties fibred in Severi-Brauer varieties over \(\mathbb{P}_K^1\). Archimedean real closed field; function field; fibration; Hasse principle; Severi-Brauer varieties Fibrations, degenerations in algebraic geometry, Algebraic functions and function fields in algebraic geometry, Semialgebraic sets and related spaces, Other nonalgebraically closed ground fields in algebraic geometry Fibration in Severi-Brauer varieties above the projective line over the function field of a real curve | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In recent years, multiview varieties have been extensively studied from the viewpoint of algebraic geometry. A multiview variety \(X\) is the Zariski closure of all images from a sequence of \(n\) ``cameras'' which are typically projective maps onto low dimensional image spaces (projective planes in the classical scenary of photogrammetry or computer vision). The aim is to use knowledge about \(X\) for the reconstruction of the cameras, up to projective transformation, from measured data on~\(X\).
In this paper, the authors prove that the reconstruction is unique unless one uses \(n+1\) cameras onto images spaces of dimension one in which case two solutions exist. This is a known result by \textit{R. I. Hartley} and \textit{F. Schaffalitzky} [Lect. Notes Comput. Sci. 3021, 363--375 (2004; Zbl 1098.68775)] but with a new algebro-geometric proof. The authors also prove that the map that sends a camera configuration to the multiview variety is dominant onto an irreducible component of the Hilbert schemes of projective subschemes. computer vision; algebraic vision; multiview geometry; projective reconstruction; Hilbert scheme Geometric aspects of numerical algebraic geometry, Rational and birational maps, Machine vision and scene understanding, Numerical aspects of computer graphics, image analysis, and computational geometry, Parametrization (Chow and Hilbert schemes) Projective reconstruction in algebraic 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 Riemann-Roch theorem; algebraic function fields Algebraic functions and function fields in algebraic geometry Eine Vorbereitung auf den Riemann-Rochschen Satz für algebraische Funktionenkörper | 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 Very special finite subsets of a projective plane over a field \(K\) are considered. These subsets, named \(K\)-configurations, have \(d_ 1>0\) of points on a line, then \(d_ 2>d_ 1\) points on a line missing the former ones, and so on. The Hilbert function of such a subset is easily computed so that it determines and is in turn determined by the integers \(d_ i\). A even more special type of finite sets of points, the \(n\)- regular \(K\)-configurations, are also considered: the Hilbert functions of some of them are shown to be those of the complete intersections of plane curves. \(K\)-configurations; Hilbert function Projective techniques in algebraic geometry, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Configuration theorems in linear incidence geometry About \(K\)-configurations in \(\mathbb{P}^ 2\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Given a Riemann surface \(X\) of genus \(g>0\), the author defines a complex-valued function \(f: J(X)\to \mathbb{C}\) on the Jacobian of \(X\), in terms of the \(\theta\)-function, and proceeds to establish a new relation, involving \(f\), between the Green function and the Faltings \(\delta\)-invariant. If \(g=1\), the function \(f\) is a constant; if \(g=2\), it coincides with the function introduced by \textit{J.-B. Bost} [cf. C. R. Acad. Sci., Paris, Sér. I 305, 643-646 (1987; Zbl 0638.14016)]. Riemann surface; Jacobian; Green function; Faltings \(\delta\)-invariant Guàrdia, J.: Analytic invariants in Arakelov theory for curves. C. R. Acad. sci. Paris ser. I 329, 41-46 (1999) Arithmetic varieties and schemes; Arakelov theory; heights, Jacobians, Prym varieties, Riemann surfaces; Weierstrass points; gap sequences, Theta functions and curves; Schottky problem, Theta functions and abelian varieties Analytic invariants in Arakelov theory for curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems \textit{S. Lang} [Bull. Am. Math. Soc., New Ser. 14, 159--205 (1986; Zbl 0602.14019)] formulated the following function-field version of the Mordell conjecture: if \(\pi: \mathcal X \to Y\) is a projective surjective morphism of complex algebraic varieties whose generic fiber is of general type, then if \(\pi\) is not birationally isotrivial, there is a proper subscheme of \(\mathcal X\) that contains the images of all sections of \(\pi\). \textit{Yu. I. Manin} [Izv. Akad. Nauk SSSR, Ser. Mat. 27, 1395--1440 (1963; Zbl 0166.16901)] and \textit{H. Grauert} [Publ. Math., Inst. Hautes Étud. Sci. 25, 363--381 (1965; Zbl 0137.40503)] solved this for families of curves, and this paper deals with a certain higher-dimensional case:
Main Theorem. Let \(C\) be a curve of genus at least \(2\) and let \(\lambda\) be a divisor with a large degree. For \(n\geq 3\) and \(d\) sufficiently large, let \(\mathcal X\) be the family of hypersurfaces of \(\mathbb P^{n}\) given by a section of an ample line bundle \(L = \lambda \boxtimes \mathcal O_{\mathbb P^{n}}(d)\) on \(C\times \mathbb P^{n}\). Then for a general such \(\mathcal X/C\), Lang's conjecture holds.
Here, `general' means that the family is chosen outside a proper algebraic subset of the parameter space, thus automatically removing the birationally-isotrivial case. Grauert's proof and the proof for the family whose smooth members are hyperbolic [\textit{J. Noguchi}, Math. Ann. 258, 207--212 (1981; Zbl 0459.14002)] both use first-order differential equations, but this paper follows [\textit{J.-P. Demailly}, Proc. Symp. Pure Math. 62(pt.2), 285--360 (1997; Zbl 0919.32014)] and shows that there is a nontrivial differential equation of order \(n\) that is satisfied by the images of sections of such a family. Then adapting techniques similar to \textit{Y.-T. Siu} [in: The legacy of Niels Henrik Abel. Berlin: Springer. 543--566 (2004; Zbl 1076.32011)] and [\textit{S. Divero}, \textit{J. Merker} and \textit{E. Rousseau}, Invent. Math. 180, No. 1, 161--223 (2010; Zbl 1192.32014)], the Zariski-non-density of images of sections follows by constructing vector fields which isolate the coefficients of the differential equations.
To obtain the desired differential equation, the author constructs a line bundle \(L\) on the jet space \(\mathcal X_{n}\) for the sections so that {\parindent=6mm \begin{itemize}\item[(1)] \(L = A - B\), where \(A\) and \(B\) are nef such that \(A^D > D A^{D-1}\cdot B\) (here \(D = \dim \mathcal X_{n}\)). \item[(2)] for any section \(s\) of \(\mathcal X\to C\), its lift to \(n\)-jet \(s_{n}: C\to \mathcal X_{n}\) pulls \(L\) back to a negative-degree divisor.
\end{itemize}} Then it follows from algebraic Morse inequalities that \(L\) is big, and for any global section \(\sigma\) of a power of \(L\), \(s_{n}^*\sigma = 0\). Thus, the image of \(s_{n}\) for any section of \(\mathcal X\to C\) lies inside the zero of \(\sigma\), giving us the differential equation. The author constructs very specific \(A\) and \(B\), with \(B\) containing the pullback of \(\lambda\) as a subdivisor. To check nefness, he uses C. Voisin's idea that the nef cone and the pseudo-effective cone agree for a very general family. With this setup, (2) holds for \(\deg \lambda \gg 0\). The intersection numbers are estimated explicitly using Segre classes, and (1) is verified for \(d\gg 0\) and \(\deg(\mathcal X\to \mathbb P^n) \gg 0\).
As a corollary of this method (it works for a finite cover of \(C\)), the author obtains a height inequality for sections whose \(n\)-jets do not lie inside a proper subvariety. This is a higher-dimensional analog of Proposition 3.1 in [\textit{P. Vojta}, Compos. Math. 78, No. 1, 29--36 (1991; Zbl 0731.14015)]. Moreover, in the appendix, the author shows by explicit calculations that one cannot find a differential equation of order \(\leq 2\) satisfied by the images of sections of such a family of surfaces (\(n=3\)). This demonstrates the optimality of order-\(n\), at least via this holomorphic Morse inequality method. Lang's conjecture; function fields; hypersurfaces; algebraic Morse inequalities; jet spaces Christophe Mourougane, ''Families of hypersurfaces of large degree'', J. Eur. Math. Soc. (JEMS)14 (2012) no. 3, p. 911-936 | Rational points, Hypersurfaces and algebraic geometry, Heights, Families, moduli, classification: algebraic theory Families of hypersurfaces of large degree | 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 classify symplectic leaves of the regular part of the projectivization of the space of meromorphic endomorphisms of a stable vector bundle on an elliptic curve, using the study of shifted Poisson structures on the moduli of complexes from their previous work [Adv. Math. 338, 991--1037 (2018; Zbl 1400.53070)]. This Poisson ind-scheme is closely related to the ind Poisson-Lie group associated to Belavin's elliptic $r$-matrix, studied by some authors. Their result leads to a classification of symplectic leaves on the regular part of meromorphic matrix algebras over an elliptic curve, which can be viewed as the Lie algebra of the above-mentioned ind Poisson-Lie group. The authors also describe the decomposition of the product of leaves under the multiplication morphism and show the invariance of Poisson structures under autoequivalences of the derived category of coherent sheaves on an elliptic curve. This paper is organized as follows. Section 1 is an introduction to the subject. Section 2 concerns a brief review of the construction of the derived moduli stack of bounded complexes of vector bundles and its shifted Poisson structure in [loc. cit.]. In Section 3 the authors prove that the coarse moduli spaces of the derived symplectic leaves are smooth schemes and show that the classical shadow of the $0$-shifted symplectic structure descends to a symplectic structure on the coarse moduli spaces and the classification results are proved. In Section 4, the authors classify all symplectic leaves for rank $2$ case without assuming regularity and give some examples of leaves. In Section 5, they prove invariance of the Poisson structure under auto equivalences. moduli of vector bundles; Poisson manifolds; Poisson groupoids and algebroids; integrable systems; algebraic curves Algebraic moduli problems, moduli of vector bundles, Poisson manifolds; Poisson groupoids and algebroids, Applications of Lie algebras and superalgebras to integrable systems, Relationships between algebraic curves and integrable systems Shifted Poisson geometry and meromorphic matrix algebras over an elliptic curve | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems We give an overview of known and new techniques on how one can obtain explicit equations for candidates of good towers of function fields. The techniques are founded in modular theory (both the classical modular theory and the Drinfeld modular theory).
In the classical modular setup, optimal towers can be obtained, while in the Drinfeld modular setup, good towers over any nonprime field may be found. We illustrate the theory with several examples, thus explaining some known towers as well as giving new examples of good explicitly defined towers of function fields. towers of function fields; Drinfeld modules; curves with many points Algebraic functions and function fields in algebraic geometry, Families, moduli of curves (algebraic), Arithmetic theory of algebraic function fields, Computational aspects of algebraic curves Good towers of function fields | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For the entire collection see Zbl 0587.00027.]
It is well-known that in general there exist no global continuous canonical forms for most moduli problems of interest in linear control systems. The author suggests looking for piecewise continuous forms of various kinds. He remarks that for the class of minimal systems under the canonical action of the general linear group, no such forms are known. Then he proceeds to establish the existence of the desired forms, using facts from real algebraic geometry. moduli spaces of linear control systems; piecewise continuous forms Canonical structure, Minimal systems representations, Linear systems in control theory, Families, moduli, classification: algebraic theory, Controllability, Algebraic methods Minimal cellular parametrizations and moduli for linear dynamical 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 The author proves the following extension to families of open curves of Manin's finiteness theorem for rational points of algebraic curves over function fields [\textit{Yu. I. Manin}, Transl., Ser. 2, Am. Math. Soc. 50, 189-234 (1966); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 27, 1395-1440 (1963; Zbl 0166.169)], well-known as the function-field analog of the Mordell conjecture:
Theorem 1. Let \(f: X\to S\) be a family of curves of general type, and let \(\bar f: \bar X\to \bar S\) be its compactification. If the set \({\mathcal H}(S,f)\) of holomorphic sections of f is infinite, then there is a (shared) bimeromorphic trivialization of the families f and \(\bar f,\) mapping all but perhaps a finite number of sections in \({\mathcal H}(S,f)\) to constant sections and the horizontal curve \(A_ h(\bar f)\) to the union of a finite set of constant sections.
Here S denotes a smooth complex curve, Zariski-open in its smooth compactification \(\bar S,\) and \(A_ h(\bar f)\) is the complement of X in the (reduced irreducible normal complex compact) surface \(\bar X.\) Especially, the cases of families of punctured elliptic curves present a function field analog of the Siegel-Mahler theorem for elliptic curves over number fields [see \textit{S. Lang}, Publ. Math., Inst. Hautes Étud. Sci. 6, 27-43 (1960; Zbl 0112.134)]. For the proof the author needs the techniques of resolution of singularities, semistable reduction and the meromorphic domination of family pairs \((f,\bar f)\) by pairs \((g,\bar g)\) with the property of ``relative hyperbolic embeddedness''. The latter is due to the author in his previous paper [``Hyperbolicity criteria and families of curves'', Teor. Funkts., Funkts. Anal. Prilozh. 52, 40-54 (1989)]. An important role plays the existence of a complex structure (Douady) on the space of sections. Then compactness of the space of sections on the open part (proposition 2.1) and local finiteness (lemma 4.4) are verified and used to find a trivialization of the given family of curves. In the last section 5 there is given an alternative proof by reduction to Manin's theorem following an idea of Parshin. families of open curves; function-field analog of the Mordell conjecture Arithmetic ground fields for curves, Families, moduli of curves (analytic), Rational points, Arithmetic varieties and schemes; Arakelov theory; heights A function-field analog of the Mordell conjecture: A non-compact version | 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 \(X\to B\) be an elliptic surface and \({\mathcal M} (a,b)\) the moduli space of torsion-free sheaves on \(X\) which are stable of relative degree zero with respect to a polarization of type \(aH+b\mu\), \(H\) being the section and \(\mu\) the elliptic fibre \((b\gg 0)\). We characterize the open subscheme of \({\mathcal M}(a,b)\) which is isomorphic via the relative Fourier-Mukai transform, with the relative compactified Simpson-Jacobian of the family of those curves \(D\hookrightarrow X\) which are flat over \(B\). This generalizes and completes earlier constructions due to \textit{R. Friedman}, \textit{J. Morgan} and \textit{E. Witten} [J. Algebr. Geom. 8, 279--401 (1999; Zbl 0937.14004)]. We also study the relative moduli scheme of torsion-free and semistable sheaves of rank \(n\) and degree zero on the fibres. The relative Fourier-Mukai transform induces an isomorphism between this relative moduli space and the relative \(n\)th symmetric product of the fibration. These results are relevant in the study of the conjectural duality between \(F\)-theory and the heterotic string. spectral covers; compactified Jacobian; elliptic surface; relative Fourier-Mukai transform Hernández~Ruipérez, D.; Muñoz~Porras, J.M., Stable sheaves on elliptic fibrations, J. geom. phys., 43, 163-183, (2002) Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Algebraic moduli problems, moduli of vector bundles, Elliptic surfaces, elliptic or Calabi-Yau fibrations Stable sheaves on elliptic fibrations | 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 an epoch-making paper from 1994, \textit{A. Beilinson} and \textit{P. Deligne} gave an interpretation of polylogarithmic functions in terms of variations of mixed Hodge structures, which led them to a motivic proof of the famous Zagier conjecture based upon Deligne's theory of mixed motives [in: Motives. Proceedings of the summer research conference on motives, Proc. Symp. Pure Math. 55, 97--121 (1994; Zbl 0799.19004)].
The aim of the present extensive and profound paper is to generalize various results of Beilinson and Deligne to the author's theory of iterated integrals within the framework of cosimplicial objects, mixed motives, and monodromy representations of fundamental groups. The motivic character of iterated integrals, which appear as horizontal sections of fiber bundles equipped with a Gauss-Manin connection and satisfy certain basic functional equations, has been pointed out before by the author [in: Algebraic \(K\)-theory and algebraic topology, NATO ASI Ser., Ser. C, Math. Phys. Sci. 407, 287--327 (1993; Zbl 0916.14006)].
These facts suggest that the Zagier conjecture generalizes to iterated integrals on algebraic varieties, and that there should be a motivic proof analoguous to the original one by Beilinson and Deligne. In this regard, the present paper is thought as being the first part of a huge and general program to work out these ideas in full detail, with further generalizations to follow. As the author points out, the present paper is a revised version of his foregoing preprint [Mixed Hodge structures and iterated integrals, Prépublication de l'Université de Nice-Sophia Antipolis, 1997]. In general, he has kept the exposition nearly self-contained and pleasantly detailed, with as few quotations as possible, full proofs, significant examples, and a list of related problems that are still open.
Finally, as the author also points out in the introduction to this paper, his generalized approach follows the original one by Beilinson and Deligne rather closely, mutatis mutandis, which certainly makes the underlying higher principles of the entire theory more apparent. mixed Hodge structures; polylogarithmic functions; iterated integrals; monodromy representations; Zagier conjecture; cosimplicial objects; mixed motives Wojtkowiak, Z.: Mixed Hodge structures and iterated integrals I. Motives, polylogarithms and Hodge theory (I), 121-208 (2002) Variation of Hodge structures (algebro-geometric aspects), Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects), Polylogarithms and relations with \(K\)-theory, Other functions defined by series and integrals, Transcendental methods, Hodge theory (algebro-geometric aspects), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Transcendental methods of algebraic geometry (complex-analytic aspects) Mixed Hodge structures and iterated integrals. 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 Wegen der nicht völligen Dualität scheint es dem Verfasser bei der Einleitung in die analytische Geometrie der Ebene unzulässig, die cartesischen Punktcoordinaten und die sogenannten Plücker'schen Liniearcoordinaten einander gegenüberzustellen; andererseits sind ihm die allgemeinen projectivischen Coordinaten für den Anfänger zu abstract. Er entwickelt daher einen besonderen Fall der letzteren, der sich auf sehr einfache Weise ableiten lässt, und sich übrigens aus dem allgemeinen Falle ergiebt, wenn man die ``Einheitslinie'' und zugleich den ``Einheitspunkt'' unendlich mitrücken lässt. von ihm steigt er zu den allgemeinen Coordinaten auf. Aehnliches gilt im Raume. Coordinate systems; Plücker coordinates Projective analytic geometry, General theory of linear incidence geometry and projective geometries, Projective techniques in algebraic geometry About the coordinates of points and lines on the plane and of the points and planes in space. | 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 function fields Arithmetic theory of algebraic function fields, Algebraic functions and function fields in algebraic geometry Differentiation of algebraic 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 main result of the article is that the Gromov invariants for the scheme of morphisms \(\text{Mor}_ d (C, \text{G} (r,k))\) of sufficiently high degree \(d\) from a Riemannian surface \(C\) into the grassmannian \(\text{G} (r,k)\) of complex \(r\)-planes in \(\mathbb{C}^ k\), can be rigorously defined for the special Schubert cycles, and are realized as an intersection of Chern classes on a projective scheme.
It follows immediately that the Gromov invariants do not depend on the complex structure of the Riemann surfaces, and the algebraic definition of the invariants make them more accessible for computations. -- The author also gives a relation between the Gromov invariants between Riemann surfaces of different genus, enabling him to treat the case \(r=2\).
The main technique of the article is to use the Quot schemes of trivial bundles on Riemann surfaces to obtain a compactification of the scheme \(\text{Mor}_ d (C, \text{G} (r,k))\). The relevant cycles become the intersection with \(\text{Mor}_ d (C, \text{G} (r,k))\) of Chern classes on the Quot schemes. Gromov invariants; scheme of morphisms; special Schubert cycles; Chern classes; trivial bundles on Riemann surfaces; Quot schemes Bertram A.: Towards a Schubert calculus for maps from a Riemann surface to a Grassmannian. Internat. J. Math. 5, 811--825 (1994) Grassmannians, Schubert varieties, flag manifolds, Riemann surfaces; Weierstrass points; gap sequences, Factorization systems, substructures, quotient structures, congruences, amalgams, Schemes and morphisms Towards a Schubert calculus for maps from a Riemann surface to a Grassmannian | 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 Parabolic vector bundles (and later parabolic torsion free sheaves) were studied for their own interest and to prove theorems (e.g. a rationality theorem) for moduli of vector bundles (\textit{V. B. Mehta} and \textit{C. S. Seshadri}, [Math. Ann. 248, No. 3, 205--239 (1980; Zbl 0454.14006)], \textit{M. Maruyama} and \textit{K. Yokogawa}, [Math. Ann. 293, No. 1, 77--99 (1992; Zbl 0735.14008)], \textit{J. N. N. Yer} and \textit{C. T. Simpson}, [Math. Ann. 338, No. 2, 347--383 (2007; Zbl 0776.14004)]). A breakthrough and a flurry of activities arose when Biswas connected rational parabolic bundles with bundles on orbifolds \textit{I. Biswas}, Duke [Math. J. 88, No. 2, 305--325 (1997; Zbl 0955.14010)]; hence connections with algebraic stacks and root stacks (\textit{N. Borne}, [Indiana Univ. Math. J. 58, No. 1, 137--180 (2009; Zbl 1186.14016)], \textit{N. Borne}, [Int. Math. Res. Not. 2007, No. 16, Article ID rnm049, 38 p. (2007; Zbl 1197.14035)]). In the paper under review the authors give many reasons to see parabolic sheaves (even with torsion) on a scheme in the set-up of logarithmic geometry [\textit{K. Kato}, in: Algebraic analysis, geometry, and number theory, Proc. JAMI Inaugur. Conf., Baltimore/MD (USA) 1988, 191--224 (1989; Zbl 0776.14004)]. The authors give a definition of parabolic quasi-coherent sheaf with fixed rational weights on a logarithmic scheme \((X, M, \rho )\) and show that the associated cathegory is equivalent to the category of sheaves on a root stack. They associate to \((M,\rho )\) the symmetric monoidal stack on the small étale site \(X_{\text{ét}}\) whose objects are invertible sheaves with sections. They develope this theory and call them a Deligne-Faltings structure, proving many results on logarithmic schemes directly from their set-up and interpreting or improving old results in their set-up [\textit{M. C. Olson}, Ann. Sci. École Norm. Sup. (4) 36, No. 5, 747--791 (2003; Zbl 1069.14022); \textit{K. Hagihara}, K-theory 29, No. 2, 75--99 (2003; Zbl 1038.19002); \textit{W. Nizioł}, Doc. Math. 13, 505--551 (2008; Zbl 1159.19003); Adv. Math. 230, No. 4--6, 1646--1672 (2012; Zbl 1269.19003)]. parabolic sheaves; parabolic bundles; log scheme; algebraic stack; root stack; logarithmic geometry; parabolic quasi-coherent sheaf Borne, N.; Vistoli, A., Parabolic sheaves on logarithmic schemes, Adv. Math., 231, 1327-1363, (2012) Generalizations (algebraic spaces, stacks), Stacks and moduli problems, Algebraic moduli problems, moduli of vector bundles, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Monoidal categories (= multiplicative categories) [See also 19D23] Parabolic sheaves on 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 The surface obtained by blowing up \(\mathbb P^2\) at three non-collinear points is independent of the points. It is called the del Pezzo surface of degree six, and is denoted \(\mathbb B_3\). Let \(R\) be the non-commutative algebra \(\mathbb C\langle x,y\rangle/(x^5-yxy,y^2-xyx)\), given a grading by \(\deg x=1\) and \(\deg y=2\). The main result of the article is the following:
Let \(\text{Gr} R\) be the category of \(\mathbb Z\)-graded left \(R\)-modules. There is an equivalence of categories
\[
\text{Qcoh}\mathbb B_3\equiv\frac{\text{Gr}}{\text{F}\dim R}
\]
where the left-hand side is the category of quasi coherent \(\mathcal O_{\mathbb B_3}\)-modules and the right-hand side is the quotient category modulo the the full subcategory \(\text{F}\dim R\) consisting of those modules that are the sum of their finite submodules.
Let \(\mathcal L=\mathcal O(-E)\) be the invertible \(\mathcal O_{\mathbb B_3}\)-module corresponding to a \((-1)\)-curve \(E\) and \(\sigma\) an order \(6\) automorphism of \(\mathbb B_3\) that cyclically permutes the six \((-1)\)-curves on \(\mathbb B_3\). Then the result above follows from the fact that \(R\) is isomorphic to the twisted homogeneous coordinate ring
\[
B(\mathbb B_3,\mathcal L,\sigma)=\bigoplus_{n\geq 0}H^0(\mathbb B_3,\mathcal L_n)
\]
where \(\mathcal L_n=\mathcal L\otimes (\sigma^\ast)\mathcal L\otimes\cdots\otimes(\sigma^\ast)^{n-1}\mathcal L.\) Results from Artin, Tate, Van den Bergh, and Stephenson imply that \(R\), as it is isomorphic to \(B(\mathbb B_3,\mathcal L,\sigma)\), is a 3-dimensional Artin-Schelter regular algebra and therefore has specific properties of non-commutative algebras, listed in the article. The close connection between \(R\) and \(\mathbb B_3\) means that almost all aspects of the representation theory of \(R\) can be expressed in terms of the geometry of \(\mathbb B_3\).
The justification of calling \(R\) a non-commutative homogenous coordinate ring for \(\mathbb B_3\) is the similarity between the equivalence of categories and the theorem of Serre: If \(X\subset\mathbb P^n\) is the scheme-theoretic zero locus of a graded ideal \(I\) in \(S=\mathbb C[x_0,\dots,x_n]\) with its standard grading, and \(A=S/I\), then there is an equivalence of categories
\[
\text{Qcoh}X\equiv\frac{\text{Gr}A}{\text{Fdim}A}
\]
where the right hand side is the quotient category of graded \(A\)-modules by the full subcategory consisting of modules whose non-zero finitely generated submodules have support only at the origin.
The author proves that \(R\) is an iterated Ore extension. The main point is to prove that \(R\) has the same Hilbert series as the weighted polynomial ring on three variables of weight \(1,2\), and \(3\). Then the main theorem is proved by constructing the del Pezzo surface \(\mathbb B_3\) and the the twisted homogeneous coordinate ring \(B(\mathbb B_3,\mathcal L,\sigma)\) explicitly. Then a graded isomorphism from \(R\) to \(B(\mathbb B_3,\mathcal L,\sigma)\) is also explicitly given.
This is a nice article pinpointing the noncommutative algebraic geometry, and also inspiring to similar investigations. non-commutative homogeneous coordinate ring; twisted homogeneous coordinate ring; iterated Ore extension; order six automorphism Noncommutative algebraic geometry, Rings arising from noncommutative algebraic geometry, Graded rings and modules (associative rings and algebras) A non-commutative homogeneous coordinate ring for the degree six del Pezzo surface | 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 smooth compact complex variety \(X\) of dimension \(3\) such that the anticanonical divisor \(-K_X\) is the ample generator of \(\text{Pic}(X)\). Then \(X\) is called a smooth prime Fano threefold. For any curve \(C\subset X\), we say that \(C\) has degree \(d\) if \(-K_{X}\cdot C=d\).
Let \(S\) be a general surface in \(|-K_{X}|\). Then \(S\) is a smooth \(K3\) surface (this was proved by Shokurov). For any natural number \(d\), the punctual Hilbert scheme \(\text{Hilb}_dS\) of length \(d\) subschemes of \(S\) is a smooth complex symplectic variety.
If \(d=1\), then \(\text{Hilb}_dS\simeq S\). Any curve \(C\subset S\) is a Lagrangian subvariety of \(S\), and if \(C\) is smooth, then the relative Jacobian \(\mathcal{J}\) is a Lagrangian fibration (equivalently, an algebraically completely integrable system) over an open subset of the complete linear system \(|C|\).
It was observed by Thomas that any component \(\mathcal{H}\) of the Hilbert scheme of smooth curves of degree \(d\) on \(X\) is sent by the intersection map \(j\) with \(S\) to a Lagrangian subvariety \(j(\mathcal{H})\) of \(\text{Hilb}_dS\).
In the paper under review, the authors study the scheme \(F(X)\) of curves of degree \(2\) (conics) on \(X\). Suppose, in addition, that \(X\) is general in moduli. Then \(F(X)\) is a smooth irreducible surface. The structure of \(F(X)\) is also known. Let \(J(X)\) be the the intermediate Jacobian of \(X\). The authors shows that the Abel-Jacobi mapping \(F\to F(X)\) induced an isomorphism \(\text{Alb}(F(X))\simeq J(X)\). In particular, \(F(X)\simeq J(X)\) if \(-K_{X}^{3}=18\).
The authors show that \(j: F(X)\to \text{Hilb}_2S\) is an immersion, and \(j\) is injective for \(-K_{X}^3\geq 12\) (but not for \(-K_{X}^{3}<12\)). Thus, the image \(j(F(X))\) is a Lagrangian surface in the symplectic fourfold \(\text{Hilb}_2S\), and \(j(F(X))\) is smooth if \(-K_{X}^{3}\geq 12\). The authors prove that \(X\) is uniquely determined by \(S\) and \(j(F(X))\) if \(18\geq-K_{X}^{3}\geq 12\), which is no longer true if \(-K_{X}^{3}=22\), because \(F(X)\simeq\mathbb{P}^{2}\) if \(-K_{X}^{3}=22\), while \(X\) has moduli.
The authors show that the embedding \(j(F(X)) \subset \text{Hilb}_2S\) gives rise to an algebraic integrable system (a Lagrangian fibration), and they describe the fibers of this Lagrangian fibration. Fano threefolds; \(K3\) surfaces; complex symplectic varieties; Lagrangian subvarieties; integrable systems Iliev A. and Manivel L. (2007). Prime Fano threefolds and integrable systems. Math. Ann. 339: 937--955 \(K3\) surfaces and Enriques surfaces, Fano varieties, Picard schemes, higher Jacobians, Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems Prime Fano threefolds and integrable 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 The author of this paper considers the relationship between the topological entropy and the growth rate of the number of periodic points for smooth dynamical systems. His results include new results about those relationships. It is known that the topological entropy needs not being equal to the exponential growth rate of the number of periodic points. \textit{R. Bowen} [Trans. Am. Math. Soc. 154, 377--397 (1971; Zbl 0212.29103)]
had proved equality between the two for certain expansive homeomorphisms (those with topologically transitive subshifts of finite type and Axiom A systems).
The main results are as follows. If \(f\) is a \(C^{\infty}\) diffeomorphism from a compact surface \(M\) to itself with positive topological entropy \(h_{\mathrm{top}} (f)\), then for any \(\delta\) in the interval \((0, h_{\mathrm{top}}(f))\) the set of saddle \(n\)-periodic points with Lyapunov exponents \(\delta\)-away from zero has exponential growth rate in \(n\) equal to the topological entropy. Furthermore, these periodic points are equidistributed with respect to measures of maximal entropy.
Similarly, if \(f\) is a \(C^{\infty}\) interval map with positive topological entropy \(h_{top} (f)\), then the set of repelling \(n\)-periodic points with the same constraints on Lyapunov exponents has exponential growth rate in \(n\) equal to the topological entropy. Here again the periodic points are equidistributed with respect to measures of maximal entropy.
The author's strategy in the proof of the main theorem is to use the concept of local exponential growth rate and a result of \textit{A. Katok} [Publ. Math., Inst. Hautes Étud. Sci. 51, 137--173 (1980; Zbl 0445.58015)]. Any subset of periodic points with a growth rate that matches or exceeds the topological entropy, but has zero local growth rate, is equidistributed with respect to maximal measures. But its exponential growth rate is equal to the topological entropy. Local exponential growth rate is the exponential growth rate in \(n\) of the \(n\)-periodic points in an arbitrarily small \(n\)-dynamical ball. Using Katok's theorem, it is then only necessary to show that the set of saddle \(n\)-periodic points with Lyapunov exponents \(\delta\)-away from zero (and the corresponding set for interval maps) have zero local exponential growth rates for \(C^{\infty}\) surface diffeomorphisms and interval maps. entropy; hyperbolic periodic points; smooth surface dynamical systems; Yomdin's theory; semi-algebraic geometry Dynamical systems involving homeomorphisms and diffeomorphisms of planes and surfaces, Topological entropy, Dynamical systems involving maps of the circle, Dynamical systems involving smooth mappings and diffeomorphisms, Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics, Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.), Entropy and other invariants, Smooth ergodic theory, invariant measures for smooth dynamical systems, Semialgebraic sets and related spaces Periodic expansiveness of smooth surface diffeomorphisms 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 ruled residue theorem characterises residue field extensions for valuations on a rational function field. Under the assumption that the characteristic of the residue field is different from 2 this theorem is extended here to function fields of conics. The main result is that there is at most one extension of a valuation on the base field to the function field of a conic for which the residue field extension is transcendental but not ruled. Furthermore the situation when this valuation is present is characterised. valuation; residue field extension; Gauss extension; rational function field; algebraic function field; genus zero; quaternion algebra Valued fields, Transcendental field extensions, General valuation theory for fields, Algebraic functions and function fields in algebraic geometry, Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.) A ruled residue theorem for function fields of 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 Let \(\mathcal C\) be a smooth integral curve in projective \(n\)-space, and let \(\mathcal \Gamma\) be a plane projection of \(\mathcal C\) of the same degree and consider the closed subscheme \(Z\subseteq \mathcal \Gamma\) corresponding to the conductor. A method to study such a curve \(\mathcal C\) consists in studying the adjoint curves, that is the curves passing through the set \(Z\) of nodes of a general plane projection \(\Gamma\) of \(\mathcal C\). The aim of this paper is to understand better \(Z\) and its relation with the geometry of \(\mathcal C\). Let \(\nu:{\mathcal C}\rightarrow \Gamma\subseteq \mathbb P^2\) be the projection morphism. Let \(D\) be the pull-back, via \(\nu\), of a general linear section of \(\Gamma\). Proposition 4.2 relates the Hilbert function \(h_Z\) with the specialities of the multiples of \(D\). After studying the least degree \(\alpha\) of an adjoint curve, the authors get a relation between the multisecants of \(\mathcal C\) and the speciality of \(\mathcal C\) (Proposition 4.11) (according to the authors, it seems that the relations between speciality and gonality of a projective curve should be further investigated). Using the so called base-point-free pencil trick, they show that the Hilbert function \(h_Z\) of \(Z\) is of decreasing type (see Theorem 5.1); in characteristic zero, the spectrum of \(\mathcal C\) provides a more precise information about the descent of the different function \(\partial h_Z\) (see Theorem 5.10). Let \(\Delta\) be the pull-back of \(Z\) to \(\mathcal C\) and let \(\nu:{\mathcal C}\rightarrow \Gamma\subseteq \mathbb{P}^2\) be the projection morphism. They show that if \(Z\) and \(\Delta\) are reduced then they enjoy a weak form of uniform position, namely they are Cayley-Bacharach schemes (see Theorem 6.3). Moreover, they find the relations between the first cohomology of \(\mathcal C\) and \(h_Z\), which allow us to show that \(h_Z\) is independent of \(\nu\) and that \(\mathcal C\) is arithmetically Cohen-Macaulay if and only if \(\Delta\) is arithmetically Gorenstein (see Corollary 6.12). Hilbert function; adjoint ideals; conductor; Cayley-Bacharach property N. Chiarli - S. Greco - R. Notari, Postulation of adjoint ideals and geometry of projective curves, Algebra, Arithmetic and Geometry with applications (West Lafayette, IN, 2000), Springer, Berlin, 2004, 235--257 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Projective techniques in algebraic geometry, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series Postulation of adjoint ideals and geometry of projective 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 Pour \(a\in {\mathbb{C}}\), avec \(| a| >1\), soit \(T(z)=\sum_{n\geq 0}a^{-n(n-1)/2} z^ n.\) Soit \(I_ K\) l'anneau des entiers du corps quadratique imaginaire K et \(\alpha_ 1,...,\alpha_ m\) des éléments de \(K^ x\). Les auteurs donnent une mesure d'indépendance linéaire sur K des \(T(\alpha_ i):\) Si on a (i) \(a=s/r\), avec r et s dans \(I_ K\setminus \{0\}\) vérifiant \(0\leq \log | r| /\log | s| =\gamma <\gamma (m)\) (où \(\gamma\) (m) est explicite). (ii) pour \(i\neq j\), le rapport \(\alpha_ i/\alpha_ j\) n'est pas une puissance entière de a; alors, pour tout \(\epsilon >0\), il existe \(c=c(\alpha_ 1,...,\alpha_ m,m,r,s,\epsilon)>0\) tel que
\[
| A_ 0+A_ 1T(\alpha_ 1) + \cdots + A_ mT(\alpha_ m)| > cA^{-\theta (m,\gamma)-\epsilon}
\]
pour tout \((A_ 0,...,A_ m)\in I_ K^{m+1}\setminus \{0\},\) où \(A=\max | A_ i|\) et \(\theta\) (m,\(\gamma)\) est explicite. L'hypothèse (ii) est nécessaire en raison de la relation fonctionnelle \(T(az)=1+azT(z).\) Les auteurs en déduisent une mesure d'irrationalité des valeurs de la fonction thêta de Jacobi \(J(z)=\sum_{-\infty <n<+\infty}b^{-n^ 2} z^ n.\)
La démonstration est une généralisation de celle de \textit{L. Tschakaloff} [Math. Ann. 84, 100-114 (1921)] qui prouvait l'indépendance linéaire des \(T(\alpha_ i)\) avec \({\mathbb{Q}}\) à la place de K. Elle fait suite à des travaux de premier [Port. Math. 33, 1-17 (1974; Zbl 0276.10020)] et de second auteur [Sémin. Théor. Nombres 1980-1981, Exp. No.30 (1981; Zbl 0474.10027)] sur cette question. La relation fonctionnelle vérifiée par T est utilisée pour construire une fonction auxiliaire explicite. Le lemme crucial (lemme 4) utilise des estimations fines de valuations p-adiques pour assurer la non nullité d'un déterminant. Notons que l'hypothèse (i) est inutile pour établier les majorations du lemme 2. entire transcendental function; measure of linear independence over imaginary quadratic field; Jacobi theta function P. Bundschuh and I. Shiokawa, ''A measure for the linear independence of certain numbers,'' Results Math. 7 (1984), 130--144. Transcendence (general theory), Approximation by numbers from a fixed field, Theta functions and abelian varieties A measure for the linear independence of certain numbers | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems This paper concerns an interpretation of a special class of holomorphic vector bundles on \(\mathbb P^{2n+1}\) in terms of linear systems on curves. The bundles \({\mathcal E}\) studied are of rank \(2n\) and are generalizations of the bundles on \(\mathbb P^ 3\) which correspond to `t Hooft instantons on \(S^ 4\). Due to the work of \textit{S. M. Salamon} [in Global Riemannian geometry, Proc. Symp., Durham/Engl. 1982, 65--74 (1984; Zbl 0616.53022)] these bundles are also related to Yang-Mills-type equations in higher dimensions, although this is not a principal concern of the authors. The bundles have a number of properties, including the fact that \({\mathcal E}(1)\) has sections and the bundles are simple. Stability unfortunately is not proven.
This particular class of bundles was introduced by \textit{C. Okonek} and \textit{H. Spindler} [J. Reine Angew. Math. 364, 35--50 (1986; Zbl 0568.14009)] and here a study is made of the divisor of jumping lines, which is determined by a hypersurface \(Y\) in a null \(\mathbb P^{2n}\) lying in the Grassmannian \(\text{Gr}_ 2{\mathbb C}^{2n+2}\). The hypersurface \(Y\) is shown to be a Poncelet variety, namely the locus of points on a rational normal curve which are poles of a divisor \(D\) of degree \(r\) such that \(D\leq D'\) for \(D'\) in some fixed linear system of degree \((r+k)\) on the curve. holomorphic vector bundles; linear systems on curves; instantons; Yang-Mills-type equations; Grassmannian Spindler, H., Trautmann, G.: Rational normal curves and the geometry of special instanton bundles on \$\$\{\(\backslash\)mathbb\{P\}\^\{2n+1\}\}\$\$ . Math. Gottingensis 18 (1987) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Constructive quantum field theory, Special algebraic curves and curves of low genus, Divisors, linear systems, invertible sheaves, Grassmannians, Schubert varieties, flag manifolds Rational normal curves and the geometry of special instanton bundles on \(\mathbb P^{2n+1}\) | 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 paper deals with the so called hungry periodic discrete (hpd) Toda equations described by
\[
I_n^{t+M} = I_n^t + V_n^t-V_{n-1}^{t+1}\;,\;V_n^{t+1} = {{I_{n+1}^tV_n^t} \over{I_n^{t+M}}}
\]
with the periodic boundary conditions
\[
I_n^t = I^t_{n+N}\;,\;V_n^t = V^t_{n+N}
\]
and \(N\), \(M\) positive integers, \(t\) is the time variable and \(n\) is the position. The following assumption is made
\[
\prod_{n=1}^N I_n^{t+M} = \prod_{n=1}^N I_n^t \neq \prod_{n=1}^N V_n^{t+1} = \prod_{n=1}^N V_n^t
\]
to ensure the existence of a unique solution. This solution is constructed using the auxiliary \(\tau\)-(tau)-function. hungry periodic discrete (hpd) Toda equation; periodic solution; theta function Iwao, S.: Solution of the generalized periodic discrete Toda equation. II. Theta function solution. J. Phys. A \textbf{43}(15), 155208 (2010) Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Theta functions and curves; Schottky problem, Special algebraic curves and curves of low genus, Periodic solutions of difference equations Solution of the generalized periodic discrete Toda equation. II: Theta function solution | 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 In a recent paper \textit{W. A. Zúñiga-Galindo} and the author [J. Math. Sci., Tokyo 20, No. 4, 569--595 (2013; Zbl 1312.14067)] begun the study of the local zeta functions for Laurent polynomials. In this work we continue this study by giving a very explicit formula for the local zeta function associated to a Laurent polynomial \(f\) over a \(p\)-adic field, when \(f\) is weakly non-degenerate with respect to the Newton polytope of \(f\) at infinity. Laurent polynomials; Igusa's zeta function; explicit formulae; Newton polytopes; non-degeneracy conditions Zeta functions and \(L\)-functions, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Toric varieties, Newton polyhedra, Okounkov bodies An explicit formula for the local zeta function of a Laurent polynomial | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems New results are presented and a brief review is given for new methods of the theory of dynamic systems on manifolds over local fields and formal groups over local rings. For the analysis of \(n\)-dimensional manifolds and dynamic systems on such manifolds, formal structures are used, in particular, \(n\)-dimensional formal groups. Infinitesimal deformations are presented in terms of formal groups. The well-known one-dimensional case is extended and \(n\)-dimensional (\(n\geq1\)) analytic mappings of an open \(p\)-adic polydisc (\(n\)-disk) \( {D}_p^n \) are considered. The \(n\)-dimensional analogs of modules arising in formal and non-Archimedean dynamic systems are introduced and investigated and their formal-algebraic structure is presented. Rigid structures, objects, and methods are outlined. From the point of view of systems analysis, new, namely formal and non-Archimedean, faces and structures of systems, mappings and iterations of mappings between these faces and structures are introduced and investigated. formal group; local ring; commutative formal group scheme; deformation; formal module; module of differentials Dynamical systems over non-Archimedean local ground fields, Isogeny, Class field theory; \(p\)-adic formal groups, Formal groups, \(p\)-divisible groups Formal and non-Archimedean structures of dynamic systems on manifolds | 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 complex projective surfaces admitting a Poisson structure; we prove a classification theorem and count how many independent Poisson structures there are on a given Poisson surface. Poisson structures; integrable systems; ruled surfaces Bartocci, C; Macrí, E, Classification of Poisson surfaces, Commun. Contemp. Math., 7, 89-95, (2005) Rational and ruled surfaces, Poisson manifolds; Poisson groupoids and algebroids Classification of Poisson 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 This paper discusses about the locations of poles of Igusa's local zeta functions \(Z(s,f,\chi)\) associated with a non-degenerate polynomial \(f\) over a non-archimedean local field \(K\) and a character \(\chi\) of the group of units of the valuation ring of \(K\). Let \(K\) be a non-archimedean local field of arbitrary characteristic. Let \({\mathcal O}_K\) be the ring of integers of \(K\) and \({\mathcal P}_K\) its maximal ideal. Let \(\pi\) be a fixed uniformizing parameter of \(K\), and let the residue field of \(K\) be \(\mathbb{F}_q\) the field with \(q = p^r\) elements. For \(x\in K\), \(v\) denotes the valuation of \(K\) such that \(v(\pi) =1\), \(|x|_K =q^{-v(x)}\) and \(ac(x) = x\pi^{-v(x)}\). Let \(f(x)\in {\mathcal O}_K[x]\), \(x = (x_1,\dots,x_n)\) be a non-constant polynomial, and \(\chi : {\mathcal O}^\times_K\to\mathbb{C}^\times\) a character of \({\mathcal O}^\times_K\), the group of unit of \({\mathcal O}_K\). We put \(\chi(0) = 0\). Then Igusa's local zeta function associated to these data is defined to be
\[
Z(s,f,\chi)=\int_{{\mathcal O}^n_K}\chi(ac(f(x)))|f(x)|^s_K \,|dx|
\]
with \(s\in \mathbb{C}\) for \(\text{Re}(s) > 0\). Here \(|dx|\) denotes the Haar measure on \(K^n\), normalized such that \({\mathcal O}^n_K\) has measure 1. In the case of \(K\) having characteristic zero, \(Z(s,f,\chi)\) is a rational function of \(q^{-2}\).
The author of this paper gives a small set of candidates for the poles of \(Z(s,f,\chi)\) in terms of the Newton polyhedron \(\Gamma(f)\) of \(f\). He also shows that for almost all \(\chi\), the local zeta function \(Z(s,f,\chi)\) is a polynomial in \(q^{-s}\) whose degree is bounded by a constant independent of \(\chi\). The second result is a description of the largest pole of \(Z(s,f,\chi)_{\text{triv}})\) in terms of \(\Gamma(f)\) when the distance between \(\Gamma(f)\) and the origin is at most one. The author's approach is based on the \(p\)-adic stationary phase formula and Néron \(p\)-desingularization. Igusa's local zeta function Wilson A. Zúñiga-Galindo, Local zeta functions and Newton polyhedra, Nagoya Math. J.172 (2003), p. 31-58 - ISSN : 2118-8572 (online) 1246-7405 (print) - Société Arithmétique de Bordeaux Zeta functions and \(L\)-functions, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Zeta and \(L\)-functions in characteristic \(p\) Local zeta-functions and Newton polyhedra | 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 result of the paper is the existence of Galois representations associated to certain genus 2 Siegel modular forms of low weight. The proof uses the already known (Shimura, Deligne, Faltings-Chai) analogous result for sufficiently high weight and congruences with systems of Hecke eigenvalues of form of high weight. The notion of pseudorepresentation is defined and used as a tool to construct the \(\lambda\)-adic representations from sufficiently many congruences. An important application of the main result is given in the last section. There it is shown how to eliminate one of the two hypotheses needed in the construction of Galois representations associated to certain Maass forms by Blasius and Ramakrishnan. existence of Galois representations; Siegel modular forms; congruences; systems of Hecke eigenvalues; pseudorepresentation; Maass forms Taylor R., Galois representations associated to Siegel modular forms of low weight, Duke Math. J. 63 (1991), 281-332. Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms, Galois representations, Langlands-Weil conjectures, nonabelian class field theory, Modular and Shimura varieties Galois representations associated to Siegel modular forms of low weight | 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 multi-graded Hilbert scheme parametrizes all ideals with a given Hilbert function. An important subclass is the class of all toric Hilbert schemes. This paper details the main component of a toric Hilbert scheme in case it contains a point corresponding to an affine toric variety. Here the component can be viewed as the set of all flat limits of this variety.
The main tool used in the paper is a computation of the fan of the toric variety. The treatment is very self-contained and thorough. Moreover, using a Chow morphism, a connection with certain GIT quotients of the toric variety is included. Some helpful examples are provided as well. toric Hilbert scheme; fiber polytope; toric Chow quotient Chuvashova O.V., The main component of the toric Hilbert scheme, Tôhoku Math. J., 2008, 60(3), 365--382 Parametrization (Chow and Hilbert schemes), Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Toric varieties, Newton polyhedra, Okounkov bodies The main component of 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 This paper is a survey report on some recent results mainly concerning weakly normal algebraic varieties, and the linear systems contained in them. The results are fully stated, with complete references and some ideas of the proofs.
Recall that an algebraic variety X is said to be weakly normal (WN) if every birational morphism \(X'\to X\), which is also a universal homeomorphism is indeed an isomorphism. A natural problem is to understand whether the general hyperplane section of a WN variety embedded in a projective space is WN; the answer in this case is positive, and a number of different approaches are described. - The above problem has a natural generalization, obtained by adapting the classical theorem of Bertini (``the general member of a linear system S over an algebraic variety X (over the field of complex numbers) is non-singular but perhaps at the base points of S and at the singular points of X'').
There is an axiomatic approach to this problem, i.e. it is possible to show that Bertini's theorem holds for all local properties P which satisfy certain axioms. This implies Bertini's theorem for several properties, including \(P=WN\) in characteristic zero [the authors, J. Algebra 98, 171-182 (1986; Zbl 0613.14006)].
In positive characteristic it can be shown that there are WN projective varieties, whose general hyperplane sections are not WN [the authors, Proc. Am. Math. Soc. 106, No.1, 37-42 (1989; Zbl 0699.14063)]. However by using the axiomatic approach one can prove Bertini's theorem for the property \(P=WN1+S_ 2\) (where WN1 means, roughly, WN \(+\) ``well behaved in codimension 1'') and, in particular, \(P=WN+Gorenstein\) [the authors, J. Algebra 128, No.2, 488-496 (1990)]. weakly normal algebraic varieties; linear systems; birational morphism; Bertini's theorem Rational and birational maps, Divisors, linear systems, invertible sheaves, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Singularities in algebraic geometry Weakly normal algebraic varieties and Bertini theorems | 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 \(k\) sei ein endlicher algebraischer Funktionen- oder Zahlkörper, \(K\) eine endliche algebraische Erweiterung von \(k\). Der Regularitätsbereich \(\mathfrak R(K)\) von \(K/k\) ist die Menge der Primideale von \(k\), die in \(K\) mindestens einen Primfaktor vom Relativgrad 1 haben. Für Zahlkörper gilt bekanntlich (Satz von Bauer): Ist für einen Normalkörper \(N/k\), bis auf endlich viel Ausnahmen \(\mathfrak R(N)\supseteq \mathfrak R(K)\), so ist \(N\subseteq K\) und umgekehrt (\(K\) beliebig). Dieser Satz ist eine Folge des Frobeniusschen Satzes, daß es in einem Normalkörper \(N/k\) unendlich viel Primideale gibt, deren Zerlegungsgruppe eine vorgegebene zyklische Untergruppe der Galois-Gruppe \(\mathfrak G\) von \(N/k\) ist. Beide Sätze werden übertragen auf Funktionenkörper \(k\), in deren Konstantenkörper \(\Omega\) der Hilbertsche Irreduzibilitätssatz gilt [d. h.: in einem beliebigen irred. Polynom \(P(x; u_1, \ldots, u_n)\) über \(\Omega\) können die \(u_i\) auf unendlich viele Weisen so durch Elemente \(a_i\) von \(\Omega\) ersetzt werden, daß \(P(x; a_1, \ldots, a_n)\) auch irred.]. Beim Analogon des Frobeniusschen Satzes darf dann sogar eine beliebige Untergruppe von \(\mathfrak G\) vorgeschrieben werden. Anders ausgedrückt: Es gibt unendlich viele Primideale, deren Zerlegungskörper ein vorgegebener Teilkörper \(K'\) von \(N\) ist. Dies gilt allgemeiner: In einem (nicht notwendig galoisschen) \(K/k\) gibt es unendlich viele Primideale, deren zugehörige Primideale in einem vorgegebenen Teilkörper \(K'\) von \(K\) den Grad 1 in bezug auf \(k\) haben und deren Grad in bezug auf \(K'\) gleich \((K:K')\) ist.
Beweis: Aus dem Irreduzibilitätssatz folgt, für einen rein transzendenten Grundkörper \(\Omega(\eta)\) die Existenz unendlich vieler Primideale vom Grad 1 in bezug auf \(\Omega\), die in \(K\) unzerlegt bleiben. Denkt man sich \(K'\) durch
ein Element \(\eta\) erzeugt und hat man \(k = \Omega(z,x)\), \(K' = \Omega(z,\eta)\), so ergibt Anwendung dieses Existenzsatzes auf \(K/\Omega(\eta)\) die Existenz unendlich vieler \(\mathfrak P\) in \(\Omega(\eta)\), die in \(K\) unzerlegt bleiben, die aber vom Grad 1 in bezug auf \(\Omega\) sind; und von diesen weist man dann nach, daß sie von der verlangten Art sind. algebraic function fields; domain of regularity; Hilbert's irreducibility theorem Arithmetic theory of algebraic function fields, Algebraic functions and function fields in algebraic geometry Über die Kennzeichnung algebraischer Funktionenkörper durch ihren Regularitätsbereich | 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 Classical Castelnuovo's lemma shows that the number of linearly independent quadratic equations of a nondegenerate irreducible projective variety of codimension \(c\) is at most \(\binom{c+1}{2}\) and the equality is attained if and only if the variety is of minimal degree. Also a generalization of Castelnuovo's lemma by \textit{G. Fano} [Torino Mem. (2) XLIV. 335--382 (1894; JFM 25.1279.02)] implies that the next case occurs if and only if the variety is a del Pezzo variety. For curve case, these results are extended to equations of arbitrary degree respectively by \textit{J. Harris} [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 8, 35--68 (1981; Zbl 0467.14005)] and \textit{S. L'vovsky} [Math. Ann. 306, No. 4, 719--735 (1996; Zbl 0857.14031)]. This paper is intended to extend these results to arbitrary dimensional varieties and to the next cases. Hilbert function; projective varieties of low degree Park, Euisung: On hypersurfaces containing projective varieties, Forum math. (2013) Projective techniques in algebraic geometry, Varieties of low degree On hypersurfaces containing 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 \(F\) be an algebraically closed field of characteristic zero and \(GL(n,F)\) be the general linear group of \(n\times n\) invertible matrices with entries in \(F\). Given an \(n\times n\) matrix \(A\), its conjugacy class, \(\mathcal O(A)\), is the orbit of \(A\) under the conjugacy action of \(GL(n,F)\), i.e. \(\mathcal O(A)=\{U^{-1}AU \mid U \in GL(n,F)\}\). Denote by \(\mathcal C(A)\) the centralizer of \(A\). It is known that \(\dim \mathcal O(A) =n^2-\dim \mathcal C(A)\). Therefore, one gets three different formulae for \(\dim \mathcal C(A)\) using the existing three different formulae for \(\dim \mathcal O(A)\) when \(A\) is nilpotent. The author gives a rather direct proof of these formulae for \(\dim \mathcal C(A)\). Using the reduction to the nilpotent case, he then gets a formula for \(\dim \mathcal O(A)\) for an arbitrary matrix \(A\) yielding a nicer formula when \(A\) has no multiple nonzero eigenvalue. He finally gives a shorter proof of a theorem on irreducibility and dimension of a rank variety. dimension; conjugacy class of a matrix; rank variety; rank function; partition of a non-negative integer Group actions on varieties or schemes (quotients), Determinantal varieties On the dimension of certain \({\mathcal G}{\mathcal L}_n\)-invariant sets of matrices | 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 0722.00001.]
This is an overview of the work of Gieseker, Trubowitz and Knörrer on the theory of algebraic Fermi curves. Fermi curves are the 1-dimensional analogs of the usual Fermi surfaces from solid state physics. Given a potential function you have one curve for each energy-level. The density of states function plays a central role in the theory and it can be measured experimentally. This is explained in some detail in section 2 of the paper.
If one goes over to a certain discrete analogon, the Fermi curves form the real points of certain algebraic curves varying in a family parametrized by the complex projective line. Together these form an algebraic surface, the Bloch variety. The density of states function reveals itself as the period integral of a certain canonical 1-form over the cycle formed by the real Fermi curve. The Bloch variety admits a natural compactification, first described by Bättig and recalled in section 5. Deep methods from algebraic geometry then can be applied to show that generically the Bloch variety essentially uniquely determines the potential and that the density of states function essentially determines the Bloch variety (for generic potentials). In view of the preceding one might say that the experimentally determinable density of states function generically determines the potential and hence tells you all of the physics. But---of course one would need the density of states function also for complex parameters, i.e. for complex values of the energy and this does not seem to have physical meaning. algebraic Fermi curves; Bloch variety; density of states function Curves in algebraic geometry, Surfaces and higher-dimensional varieties, Schrödinger operator, Schrödinger equation, PDEs in connection with optics and electromagnetic theory, Applications of quantum theory to specific physical systems Algebraic Fermi curves [after Gieseker, Trubowitz and Knörrer] | 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 Starting from a discrete spectral problem, a discrete soliton hierarchy is derived. Some \((2+1)\)-dimensional discrete systems related to the hierarchy are proposed. The elliptic coordinates are introduced and the equations in the discrete soliton hierarchy are decomposed into solvable ordinary differential equations. The straightening out of the continuous flow and the discrete flow are exactly given through the Abel-Jacobi coordinates. As an application, explicit algebro-geometric solutions for the \((2+1)\)-dimensional discrete systems are obtained. \((2+1)\)-dimensional discrete systems; algebro-geometric solutions; elliptic coordinates; straightening out; Riemann-Jacobi inversion technique Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions, Lattice dynamics; integrable lattice equations, Relationships between algebraic curves and integrable systems, Soliton equations, Discrete version of topics in analysis Algebro-geometric solutions for some \((2+1)\)-dimensional discrete 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 Computations of stable homotopy groups are often based on spectral sequences whose starting term is algebraically describable and manageable, at least in a certain range. The authors found a way to compare two different spectral sequences used to compute stable homotopy groups in the Morel-Voevodsky motivic stable homotopy theory over a field. These are the motivic Adams spectral sequence, which is based on resolving via Voevodsky's motivic Eilenberg-MacLane spectrum, and the slice spectral sequence, which is obtained from the effectivity filtration on the motivic stable homotopy category.
The method is designed to work for motivic spectra \(E\) which satisfy the following property: There exists a natural number \(n\) and a prime number \(p\) such that the canonical map
\[
f_{(q+1)(p^n-1)} E\to f_{q(p^n-1)}E
\]
of effective covers induces the zero homomorphism on motivic homology with coefficients in the field with \(p\) elements for all integers \(q\). Examples of such are motivic Eilenberg-MacLane spectra with suitable coefficients, the (very) effective cover of the algebraic \(K\)-theory spectrum, algebraic Morava \(K\)-theory spectra, and -- at least over fields of characteristic zero -- truncated algebraic Brown-Peterson-spectra. Non-examples are given by the algebraic cobordism spectrum \(MGL\), and probably its variants \(MSL\) and \(MSp\). Speculation on the motivic sphere spectrum is included. Due to the technical setup, the authors complete the motivic stable homotopy category at the chosen prime \(p\) and the algebraic Hopf map \(\eta\).
Applications consist in relating differentials between different spectral sequences, among them the \(\rho\)-Bockstein spectral sequence over the field of real numbers, and the Hill-Hopkins-Ravenel spectral sequence for certain genuine \(C_2\)-spectra. motivic stable homotopy theory; motivic Adams spectral sequence; slice spectral sequence; algebraic slice spectral sequence Motivic cohomology; motivic homotopy theory, Stable homotopy theory, spectra, Equivariant homotopy theory in algebraic topology, General theory of spectral sequences in algebraic topology, Adams spectral sequences Algebraic slice spectral sequences | 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 On a smooth projective threefold, we construct an essentially surjective functor \(\mathcal{F}\) from a category of two-term complexes to a category of quotients of coherent sheaves and describe the fibers of this functor. Under a coprime assumption on rank and degree, the domain of \(\mathcal{F}\) coincides with the category of higher-rank PT-stable objects, which appears on one side of Toda's higher-rank DT/PT correspondence formula. The codomain of \(\mathcal{F}\) is the category of objects that appears on one side of a correspondence formula by \textit{A. Gholampour} and \textit{M. Kool} [Épijournal de Géom. Algébr., EPIGA 3, Article No. 17, 29 p. (2019; Zbl 1437.14014)], between the generating series of topological Euler characteristics of two types of quot schemes. PT-stable object; quot scheme; stable pair \(3\)-folds, Stacks and moduli problems, Derived categories of sheaves, dg categories, and related constructions in algebraic geometry A relation between higher-rank PT-stable objects and quotients of coherent sheaves | 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 prove new interesting relations among various invariants associated to complex analytic function germs (or to pairs of such germs) defined on a singular analytic set: the Milnor number, (first) Teissier number, local Euler obstruction, Brasselet number. These invariants are also related to the number of Morse critical points occurring in a partial Morsefication of the given germ, generalizing in this way the formulas due to Lê and Greuel. Euler obstruction; constructible function Dutertre, N., Grulha Jr., N.G.: Lê-Greuel type formula for the Euler obstruction and applications. Preprint (2011). arXiv:1109.5802 Singularities in algebraic geometry, Topology of analytic spaces, Singularities of vector fields, topological aspects Lê-Greuel type formula for the Euler obstruction 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 In their fundamental paper [Sel. Math., New Ser. 1, No. 3, 411--457 (1995; Zbl 0842.46040)], \textit{J.-B. Bost} and \textit{A. Connes} constructed a quantum statistical mechanical system, known as the Bost-Connes or BC system, that recovers the explicit class field theory for \(\mathbb Q\). In this paper the author uses the theory of complex multiplication for Siegel modular varieties to construct arithmetic subalgebras for BC-type systems attached to number fields containing a CM field. The construction is based on the method used by \textit{E. Ha} and \textit{F. Paugam} [IMRP, Int. Math. Res. Pap. 2005, No. 5, 237--286 (2005; Zbl 1173.82305)] and extends the construction of arithmetic subalgebras for imaginary quadratic fields by \textit{A. Connes, M. Marcolli} and \textit{N. Ramachandran} [Sel. Math., New Ser. 11, No. 3-4, 325--347 (2005; Zbl 1106.58005)]. Bost-Connes type systems; complex multiplication; Shimura varieties Yalkinoglu, B.: On Bost-\{C\}onnes type systems and complex multiplication. J. NCG (to appear). doi: 10.4171/JNCG/92 . arXiv:1010.0879v1 Complex multiplication and moduli of abelian varieties, Relations with noncommutative geometry, Class field theory, Modular and Shimura varieties, Noncommutative dynamical systems, Noncommutative geometry in quantum theory, Noncommutative geometry (à la Connes) On Bost-Connes type systems and complex multiplication | 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 expand upon works of \textit{C. A. Athanasiadis} [J. Algebr. Comb. 10, No. 3, 207--225 (1999; Zbl 0948.52012)]; \textit{P. H. Edelman} and \textit{V. Reiner} [Discrete Comput. Geom. 15, No. 3, 307--340 (1996; Zbl 0853.52013); erratum ibid. 17, 357 (1997)]; \textit{A. Postnikov} and \textit{R. P. Stanley} [J. Comb. Theory, Ser. A 91, No. 1--2, 544--597 (2000; Zbl 0962.05004)]; and \textit{M. Yoshinaga} [Invent. Math. 157, No. 2, 449--454 (2004; Zbl 1113.52039)] with significant contributions to the study of deformation of the Weyl arrangement of type \(A_2\).
Authors' abstract: ``We consider deformations of the Weyl arrangement of type \(A_2\), which include the extended Shi and Catalan arrangements. These last ones are well-known to be free. We study their sheaves of logarithmic vector fields in all other cases, and show that they are Steiner bundles. Also, we determine explicitly their unstable lines. As a corollary, some counter-examples to the shift isomorphism problem are given.'' line arrangements; logarithmic sheaves; Weyl arrangements; root systems Arrangements of points, flats, hyperplanes (aspects of discrete geometry), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Relations with arrangements of hyperplanes Logarithmic bundles of deformed Weyl arrangements of type \(A_2\) | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(p\) be a prime number, let \(K\) be a finite extension of \({\mathbb Q}_p\) with valuation ring \(R\), maximal ideal \(P\) of \(R\), and residue field \(\overline{K}=R/P\simeq{\mathbb F}_q\). For \(z\in K\) write \(|z|=q^{-\text{ord} z}\), where \(\text{ord} z\) is the valuation of \(z\). For a non-constant element \(f\in K[x_1,\ldots,x_m]\) one defines the \(p\)-adic Igusa local zeta function \(Z(s)\) associated to \(f\) and with trivial character as the \(p\)-adic integral \({Z(s):=\int_{R^m}|f(x)|^s|dx|}\), where \(|dx|\) is the Haar measure on \(K^m\) (normalized such that \(R^m\) has volume \(1\)), and where \(s\in{\mathbb C}\), \(\Re(s)>0\). More generally, for a multiplicative character \(\alpha:R^\times\to{\mathbb C}^\times\), for \(\pi\) a fixed uniformizing parameter of \(R\) and with \(\text{ac}(z):=z\pi^{-\text{ord} z}\), \(z\in K\), one defines the \(p\)-adic Igusa local zeta function \({Z(s,\alpha):=\int_{R^m}\alpha(\text{ac}(f(x)))|f(x)|^s|dx|}\). For a field \(k\) of characteristic zero consider the Grothendieck ring \(K_0(\text{Sch}_k)\) generated by symbols \([S]\) for algebraic varieties \(S\) over \(k\). In particular, let \({\mathbf L}=[{\mathbb A}_k^1]\). As an analogue to \(Z(s)\) one may construct a series \(Z_{\text{geom}}(s)\in K_0(\text{Sch}_k)[{\mathbf L}^{-1}][[{\mathbf L}^{-s}]]\) which for a fixed value of \(s\in{\mathbb N}\) can be interpreted as a Kontsevich integral. To extend this definition to the case of \(Z(s,\alpha)\) for a non-trivial character \(\alpha\), one considers a smooth connected separated scheme \(X\) of finite type over \(k\), and a multiplicative character \(\alpha\) of any finite subgroup of \(k^\times\). For a reduced subscheme \(W\) of \(X\) and a morphism \(f:X\to{\mathbb A}^1_k\), one defines the motivic Igusa function \({\int_W(f^s,\alpha)\in K_0({\mathcal M})[[{\mathbf L}^{-s}]]}\) where \(K_0({\mathcal M})\) is the Grothendieck ring of Chow motives and \(\mathbf L\) is the Lefschetz motive. It is shown that a motivic Igusa function is rational, and in case \(W=X={\mathbb A}_k^m\) and \(f\) a homogeneous polynomial, \({\int_X(f^s,\alpha)}\), \(\alpha\) trivial, satisfies a functional equation of the form \({\left(\int_Xf^s\right)^{\vee}={\mathbf L}^{-rs}\int_Xf^s\in K_0({\mathcal M}_k)[{\mathbf L}^s,{\mathbf L}^{-s}]_{\text{loc}}}\), where \({\int_Xf^s=\int_X(f^s,\text{triv})}\). For general \(\alpha\) one should replace the Grothendieck group of Chow motives by the Grothendieck group of Voevodsky's triangulated category of geometrical motives. One may relate motivic Igusa functions to nearby cycles by considering \({{{\mathbf L}^m}\over{1-{\mathbf L}}}\lim_{s\to -\infty}\int_{\{x\}}(f^s,\alpha)\), where \(x\) is a closed point of the fiber \(f^{-1}(0)\). Igusa zeta function; functional equation; vanishing cycles; nearby cycles; Kontsevich integral; motivic Igusa function J. Denef and F. Loeser, \textit{Motivic Igusa zeta functions}, \textit{J. Algebraic Geom.}\textbf{7} (1998) 505 [math/9803040]. Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Motivic cohomology; motivic homotopy theory, Zeta functions and \(L\)-functions Motivic Igusa zeta 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 number of solutions of polynomial systems appearing in geometry, robotics and vision is not given by the Bezout numbers, but is often much smaller because there are special phenomena as the presence of a conic called umbilic at infinity. (This phenomenon is a multidimensional analogue of the fact that two circles have two points in common in the affine plane and not four, because they have the cyclic points in common at infinity.) Note that this phenomenon is not only related to the particular Newton diagram of the equations considered but also to the particular arithmetic properties of their coefficients. -- This is a basic and important fact in polynomial system solving since many applications arise from this context.
Techniques to deal with this situation are developed in this paper. Several examples are studied. number of solutions of polynomial systems; robotics; vision Mourrain, B., Enumeration problems in geometry, robotics and vision, (Algorithms in algebraic geometry and applications, Progr. math., vol. 143, (1996), Birkhäuser Basel), 285-306 Enumerative problems (combinatorial problems) in algebraic geometry, Computer graphics; computational geometry (digital and algorithmic aspects), Kinematics of mechanisms and robots, Polynomial rings and ideals; rings of integer-valued polynomials, Numerical computation of solutions to systems of equations Enumeration problems in geometry, robotics and 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 In this note, the author proves that for a very ample \((2k-1)\)-spanned line bundle \(L\) on a smooth projective surface, the so-called Severi variety \(\tilde{V}_k(L)\) parameterizing all irreducible \(k\)-nodal curves \(D\in |L|\) is irreducible. Followed from the spannedness \(h^0(L)\geq 3k\), so that \(\dim \tilde{V}_k(L)=h^0(L)-k-1\) if \(h^0(L)>3k\) and \(\tilde{V}_k(L)=\emptyset\) otherwise. This result improves the same statement in [\textit{M. Kemeny}, Bull. London Math. Soc. 43, No. 1, 159--174 (2013; Zbl 1032.14005)] proved for \((3k-1)\)-very ample line bundles and is along the result in [\textit{C. Ciliberto} and \textit{Th. Dedieu}, Proc. Am. Math. Soc. 147, No. 10, 4233--4244 (2019; Zbl 1423.14192)].
Reviewer's remark: In the definition of \(k\)-spanned (\(k\)-very ample) line bundles, the rôle of \(k\) is missing. \(K3\) surface; Severi variety; nodal curve; Hilbert scheme of nodal curves Families, moduli of curves (algebraic), Surfaces and higher-dimensional varieties, Divisors, linear systems, invertible sheaves On the irreducibility of the Severi variety of nodal curves in a smooth surface | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(G_ i(X_ 1,...,X_ n)\) (1\(\leq i\leq s)\) be integral, nonlinear forms. Let p be prime and let Q be a box in \({\mathbb{R}}^ n\), with sides aligned with the coordinate axes. The author considers N(G,Q), defined as the number of integral solutions in Q of the simultaneous congruences \(G_ i\equiv 0 (mod p)\). In order to do this he assumes a condition \(P_ G(p):\) ``The highest degree part of \(\sum a_ iG_ i\) is non- singular (mod P) for all non-zero s-tuples \((a_ 1,...,a_ s)\) in \({\mathbb{F}}_ p\). Moreover the variety over \({\mathbb{F}}_ p\) defined by G is of codimension s.''
Under the assumption \(P_ G(p)\) it is shown that \(N(G,Q) \sim vol(Q)p^{-s},\) providing that both \(vol(Q)/p^{n/2+s},\) and the sides of the box Q, tend to infinity. If Q is a cube, and the variety involved is non-singular, the exponent \(n/2+s\) may be reduced by an amount \((n- 2s)/(2n-2).\)
The author also investigates the condition \(P_ G(p)\) and shows that it holds fairly frequently (in a precise sense) for \(s=2\), and rather rarely for \(s\geq 3\). Finally, he gives estimates for the frequency of integer solutions of \(G=0.\)
For example, if \(s=2\) then, apart from certain exceptional pairs of forms G, one has \(O(B^{n-4+12/(n+2)})\) integer solutions of \(G=0\) in a cube of side B, providing that the variety \(G=0\) is non-singular over \({\mathbb{C}}.\)
Estimates for exponential sums, which follow from Deligne's Riemann Hypothesis for varieties over finite fields, play a crucial role in the proofs. varieties; finite fields; small solutions of congruences; systems of forms; points in boxes M. Fujiwara,Distribution of rational points on varieties over finite fields, Mathematika35 (1988), 155--171. Forms of degree higher than two, Finite ground fields in algebraic geometry, Quadratic forms over general fields, Diophantine equations in many variables, Trigonometric and exponential sums (general theory) Distribution of rational points on varieties 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 Wenn auf einer algebraischen Curve ein \(\alpha\)-fach unendliches System von Punktguppen durch eine lineare Curvenschar ausgeschnitten wird, so kann man von den \(\mu\) Punkten einer Gruppe \(\mu-\alpha\) vermöge algebraischer Gleichungen durch die Gleichungen des Abel'schen Theorems ersetzen lassen. Die Resultate sind, wenn auch in etwas anderer Form, bekannt. Auf Seite 201 der Note ist in der sechsten Zeile ``greater'' in ``less'' zu verbessern. Linear systems; Abel's theorem Jacobians, Prym varieties On Abel's theorem. | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let consider the following question:
Fix integers \(1\leq d_1\leq d_2\leq \dots\leq d_n\) and let \({\mathcal H}\) be the Hilbert function of some finite set of distinct points in \({\mathbb P}^n\). Do there exist finite sets of distinct, reduced points \({\mathbb X},{\mathbb Y}\), such that \((i) {\mathbb X}\subset {\mathbb Y}\); (ii) the Hilbert function of \({\mathbb X}\) is \({\mathcal H}\) and (iii) \({\mathbb Y}\) is a complete intersection of type \( \{d_1, d_2, \dots, d_n\}\).
In this paper the author conjectures that it is enough to solve the above question for a class of complete intersections, namely rectangular complete intersections, and prove it for some special cases in \({\mathbb P}^2 \) and \({\mathbb P}^3\). A rectangular complete intersection of type \( \{d_1, d_2, \dots, d_n\}\), with \(d_1\geq 2\) is the subset \({\mathbb Y}\subset {\mathbb P}^n\) of \(d_1 d_2 \dots d_n\) distinct points with integer coordinates:
\[
\{ [1:b_1:\dots:b_n]/ b_i\in {\mathbb Z}, 0\leq b_1\leq d_n-1, \dots, 0\leq b_n\leq d_1-1\}.
\]
Finally as an application the author gives a family of points which has the Cayley-Bacharach Property. Hilbert-function; O-sequences; regular sequences, Cayley-Bacharach property Cooper, S. M., Subsets of complete intersections and the EGH conjecture, (Progress in Commutative Algebra 1, (2012), de Gruyter Berlin), 167-198 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Complete intersections, Linkage, complete intersections and determinantal ideals Subsets of complete intersections and the EGH conjecture | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems In the famous article [Compos. Math. 110, No. 1, 65--126 (1998; Zbl 0894.18005)] \textit{E. Getzler} and \textit{M. Kapranov} studied operadic type structures related to the moduli space of algebraic curve. Intuitively, algebraic curves with marked points can be glued along the marked points generating operations on the moduli spaces. However, when considering arbitrary genuses of curves, the classical operadic picture, in which operations are labeled by trees, is replaced by operation labeled instead by graphs. They call this operadic structure modular. Moreover, moduli of curves with marked points do have typically (for instance when considering genus \(0\) curves) an extra cyclic symmetry obtained by permuting the punctures. In the same paper, they show how to construct a Feynman transform on the category of dg-modular operads and how to compute its Euler characteristic in terms of the Wick's theorem, hence highlighting the relation of this operad with mathematical physics.
In this paper the authors present a generalization of these results for curves with marked points \(k-\log\) canonically embedded, meaning admitting a projective embedding by a complete linear system. The study of log canonical models for curves has been central in the study of moduli spaces of curves and for its relationships to the Minimal Model program. There are three results presented: first, they show that for \(k\geq 5\) the moduli spaces of \(k-\log\) canonically embedded curves assemble together in a modular operad in Deligne-Mumford stacks. Second, they show that for \(k\geq 1\) the moduli spaces of \(k-\log\) canonically embedded curves of genus \(0\) assemble together in a cyclic operad in schemes. Third, they show that for \(k\geq 2\) the moduli spaces of \(k-\log\) canonically embedded curves assemble together in a stable cyclic operad in Deligne-Mumford stacks. In order to prove these results, they construct morphisms on these moduli spaces corresponding to the gluing of two embedded curves and to the gluing of two points together on the same embedded curve.
The proofs of these statements appear correct. Would be interesting, as a follow up work, to understand weather the construction of Getzler and Kapranov of the Feynman transform could be generalized to this setting. modular operad; log-canonical Hilbert scheme Families, moduli of curves (algebraic), , Parametrization (Chow and Hilbert schemes) Modular operads of embedded curves | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems [For the entire collection see Zbl 0702.00008.]
This article is a survey recalling some properties of Dirichlet series associated with a polynomial. Most of the Dirichlet series we will study do not have a functional equation. We will be interested in their analytic continuation to the whole complex plane, their poles, their residues at simple poles and their values at negative integers. survey; Riemann zeta-function; Hurwitz zeta-function; Barnes zeta function; Shintani zeta-function; Mellin zeta-function; Dirichlet series associated with a polynomial Cassou-Noguès, P.: Dirichlet series associated with a polynomial. Springer proc. Phys. 47, 244-252 (1990) Other Dirichlet series and zeta functions, Zeta functions and \(L\)-functions of number fields, Zeta functions and \(L\)-functions, Toric varieties, Newton polyhedra, Okounkov bodies, Representations of entire functions of one complex variable by series and integrals, Analytic continuation of functions of one complex variable Dirichlet series associated with a polynomial | 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 Fix an algebrically closed field \(k\) of characteristic zero. Let \(R=k[x_0,x_1,y_0,y_1]\) be the \(\mathbb{N}^2-\)graded polynomial ring with \(\deg(x_i)=(1,0)\) for \(i=0,1\) and \(\deg(y_i)=(0,1)\) for \(i=0,1\). The ring \(R\) is the coordinate ring of \(\mathbb{P}^1 \times \mathbb{P}^1\). Consider now a set of points \(X=\{P_1, \dots, P_s\} \subset \mathbb{P}^1 \times \mathbb{P}^1\) and fix positive integers \(m_1, \dots, m_s\). In this paper the authors study some of the properties of the scheme \(Z=m_1P_1+\cdots m_sP_s\) of fat points. In particular, the main theorem of the paper shows how to compute the degree of a separator of \(P_i\) of multiplicity \(m_i\) directly from the combinatorics of the scheme \(Z\), provided \(Z\) is ACM.
Before to state such theorem we need to recall some definition.
Let \(X\) be a set of \(s\) distinct points in \(\mathbb{P}^1 \times \mathbb{P}^1\). Let \(\pi_1:\mathbb{P}^1 \times \mathbb{P}^1 \rightarrow \mathbb{P}^1\) denote the projection morphism defined by \(P \times Q \mapsto P\). Let \(\pi_2:\mathbb{P}^1 \times \mathbb{P}^1 \rightarrow \mathbb{P}^1\) be the other projection morphism. The set \(\pi_1(X) = \{P_1,\ldots,P_a\}\) is the set of \(a \leq s\) distinct first coordinates that appear in \(X\). Similarly, \(\pi_2(X) = \{Q_1,\ldots,Q_b\}\) is the set of \(b \leq s\) distinct second coordinates. For \(i = 1,\ldots,a\), let \(L_{P_i}\) be the degree \((1,0)\) form that vanishes at all the points with first coordinate \(P_i\). Similarly, for \(j = 1,\ldots,b\), let \(L_{Q_j}\) denote the degree \((0,1)\) form that vanishes at points with second coordinate \(Q_j\).
Let \(D:=\{(x,y) ~|~ 1 \leq x \leq a, 1 \leq y \leq b\}.\) If \(P \in X\), then \(I_{P} = (L_{R_i},L_{Q_j})\) for some \((i,j) \in D.\) So, we can write each point \(P \in X\) as \(P_i \times Q_j\) for some \((i,j)\in D\). \vskip0.3cm Definition. Suppose that \(X\) is a set of distinct points in \(\mathbb{P}^1 \times \mathbb{P}^1\) and \(|\pi_1(X)| = a\) and \(|\pi_2(X)| = b\). Let \(I_{P_i \times Q_j} = (L_{R_i},L_{Q_j})\) denote the ideal associated to the point \(P_i \times Q_j \in X\). For each \((i,j) \in D\), let \(m_{ij}\) be a positive integer if \(P_i \times Q_j \in X\), otherwise, let \(m_{ij} = 0\). Then we denote by \(Z\) the subscheme of \(\mathbb{P}^1 \times \mathbb{P}^1\) defined by the saturated bihomogeneous ideal
\[
Iz = \bigcap_{(i,j) \in D} I_{P_i\times Q_j}^{m_{ij}}
\]
We say \(Z\) is a \textit{fat point scheme} or \textit{a set of fat points} of \(\mathbb{P}^1 \times \mathbb{P}^1\). The integer \(m_{ij}\) is called the \textit{multiplicity} of the point \(P_i \times Q_j\). The \textit{support} of \(Z\), written \(\mathrm{supp}(Z)\), is the set of points \(X\).
\vskip0.3cm Definition. A fat point scheme is said to be \textit{arithmetically Cohen-Macaulay} (ACM for short) if the associated coordinate ring is Cohen-Macaulay.
\vskip0.3cm Definition. Let \(Z = m_1P_1 + \cdots + m_iP_i + \cdots + m_sP_s\) be a set of fat points in \(\mathbb{P}^1 \times \mathbb{P}^1\). We say that \(F\) is a \textit{separator} of the point \(P_i\) of multiplicity \(m_i\) if \(F \in I_{P_i}^{m_i-1} \setminus I_{P_i}^{m_i}\) and \(F \in I_{P_j}^{m_j}\) for all \(j \neq i\). \vskip0.3cm If we let \(Z' = m_1P_1 + \cdots + (m_i-1)P_i + \cdots + m_sP_s\), then a separator of the point \(P_i\) of multiplicity \(m_i\) is also an element of \(F \in I_{Z'}\setminus I_Z\). The set of minimal separators are defined in terms of the ideals \(I_{Z'}\) and \(I_Z\).
\vskip0.3cm Definition. A set \(\{F_1,\ldots,F_p\}\) is a \textit{set of minimal separators} of \(P_i\) of multiplicity \(m_i\) if \(I_{Z'}/I_Z = (\overline{F}_1,\ldots,\overline{F}_p)\), and there does not exist a set \(\{G_1,\ldots,G_q\}\) with \(q < p\) such that \(I_{Z'}/I_Z = (\overline{G}_1,\ldots,\overline{G}_q)\).
\vskip0.3cm Definition. The \textit{degree of the minimal separators} of \(P_i\) of multiplicity \(m_i\), denoted \(\deg_Z(P_i)\), is the tuple
\[
\deg_Z(P_i) = (\deg F_1,\ldots,\deg F_p)
\]
\vskip0.1cm where \(\deg F_i \in \mathbb N^2\) and \(F_1,\ldots,F_p\) is any set of minimal separators of \(P_i\) of multiplicity \(m_i\). \vskip0.3cm For a general fat point scheme \(Z\subset \mathbb{P}^1 \times \mathbb{P}^1\), there is no known formula for \(p=|\deg(P)|\). However, if \(Z\) is ACM, then \(p\) can be computed. As a matter of fact, the main result of this paper is a formula to compute the degree of a minimal separator for each fat point in an ACM fat point scheme in \(\mathbb{P}^1 \times \mathbb{P}^1\).
{ Theorem} Let \(Z \subset \mathbb{P}^1 \times \mathbb{P}^1\) be an ACM set of fat points with \(a=|\pi_1(\mathrm{supp}(Z))|\) and \(b=|\pi_2(\mathrm{supp}(Z))|\). Suppose \(P_i \times Q_j \in \mathrm{supp}(Z)\) is a point with multiplicity \(m_{ij}\). Set
\[
a_{\ell} = \sum_{s=1}^a (m_{sj} - \ell)_+
\]
and
\[
b_{\ell} = \sum_{p=1}^b (m_{ip} - \ell)_+
\]
for \(\ell = 0,\ldots,m_{ij}-1\), where \((m-t)_+=\max\{0, m-t\}\). Then
\[
\deg_Z(P_i \times Q_j) = \{(a_{m_{ij}-1-\ell}-1,b_{\ell}-1) ~|~ \ell = 0,\ldots,m_{ij}-1\}.
\]
separators; fat points; Cohen-Macaulay; Hilbert function Guardo, Elena; Van Tuyl, Adam, Separators of arithmetically Cohen-Macaulay fat points in \(\mathbf{P}^1\times\mathbf{P}^1\), J. Commut. Algebra, 4, 2, 255-268, (2012) Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Syzygies, resolutions, complexes and commutative rings, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Separators of arithmetically Cohen-Macaulay fat points \(\mathbb P^1\times \mathbb P^1\) | 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 F. Klein studied simple singularities, classifying them as quotients of $\mathbb{C}^2$ by the action of a finite subgroup $\Gamma \subseteq \mathrm{SU}_2$. P. Du Val showed that the exceptional divisors of the minimal resolution of the isolated singularity of such a quotient form an arrangement of projective lines whose dual graph is a simply-laced Dynkin diagram $\Delta(\Gamma)$; thus the quotient $\mathbb{C}^2/\Gamma$ is called a simple singularity of type $\Delta(\Gamma)$. P. Slodowy then extended the definition of a simple singularity to the non simply-laced types by adding a second finite subgroup $\Gamma' \subseteq \mathrm{SU}_2$ such that $\Gamma'\supseteq \Gamma$ as a normal subgroup; $\Gamma'/\Gamma =\Omega$ acts on $\mathbb{C}^2/\Gamma$ and this action can be lifted to the minimal resolution of the singularity, inducing an action on the exceptional divisors, which corresponds to a group of automorphisms of the Dynkin diagram of $\mathbb{C}^2/\Gamma$. \par A deformation of a simple singularity $(X_0,\Omega)$ is an $\Omega$-equivariant deformation of the singularity $X_0$ with a trivial action of the automorphism group $\Omega$ on the base space. Setting $\pi : X \rightarrow Y$ as a deformation of $X_0$, a deformation $\psi : X' \rightarrow Y'$ of $X_0$ is induced from $\pi$ by a morphism $\varphi : Y' \rightarrow Y$ if there exist a morphism $\Phi :X' \rightarrow Y'$ such that $\pi\circ \Phi= \varphi\circ\psi$, and given $X_0 \stackrel{i}{\hookrightarrow} X$ and $X_0\stackrel{j}{\hookrightarrow} X'$, $\Phi\circ j=i$. \par A semiuniversal deformation $\pi_0 : X \rightarrow Y$ of a simple singularity $(X_0,\Omega)$ is a deformation of $(X,\Omega)$ such that any other deformation $\psi : X' \rightarrow Y'$ of $(X,\Omega)$ is induced from $\pi_0$ by an $\Omega$-equivariant morphism $\varphi : Y' \rightarrow Y$ with a uniquely determined differential $d_{y'} \varphi :T_{y'} Y' \rightarrow T_y Y$. The quotient of a semiuniversal deformation of a simple singularity of inhomogeneous type $B_r$ ($r \geq 2$), $C_r$ ($r \geq 3$), $F_4$ or $G_2$ by the natural symmetry of the associated Dynkin diagram is a deformation of a simple singularity of homogeneous type $X=D_s$, $E_6$ or $E_7$. \par Letting $\alpha : X_\Gamma \rightarrow \mathfrak{h}/W$ to be the semiuniversal deformation of a simple singularity of type $\Delta(\Gamma)=A_{2r-1}$ ($r \geq 2$), $D_{r}$ ($r \geq 4$) or $E_6$ obtained by the construction of H. Cassens and P. Slodowy, and $\mathfrak{h}$ and $W$ being the Cartan subalgebra and the associated Weyl group of the simple Lie algebra $\mathfrak{g}$ of the same type, respectively, they showed that $\Omega$ of the Dynkin diagram of $\mathfrak{g}$ acts on $X_\Gamma$ and $\mathfrak{h}/W$ such that $\alpha$ is $\Omega$-equivariant. Slodowy then showed that taking the restriction $\alpha^\Omega$ of $\alpha$ over the $\Omega$-fixed points of $\mathfrak{h}/W$ leads to a semiuniversal deformation of a simple singularity, which is inhomogeneous. As $\alpha$ is $\Omega$-equivariant, there is an action of $\Omega$ on every fiber of $\alpha^\Omega$, and the quotient leads to a new morphism $\overline{\alpha^\Omega}$, which is a non-semiuniversal deformation of a simple singularity of homogeneous type $\Delta(\Gamma')$. \par Let $\Gamma$ be a finite subgroup of $\mathrm{SU}_2$, $R$ its regular representation, $N$ its natural representation as a subgroup of $\mathrm{SU}_2$, and $\Delta(\Gamma)$ the associated Dynkin diagram. If $\Omega$ acting on $M(\Gamma)=(\mathrm{End}(R) \otimes N)^\Gamma$ is symplectic, then $\widetilde{\alpha}:X_{\Gamma}\times_{\mathfrak{h}/W}\mathfrak{h}\rightarrow \mathfrak{h}$ and $\alpha:X_{\Gamma}\rightarrow \mathfrak{h}/W$ can be made into $\Omega$-equivariant maps (Theorem 1.4, page 388): letting $M(\Gamma)$ to be the representation space of a McKay quiver built on a Dynkin diagram of type $A_{2r-1}$, $D_{r}$ or $E_6$, there exists a symplectic action of $\Omega=\Gamma'/\Gamma$ on $M(\Gamma)$, inducing the natural action on the singularity $\mathbb{C}^2/\Gamma$; this action then turns $\alpha$ into an $\Omega$-equivariant morphism. \par After A. Caradot shows that the morphism $\alpha^{\Omega} : X_{\Gamma, \Omega} \rightarrow (\mathfrak{h}/W)^{\Omega}$ is $\Omega$-invariant, it follows that $\Omega$ acts on each fiber of $\alpha^{\Omega}$, and the fibers can be quotiented. Furthermore it is known that $(\alpha^{\Omega})^{-1}(\overline{0}) = X_{\Gamma,0}=\mathbb{C}^2/\Gamma$, and thus $(\alpha^{\Omega})^{-1}(\overline{0})/\Omega=X_{\Gamma,0}/\Omega = (\mathbb{C}^2/\Gamma)/(\Gamma'/\Gamma) \cong \mathbb{C}^2/\Gamma'$, which is a simple singularity since $\Gamma'\subseteq \mathrm{SU}_2$ is finite. Thus, $\overline{\alpha^{\Omega}} : X_{\Gamma, \Omega} /\Omega \rightarrow (\mathfrak{h}/W)^{\Omega}$ is a deformation of the simple singularity $\mathbb{C}^2/\Gamma'$ of type $\Delta(\Gamma')$, where the deformation $\overline{\alpha^{\Omega}}$ is obtained through $\Delta(\Gamma)-\Delta(\Gamma,\Gamma')-\Delta(\Gamma')$-procedure. \par Caradot also studies the regularity of the fibers of $\overline{\alpha^{ \Omega}}$ (Theorem 2.3, page 390): assuming $\alpha^\Omega$ is the semiuniversal deformation of a simple singularity of inhomogeneous type $B_r$ ($r \geq 2$), $C_r$ ($r \geq 3$), $F_4$ or $G_2$, every fiber of the quotient $\overline{\alpha^\Omega}$ is singular. \par Finally, after stating a conjecture (Conjecture 3.1, page 396) that there exists a subset $\Theta$ of simple roots of the root system of type $\Delta(\Gamma')$ such that the Dynkin diagram associated to the singular configuration of any fiber of $\overline{\alpha^\Omega}$ is a subdiagram of the Dynkin diagram of type $\Delta(\Gamma')$ containing the vertices associated to $\Theta$, the author proves the conjecture for the types $A_3-B_2-D_4$, $A_5-B_3-D_5$, $D_4-C_3-D_6$, $D_4-G_2-E_6$, $D_4-G_2-E_7$, and $E_6-F_4-E_7$. (Theorem 3.2, page 397). deformations of simple singularities; simple root systems; simple singularities of inhomogeneous types; singular configurations Deformations of singularities, Root systems, Representation theory for linear algebraic groups Root systems and quotients of deformations of simple 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 In physics and mathematics literature there has been recent interest in a class of Riemann-Hilbert problems that are naturally suggested by the form of the wall-crossing formula in Donaldson-Thomas (DT) theory. These problems involve piecewise holomorphic maps from the complex plane to the group of automorphisms of a Poisson algebraic torus, with discontinuities along a collection of rays prescribed by the DT invariants. The authors consider the special case of a refined BPS structure satisfying certain conditions. The basic example is the one arising from the refined DT theory of the \(A_1\) quiver. The article proposes an explicit solution to the corresponding quantum Riemann-Hilbert problem in terms of products of modified gamma functions. The solutions are also written in adjoint form using a modified version of the Barnes double gamma function; the latter arises in expressions for the partition functions of supersymmetric gauge theories. Riemann-Hilbert problem; Donaldson-Thomas theory; Barnes double gamma function Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Boundary value problems in the complex plane, Quantum groups and related algebraic methods applied to problems in quantum theory, Other special functions A quantized Riemann-Hilbert problem in Donaldson-Thomas theory | 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 hard square and hard hexagon models can be obtained from the Ising model in a magnetic field \(H\) in the limit \(H\to\infty\) for the square and triangular lattices respectively, and thus it is natural to study the question of analyticity in these two models. However, unlike the Ising model at \(H = 0\), where both the square and triangular lattices have been exactly solved, the hard hexagon model is exactly solved by R. J. Baxter, whereas the hard square model is not. Thus, the comparison of these two models is the ideal place to study the relation of integrability to the analyticity properties of the free energy in the complex plane. In the paper the authors compare the integrable hard hexagon model with the non-integrable hard squares model by means of partition function roots and transfer matrix eigenvalues. Partition functions for toroidal, cylindrical, and free-free boundary conditions up to sizes \(40 \times 40\) and transfer matrices up to 30 sites are considered. It is shown that for all boundary conditions the hard squares roots lie in a bounded area of the complex fugacity plane along with the universal hard core line segment on the negative real fugacity axis. The density of roots on this line segment exhibits much greater structure than the corresponding density of hard hexagons. Moreover, the special point \(z=-1\) of hard squares is studied, where all eigenvalues have unit modulus. Some conjectures of the partition functions at \(z=-1\) are given. hard square model; hard hexagon model; partition function zeros; equimodular curves Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics, Relationships between algebraic curves and integrable systems, Eigenvalues, singular values, and eigenvectors, Moduli and deformations for ordinary differential equations (e.g., Knizhnik-Zamolodchikov equation), Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies, General theory of ordinary differential operators Integrability versus non-integrability: hard hexagons and hard squares compared | 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 proves a conjecture of Deligne for a smooth variety \(X\) over \(\mathbb F_p\) (which ideally should hold for any normal scheme of finite type over \(\mathbb F_p\)):
Given a number field \(E\), fix a non-Archimedean place \(\lambda\) of \(E\) which is relatively prime to \(p\). Then consider the irreducible lisse \(\bar E_\lambda\)-sheaves on \(X\) satisfying the following two conditions:
1. the determinant has finite order;
2. for every closed point \(x\in X\), the characteristic polynomial of the Frobenius \(F_x\) has coefficients in \(E\).
The paper proves that the set of isomorphism classes of the above \(\bar E_\lambda\)-sheaves is independent of \(\lambda\).
For \(X\) of dimension 1, this result has been proven by \textit{L. Lafforgue} as an instance of the Langlands conjecture for GL\((n)\) over function fields [Invent. Math. 147, No. 1, 1--241 (2002; Zbl 1038.11075)]. The author reduces the higher-dimensional case to the known one using a general method due to \textit{G. Wiesend} based on Hilbert irreducibility [J. Number Theory 121, No. 1, 118--131 (2006; Zbl 1120.14014)]. Deligne conjecture; lisse sheaf; \(\ell\)-adic representation; independence of \(\ell\); Langlands conjecture; local system; arithmetic scheme; Hilbert irreducibility; weakly motivic Drinfeld, V., On a conjecture of Deligne, Moscow Math. J., 12, 515-542, (2012) Finite ground fields in algebraic geometry, Varieties over global fields On a conjecture of Deligne | 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 theme of this paper is a trial of generalization of the work ``Zeta functions of simple algebras'' by \textit{R. Godement} and \textit{H. Jacquet} [Lect. Notes Math. 260 (1972; Zbl 0244.12011)] from a viewpoint of the theory of prehomogeneous vector spaces. The authors consider the case of a family of complex symmetric spaces \(G/H\) which are obtained as Zariski open subsets of a vector space. The family of symmetric spaces corresponds to the prehomogeneous vector spaces of commutative parabolic type. For example, in Godement and Jacquet's case, they considered \(GL_ n (\mathbb{C}) \simeq GL_ n (\mathbb{C}) \times GL_ n (\mathbb{C})/GL_ n (\mathbb{C})\). It is embedded in the space of \(n \times n\) complex matrices \(M_ n (\mathbb{C})\). The action \(GL_ n (\mathbb{C}) \times GL_ n (\mathbb{C})\) on \(M_ n (\mathbb{C})\) is a typical example of prehomogeneous vector space of commutative parabolic type. If \(\pi\) is a a generic representation of the principal spherical series of \(G\) which has a natural \(H\)-invariant generalized vector, we define \(C^ \infty\) and \(H\)-invariant coefficients of \(\pi\) and the zeta function associated to these coefficients. We obtain an explicit functional equation for this zeta function. symmetric spaces; prehomogeneous vector space of commutative parabolic type; zeta function Bopp, N.; Rubenthaler, H.: Fonction zêta associée à la série principale sphérique de certains espaces symétriques. Ann. sci. École norm. Sup. (4) 26, No. 6, 701-745 (1993) Homogeneous spaces and generalizations, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects) Zeta function associated to the spherical principal series of some symmetric 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 We adapt \textit{M. F. Atiyah}'s \(L^2\)-index theory [Astérisque 32-33, 43-72 (1976; Zbl 0323.58015)] to treat coherent sheaves on algebraic manifolds and use it as a tool to investigate certain questions posed by \textit{J. Kollár} [``Shafarevich maps and automorphic forms'' (1995; Zbl 0871.14015); chapter 18]. Let \(X\) be a connected projective algebraic compact complex manifold. We prove that, if \(L\) is a big and nef divisor on \(X\), such that the restriction of \(K_X+L\) to the general fiber of a Shafarevich map is effective, \(K_X+L\) is effective. Let \(X\) be a connected Kähler manifold such that some big cohomology class of type \((1,1)\) is in the image of \(H^2(\pi_1(X),\mathbb{R})\). We prove that \(\chi(X, K_X) \geq 0\). Furthermore, if \(\chi(X, K_X)\) is not \(0\), the universal covering space of \(X\) carries a non trivial \(L^2\) holomorphic form of maximal degree. If \(\chi(X, K_X)\) is zero, we prove that zero belongs to the spectrum of the Laplace-Beltrami operator on the middle degree forms, provided the fundamental group has subexponential growth. \(L^2\) index theorem; Nadel's vanishing theorem; adjoint bundle; adjoint linear systems; coherent sheaves; Shafarevich map; fundamental group DOI: 10.5802/aif.1670 Divisors, linear systems, invertible sheaves, Homotopy theory and fundamental groups in algebraic geometry, Sheaves and cohomology of sections of holomorphic vector bundles, general results, Vanishing theorems, Coverings in algebraic geometry, Transcendental methods of algebraic geometry (complex-analytic aspects) \(L^2\) adjoint linear 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 A characteristic condition is given on a zero-dimensional differentiable 0-sequence \(H=(h_ i)_{i\geq 0}\), \(h_ 1\leq 3\), in order to be the Hilbert function of a generic plane section of a reduced irreducible curve of \({\mathbb{P}}^ 3\), hence of points of \({\mathbb{P}}^ 2\) with the uniform position property. In this way an answer is given to some question stated by \textit{J. Harris} in 1982. The result is obtained by constructing a smooth irreducible arithmetically Cohen-Macaulay curve in \({\mathbb{P}}^ 3\) whose generic plane section has an assigned Hilbert function satisfying that condition. Hilbert function of a generic plane section of a reduced irreducible curve; uniform position; arithmetically Cohen-Macaulay curve; space curve Maggioni, R.; Ragusa, A., The Hilbert function of generic plane sections of curves of \(\mathbf{P}^3\), Invent. Math., 91, 2, 253-258, (1988) Special algebraic curves and curves of low genus, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.), Projective techniques in algebraic geometry The Hilbert function of generic plane sections of curves of \({\mathbb P}^ 3\) | 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 Cayley Bacharach-type theorems assert that an algebraic hypersurface going through a suitable part of a zero-dimensional complete intersection of algebraic hypersurfaces, does go through the whole complete intersection. Best known example is the classical theorem asserting that a plane cubic curve through eight intersection points of two other plane cubics needs to go through the ninth one. In the present paper the Cayley-Bacharach property is related to adjoint systems and, as a consequence, a Cayley-Bacharach theorem true in arbitrary complex smooth projective algebraic varieties is proved. Calay-Bacharach-type theorems; adjoint systems Tan S L. Cayley-Bacharach property of an algebraic variety and Fujita's conjecture. J Algebraic Geometry, 2000, 9: 201--222 Adjunction problems, Hypersurfaces and algebraic geometry, Complete intersections Cayley-Bacharach property of an algebraic variety and Fujita's conjecture | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The theory of \(p\)-adic uniformization is a theory describing certain algebraic varieties as quotients of a \(p\)-adic upper half space, or more generally, of the analytic space of a formal scheme defined over the ring of integers of a finite extension \(K\) of the \(p\)-adic numbers (or a local non Archimedean field). In this theory one attempts to replace the classical complex upper half space by a \(p\)-adic analogue: Optimally, by the \(n\) dimensional projective space over a complete algebraic closure of the \(p\)-adic numbers from which the \(K\) rational hyper-planes are removed. The paper begins with a short historical survey and describes the state of the art. The theory originated with \textit{J. Tate} followed by \textit{D. Mumford} [Compos. Math. 24, 129--174 (1972; Zbl 0228.14011)] who considered the case of curves over a complete local ring with reduction consisting of rational curves. It was then expanded by \textit{I. V. Cherednik} [Math. USSR, Sb. 29 (1976), 55--78 (1978); translation from Mat. Sb., Nov. Ser. 100(142), 59--88 (1976; Zbl 0329.14015)] who considered the case of a Shimura curve associated to a quaternion algebra over the rational numbers that is ramified at \(p\). Its special fiber is of the form considered by Mumford. The theory was much expanded by \textit{V. G. Drinfel'd} [Funct. Anal. Appl. 10, 107--115 (1976); translation from Funkts. Anal. Prilozh. 10, No.~2, 29--40 (1976; Zbl 0346.14010)] and \textit{J. F. Boutot} and \textit{H. Carayol} [in: Courbes modulaires et courbes de Shimura, C. R. Sémin., Orsay 1987/88, Astérisque 196/197, 45--158 (1991; Zbl 0781.14010)], who interpreted the covering space as a moduli space for a certain class of \(p\)-divisible groups. Drinfel'd also pointed out to deep connections between \(p\)-adic uniformization and Langlands' conjectures. The author explains the appearance of \(p\)-adic uniformization in the theory of Shimura curves, and the extensions found by \textit{M. Rapoport} and \textit{Th. Zink} [``Period spaces for \(p\)-divisible groups'', Ann. Math. Stud. 141 (1996; Zbl 0873.14039)] who showed, e.g., that the formal neighborhood of the supersingular locus in a typical moduli problem of abelian varieties is parameterized by a formal scheme. To explain this result there is a discussion of moduli problems of isogeny class of \(p\)-divisible groups with extra structure. The particular case considered by Drinfeld (loc. cit.) is discussed in detail, including the connection to the Bruhat-Tits building. A merit of this case is that one obtains an explicit description of the uniformizing formal scheme.
In the last sections of the paper the author discusses the theory of Shimura curves and the application of \(p\)-adic uniformization to it. The link to the case of moduli of \(p\)-divisible groups is that when \(p\) ramifies in the quaternion algebra all the abelian varieties with quaternionic multiplication (and suitable extra structure) over an algebraically closed field of characteristic \(p\) form a single isogeny class. \(p\)-adic uniformization; Shimura variety; moduli space; \(p\)-divisible group; formal group; Langlands conjectures; Shimura curves; Bruhat-Tits building; uniformizing formal scheme Modular and Shimura varieties, Rigid analytic geometry, Arithmetic aspects of modular and Shimura varieties, Formal groups, \(p\)-divisible groups \(p\)-adic uniformization of Shimura varieties | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(X\) be a compact symplectic manifold acted on by a compact Lie group \(K\) with a moment map \(\mu:X\to\text{Lie} (K)^*\). In [\textit{F. C. Kirwan}, Cohomology of quotients in symplectic and algebraic geometry, Math. Notes 31, Princeton (1984; Zbl 0553.14020)] it is shown that \(X\) has a natural stratification such that the associated spectral sequence for cohomology with rational coefficients degenerates at the \(E_2\) term. The authors of this note observe that the same proof shows that there is an integer \(N\), depending on the action of \(K\) on \(X\), such that the spectral sequence of the stratification degenerates for any complex oriented cohomology theory in which \(N\) is inverted. They pose two interesting problems and discuss their consequences. The first is whether the same result is still true if the assumption of complex orientability is removed. The second applies to the algebraic case when \(X\) is a projective variety acted on linearly by a reductive group \(G\) (which over the complex numbers can be thought of as the complexification of \(K)\), and asks how the degeneration of the spectral sequence for \(l\)-adic cohomology relates to the action of Galois groups. symplectic manifold; Hamiltonian group action; stratification; spectral sequence for cohomology Compact Lie groups of differentiable transformations, \(p\)-adic cohomology, crystalline cohomology, Generalized (extraordinary) homology and cohomology theories in algebraic topology, Spectral sequences in algebraic topology The naturality in Kirwan's decomposition | 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 result of the paper is the following: given a finite group \(G\) and a Mackey functor \(M\), the authors construct a functor from \(G\)-spaces to topological abelian groups, \(X\mapsto F^G(X^+,M)\), such that \(\pi_q (F^G(X^+,M))\) is naturally isomorphic to the Bredon homology group \(H^G_q(X;M)\) (for \(X\) having the homotopy type of a \(G\)-\(CW\)-complex). This generalizes some of the results of \textit{P. Lima-Filho} [Math. Z. 224, No.~4, 567--601 (1997; Zbl 0882.55008)] and \textit{P. F. dos Santos} [J. Pure Appl. Algebra 183, No.~1--3, 299--312 (2003; Zbl 1032.55010)] corresponding to the case where \(M\) is obtained from the \(\mathbb{Z}[G]\)-module \(\mathbb{Z}\), and from a general \(\mathbb{Z}[G]\), respectively. A similar result can be found in \textit{Z. Nie} [Bull. Lond. Math. Soc. 39, No.~3, 499--508 (2007; Zbl 1197.55002)].
As the authors explain, their result can be considered as a version for Bredon homology of the classical Dold-Thom Theorem: for a topological space \(X\) with the homotopy type of a \(CW\)-complex there is a natural isomorphism between \(\pi_q(\mathbb{Z}\cdot X)\) and \(H_q(X;\mathbb{Z})\), where \(\mathbb{Z}\cdot X\) denotes the free abelian group on \(X\) endowed with an appropriate topology.
In Section~3, a transfer map \(t_p: F^G(X^+,M)\to F^G(E^+, M)\) associated to an \(G\)-equivariant covering map \(p:E\to X\) is defined. In the last section, a formula for the composite of the transfer and the projection of a \(G\)-equivariant map is proved. Also, a characterization of the Mackey functors for which this formula is analogous to the nonequivariant one is given. coefficient systems; equivariant homology DOI: 10.1080/00927870701716074 Equivariant homology and cohomology in algebraic topology, Equivariant homotopy theory in algebraic topology, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies) Topological Abelian groups and equivariant homology | 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 Erläuterungen und Zusätze zu einer Reihe von Aufsätzen des Verfassers, welche theils im Cambridge and Dublin J. (V. VI-VIII) theils im Quart. J. V. VI. veröffentlicht sind. Dieselben beziehen sich auf folgende Probleme:
1) ``Integralization'' einer gegebenen rationalen Function von einer algebraischen Function \(x\) \(n\)ter Ordnung d. h. die Darstellung derselben als ganze Function \((n-1)^{\text{ten}}\) Grades von \(x\) mit Coefficienten, welche aus den Coefficienten der Gleichung \(f(x, u)=0\) rational zusammengesetzt sind. Insbesondere werden die Differentialquotienten \(\frac{dx}{du}, \frac{d^2 x}{du^2}\) etc. in dieser Form gegeben.
2) Die Ermittelung von Werthsystemen, welche ein unbestimmtes Gleichungs-System erfüllen (multilinear solution). Ausser einigen Bemerkungen über die Auflösung von solchen cubischen Gleichungen ist folgender Satz über ein System von \(n\) homogenen quadratischen Gleichungen angeführt: Wenn die Zahl der darin auftretenden Unbekannten nicht unter \(E\left( \frac {n^2+3}{2} \right)\) herabsinkt, -- dieses Symbol bezeichnet die grösste in \(\frac {n^2+3}{2}\) enthaltene ganze Zahl -- so kann ein den gegebenen Gleichungen genügendes Werthsystem durch Auflösung von nur quadratischen und biquadratischen Gleichungen gefunden werden. -- Den Beweis dieses Satzes hat Hr. Cockle in Cambr. D. J. VIII. p. 55 gegeben.
3) Bildung von Differential-Gleichungen für eine gegebene algebraische Function \(z\) von zwei unabhängigen Veränderlichen \(x\), \(y\), welche vom Verfasser als ``partial differential resolvents'' bezeichnet werden. ``Coresolvents'' heissen jede solche Differentialgleichung und die Gleichung \(f(z, x, y)=0\), zusammen aufgefasst. -- Solche Differentialgleichungen ergeben sich aber nach dem unter 1) auseinandergesetzten Principe. Coresolvents; integralization; rational function; entire function; algebraic function; differential quotient; multilinear solution; system of \(n\) homogeneous quadratic equations; unknowns; differential equations Algebraic functions and function fields in algebraic geometry, Real rational functions, Equations in general fields, Multilinear algebra, tensor calculus, Implicit ordinary differential equations, differential-algebraic equations, Research exposition (monographs, survey articles) pertaining to real functions, Continuity and differentiation questions Third Chapter on Coresolvents. | 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(x) be a rational function field in one variable. The author shows that the set \(\Phi_ n(k)\) of all fields F between k and k(x) with [k(x) : F]\(=n\) is a nonsingular irreducible affine algebraic variety of dimension 2n-2. The proof is based on properties of the quadratic ``Bezout form'' associated with any rational function. If \(n\geq 3\) and if k is algebraically closed of characteristic zero, then the quotient space of all ``non-polynomial'' subfields modulo the group \(PGL_ 2k\) is a quasi-affine variety of dimension 2n-5. variety of subfields; stable points; automorphism action; moduli space; rational function field; affine algebraic variety; Bezout form Arithmetic theory of algebraic function fields, Transcendental field extensions, Group actions on varieties or schemes (quotients), Algebraic functions and function fields in algebraic geometry The variety of subfields of k(x) | 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 major problem in computational number theory is to compute the zeta function of a curve $X$ over a finite field $\mathbb{F}_{p^a}$. The paper under review extends previously known Kedlaya type algorithms from [\textit{P. Gaudry} and \textit{N. Gürel}, Exp. Math. 12, No. 4, 395--402 (2003; Zbl 1076.11038)] (see also \textit{C. Gonçalves} [Algorithmic arithmetic, geometry, and coding theory, Contemp. Math. 637, 145--172 (2015; Zbl 1343.14016)]) in the case of superelliptic curves $y^r=f(x)$ to the case of singular superelliptic curves. A big part of the work is to control the $p$-adic precision necessary to work out the computations. In this case, the complexity is $O(a^{3+\epsilon} g^{5+\varepsilon})$ where $g$ is the genus of the curve ($p$ begin looked at as a constant). This is slightly faster than the more general algorithm of \textit{J. Tuitman} [Finite Fields Appl. 45, 301--322 (2017; Zbl 1402.11097)] but the author honestly asserts that the implementation of the later is faster in practice. zeta function; \(p\)-adic algorithm; superelliptic curves Singularities of curves, local rings, Curves over finite and local fields, Computational aspects of algebraic curves, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture) \(p\)-adic point counting on singular superelliptic 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 classical notion of normalization is not sufficient to address many natural questions one encounters in the theory of complex spaces and in algebraic geometry. To remedy this, two notions related to normalization were introduced about 50 years ago, namely, \textit{weak normalization} and \textit{seminormalization}. However, these by now standard concepts, are not adequate for problems in real algebraic geometry. The main difficulty is to capture the behavior of algebraic varieties defined over \(\mathbb{R}\) near their real loci. Therefore it is natural to focus on affine real algebraic sets. The set of central points of a real algebraic set X, denoted by \( \operatorname{Cent}X\), is the closure in the Euclidean topology of the set of non-singular points of \(X\). In general, \(\operatorname{Cent}X\) is different from \( X\). The authors of the paper under review construct the weak normalization and seminormalization of \(X\) relative to \(\operatorname{Cent}X\). The idea is as follows. The ring \(\mathcal{P}(X)\) of polynomial functions on \(X\) is a subring of the ring \(\mathcal{K}^{0}(\operatorname{Cent}X)\) of rational functions on \(X\) admitting continuous extensions to \(\operatorname{Cent}X\). The integral closure of \(\mathcal{P}(X)\) in \(\mathcal{K}^{0}(\operatorname{Cent}X)\) is a finite module over \(\mathcal{P}(X)\), and therefore it coincides with the polynomial ring of a real algebraic set, denoted \(X^{w_{c}}\). Moreover, there is a natural finite birational polynomial morphism \(\pi ^{w_{c}}:X^{w_{c}}\rightarrow X\) which induces a homeomorphism for the Euclidean topology between the central loci. This morphism is called the \textit{weak normalization of} X \textit{relative to} \(\operatorname{Cent}X\). One of the main results is the following universal property (see Theorem 4.8): Let \(X\) be a real algebraic set. Let \(Y\) be a real algebraic set equipped with a finite birational morphism \(\pi :Y\rightarrow X\). Then \(\pi \) induces a bijection from \(\operatorname{Cent}Y\) onto \(\operatorname{Cent}X\) if and only if \( \pi ^{w_{c}}:X^{w_{c}}\rightarrow X\) factorizes through \(\pi \). To construct the \textit{seminormalization of} \(X\) \textit{relative to} \(\operatorname{Cent}X,\) one proceeds in a similar manner, replacing \(\mathcal{K}^{0}(\operatorname{Cent} X)\) by the ring \(\mathcal{R}^{0}(\operatorname{Cent}X)\) of hereditarily rational functions on \(X\) admitting continuous extensions to \(\operatorname{Cent}X.\) The authors establish many useful algebraic and geometric properties of their constructions. real algebraic set; seminormalization; weak-normalization; continuous rational function; hereditarily rational function Real algebraic sets, Integral closure of commutative rings and ideals Weak and semi normalization in real algebraic geometry | 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 his book ``Vorlesungen über algebraische Geometrie'' (Leipzig 1921; F 48.068701), \textit{F. Severi} has asserted with an incomplete proof that the subscheme \({\mathcal I}_{d, g,r}'\) which is the union of the irreducible components of the Hilbert scheme \({\mathcal H}_{d, g,r}\) whose general points correspond to smooth, irreducible, and nondegenerate curves of degree \(d\) and genus \(g\) in \(\mathbb{P}^r\) is irreducible if \(d\geq g+r\). Also \textit{J. Harris} [``Curves in projective space'', Sem. Math. Sup. 85 (Montreal 1982; Zbl 0511.14014)] has conjectured that \({\mathcal I}_{d,g,r}\) is irreducible if the Brill-Noether number \(\rho (d,g, r): =g-(r+1) (g-d+r)\) is positive.
In this paper we demonstrate various reducible examples of the subscheme \({\mathcal I}_{d,g,r}'\) with positive Brill-Noether number. Indeed an example of a reducible \({\mathcal I}_{d,g,r}'\) with positive \(\rho (d,g,r)\), namely the example \({\mathcal I}_{2g-8,g,g-8}'\) (or other variations of it), has been known to some people (including the author), but it seems to have first appeared in the literature in a paper by \textit{D. Eisenbud} and \textit{J. Harris} [Ann. Sci. Éc. Norm. Supér., IV. Sér. 22, No. 1, 33-53 (1989; Zbl 0691.14006)]. The purpose of the paper under review is to add a wider class of examples to the list of such reducible examples by using general \(k\)-gonal curves. We also show that \({\mathcal I}_{d,g,r}'\) is irreducible for the range of \(d\geq 2g-7\) and \(g-d+r \leq 0\). Throughout we will be working over the field of complex numbers. reducible Hilbert scheme; F 48.068701; irreducible components of the Hilbert scheme; positive Brill-Noether number Keem, C, Reducible Hilbert scheme of smooth curves with positive brill-Noether number, Proc. Amer. Math. Soc., 122, 349-354, (1994) Parametrization (Chow and Hilbert schemes), Curves in algebraic geometry Reducible Hilbert scheme of smooth curves with positive Brill-Noether number | 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 height function on an abelian variety is addressed via an analogue of Mahler's measure function. It has previously been shown that the measure of a suitable polynomial yields the canonical height function on an elliptic curve; such work is generalised here to demonstrate that the canonical height on a higher-dimensional abelian variety may also be pursued from this viewpoint. Effective forulae which make use of the group law are given for the computation of local measures, and it is shown how these give rise to the construction of Riemann-style integrals on an abelian variety.
A descriptive treatment of local and canonical height functions of abelian varieties may be found in [\textit{J.-P. Serre}, Lectures on the Mordell-Weil theorem, Vieweg, Braunschweig (1997; Zbl 0863.14013)], and \textit{D. W. Boyd}'s paper [Can. Math. Bull. 24, 453-469 (1981; Zbl 0474.12005)] provides a useful overview of properties of the classical measure. Proofs of the theorems presented in the present paper are to be found in [\textit{B. ní Fhlathúin}, Mahler's measure on an abelian variety, Ph.D. thesis, Univ. East Anglia (1995)]. Examples of measures on an elliptic curve have been calculated using the PARI number theory package, which incorporates the elliptic group law. height function; abelian variety; Mahler's measure function; computation of local measures; Riemann-style integrals Varieties over global fields, Arithmetic varieties and schemes; Arakelov theory; heights The height on an abelian variety | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let C be a complete, smooth curve over the field \(F_ q\). Let \(\infty\) be a fixed prime of C and A the ring of functions regular away from \(\infty\). Let \(\phi\) be a rank one Drinfeld A-module defined over a finite extension of the fraction field k of A. Let \({\mathcal P}\subset A\) be a prime and let \(\phi\) [\({\mathcal P}]\) be the \({\mathcal P}\) torsion points of \(\phi\). As is well-known \(\phi\) [\({\mathcal P}]\) is isomorphic to A/\({\mathcal P}\). Choose one such isomorphism \(\psi\) :A/\({\mathcal P}\to \phi [{\mathcal P}]\) (for example one can use the exponential function of \(\phi\)). Now let L be an extension of k which contains both \(\phi\) [\({\mathcal P}]\) and a constant field extension isomorphic to A/\({\mathcal P}\). Choose an \(F_ q\)- linear injection \(\chi\) : A/\({\mathcal P}\to L\); all other such injections are \(q^ j\)-th powers of \(\chi\). To \(\psi\),\(\chi\) the author associates the ``basic Gauss sum'':
\[
\sum_{z\in A/{\mathcal P}^*}\chi (z^{- 1})\psi (z).
\]
For characters not coming in this fashion, the author defines the Gauss sum by multiplying the basic Gauss sums in the fashion of L. Carlitz using q-adic expansions. Such sums were originally studied by the author in the case \(A=F_ q[T]\) [Invent. Math. 94, 105-112 (1988; Zbl 0629.12014)] and, quite amazingly, all the results one would expect from classical theory hold.
In the paper being reviewed, the author examines the Gauss sums in more general contexts. In particular he handles non-polynomial class {\#}1 rings A. One already sees in this situation that the results are not quite so simple as in the polynomial case. The author also discusses the case of composite moduli. Here, unlike the classical theory, one obtains an addition rule where things can also be quite unruly. - Perhaps the neatest result in the paper appears in the appendix. In it, the author in a few short lines establishes (in the prime case described above) that the Gauss sums are never zero! function fields of one variable over finite fields; Gauss sum; non- polynomial class {\#}1 rings Thakur D. : Gauss sums for function fields , J. Number Theory 37 (1991) 242-252. Arithmetic theory of algebraic function fields, Other character sums and Gauss sums, Drinfel'd modules; higher-dimensional motives, etc., Finite ground fields in algebraic geometry Gauss sums for 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 provides new upper bounds for the so called symmetric tensor rank of multiplication in finite extensions \(\mathbb F_{q^n}\) of a finite field \(\mathbb F_q,\,\, q=p^m\). Associated with the bilinear multiplication \(\mathbb F_{q^n}\times \mathbb F_{q^n}\rightarrow \mathbb F_{q^n}\) we get a tensor \(T=\sum_{i=1}^r x_i^*\otimes y_i^*\otimes c_i\in \mathbb F_{q^n}^*\otimes \mathbb F_{q^n}^*\otimes \mathbb F_{q^n}\). The minimum \(r\) in such an expression it is called the tensor rank of the multiplication in \(\mathbb F_{q^n}\) while the minimum \(r\) for a decomposition \(T=\sum_{i=1}^r x_i^*\otimes x_i^*\otimes c_i\) it is called the the symmetric tensor rank \(\mu_q^{\mathrm{sym}}\) of the multiplication.
Upper bounds for \(\mu_q^{\mathrm{sym}}\) can be deduced from results of \textit{D. Chudnovsky} and \textit{G. Chudnovsky} [J. Complex. 4, No. 4, 285--316 (1988; Zbl 0668.68040)]. In particular there exists a constant \(C_q\) such that \(\mu_q^{\mathrm{sym}}\leq C_qn\). Theorem 5 gathers the best previous estimations for \(C_q\). The aim of the present paper is to improve these results in the case \(q=p, p^2\) and \(p\geq 5\).
The used tools are the construction of families of modular curves \(\{X_i\}\) with increasing genus \(g_i\) attaining the Drinfeld-Vladut bound and such that \(\lim_{i\rightarrow \infty}g_{i+1}/g_i =1\) as well as bounds on gaps between two consecutive primes (Theorem 6).
Section 2.1 studies the case \(q=p^2\) (Proposition 7) and Section 2.2 the case of prime fields (Proposition 10). finite fields; symmetric tensor rank; algebraic function field; tower of function fields; modular curve; Shimura curve Computational aspects of algebraic curves, Effectivity, complexity and computational aspects of algebraic geometry, Number-theoretic algorithms; complexity, Computational aspects of field theory and polynomials Dense families of modular curves, prime numbers and uniform symmetric tensor rank of multiplication in certain 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 In Invent. Math. 95, No. 1, 1-11 (1989; Zbl 0676.14009), \textit{G. Ellingsrud} and \textit{C. Peskine} proved that the Hilbert scheme of (smooth) non-general-type surfaces in \(\mathbb{P}^ 4\) consists only of a finite number of components. In particular, there is a maximum degree \(d_ 0\) for such surfaces. There are known examples of non-general-type surfaces of degree 15, so that \(d_ 0 \geq 15\). Although there is not an explicit upper bound for \(d_ 0\) in the quoted paper, the bound that could be obtained from the proof would be extremely high (about 10,000). In the paper under review, the authors prove that \(d_ 0 \leq 105\). They use essentially the same kind of inequalities as Ellingsrud and Peskine, but improve some of them by using the machinery of initial ideals. More precisely, for a codimension-two subvariety in a projective space, they get a set of (numerical) invariants for the generic initial ideal associated to its homogeneous ideal (in the case of curves, these invariants correspond to the numerical character introduced by Gruson and Peskine). This is what allows them to give the required bounds for the sectional genus and the Euler-Poincaré characteristic of (the structure sheaf of) a surface in \(\mathbb{P}^ 4\) in terms of the degree and the postulation, which is the main ingredient to get a bound for \(d_ 0\). With this method, the upper bound for the sectional genus coincides with the one obtained by Gruson and Peskine (and then used in the proof by Ellingsrud and Peskine), while the lower bound for the Euler-Poincaré characteristic is better than the one used by Ellingsrud and Peskine.
Finally, it could be worth to mention that, using also initial ideas, \textit{Shelly Cook} seems to have succeeded in improving the previous bound for \(d_ 0\). The last reference the reviewer had was that she succeeded in proving \(d_ 0 \leq 76\). number of components of Hilbert scheme; non-general-type surfaces; initial ideals; codimension-two subvariety Braun, R; Floystad, G, A bound for the degree of smooth surfaces in \({\mathbb{P}}^4\) not of general type, Compositio Math., 93, 211-229, (1994) Surfaces of general type, Low codimension problems in algebraic geometry, Projective techniques in algebraic geometry A bound for the degree of smooth surfaces in \(\mathbb{P}^ 4\) not of general 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 Let \({\mathbb P}^{2[n]}\) be the Hilbert scheme parametrizing zero dimensional subschemes of \({\mathbb P}^2_{\mathbb C}\) of length \(n\). \({\mathbb P}^{2[n]}\) is a smooth irreducible projective variety of dimension \(2n\). In this paper, the authors study the birational geometry of \({\mathbb P}^{2[n]}\). They show that \({\mathbb P}^{2[n]}\) is a Mori-Dream space (in particular \(R(D):=\bigoplus _{m\geq 0}H^0(\mathcal O (mD))\) is finitely generated for any integral divisor \(D\) on \({\mathbb P}^{2[n]}\)). They characterize the effective cone (for many values of \(n\)), and investigate its stable base locus decomposition (into finitely many rational polyhedral cones) and the birational models (corresponding to \({\text{Proj}}(R(D))\) for \(D\) in the big cone). For \(n\leq 9\) they determine the Mori cone decomposition of the cone of big divisors corresponding to different birational models \({\text{Proj}}(R(D))\) and the birational maps (flips and divisorial contractions) between models of adjacent chambers (wall crossings). They also give a modular interpretation in terms of the moduli spaces of Bridgeland semi-stable objects and a description as a moduli space of quiver representations using G.I.T. Hilbert scheme; minimal model program; quiver representations; Bridgeland stability conditions Arcara, D.; Bertram, A.; Coskun, I.; Huizenga, J., The minimal model program for the Hilbert scheme of points on \(\mathbb{P}^2\) and Bridgeland stability, Adv. Math., 235, 580-626, (2013) Minimal model program (Mori theory, extremal rays), Parametrization (Chow and Hilbert schemes), Algebraic moduli problems, moduli of vector bundles, Stacks and moduli problems The minimal model program for the Hilbert scheme of points on \(\mathbb P^2\) and Bridgeland stability | 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 \(P_1,\ldots,P_s\) be distinct points in projective \(n\)-space, and let \(p_1,\ldots,p_s\) be the corresponding prime ideals (of height \(n\) generated by \(n\) linear forms). The linear system of all forms of degree \(d\) with multiplicity at least \(\alpha_i\) at each \(P_i\) is the \(d\)-th homogeneous piece of the ideal \(I=p_1^{\alpha_1}\cap \cdots\cap p_s^{\alpha_s}\). This paper calculates the Hilbert function of \(I\) for various configurations of points. For example, an algorithm for computing the Hilbert function is given for \(n=2\) and \(s=3\). The paper concludes with some intriguing problems.
[For the entire collection see Zbl 0584.00015.] ideal of points in projective n-space; multiplicity; Hilbert function Davis, E.; Geramita, A. V.: The Hilbert function of a special class of 1-dimensional C.M. Graded algebras. Queen's papers in pure and applied mathematics no. 67 (1984) Multiplicity theory and related topics, Relevant commutative algebra, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Projective techniques in algebraic geometry The Hilbert function of a special class of 1-dimensional Cohen-Macaulay graded algebras | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author shows among other results the Deligne conjecture
\[
\text{Lef(Fr}^{n }\cdot b,K) =\sum_{D\in\Pi_{0}(\text{Fix Fr}^{n}\cdot b)} \text{ naive.loc}_{p}( \text{Fr}^{n}\cdot b,K)\tag{1}
\]
for \(U\) an open subset of a proper scheme \(X\), for \(b: V\to U\times_{k}U\) a correspondence (\(k\) the algebraic closure of a finite field) for \(K\) an element of the derived category \(D^{b}_{c}(U,\overline{\mathbb Q}_{l})\) of bounded complexes with constructible cohomology sheaves, for the corresponding Frobenius map Fr and if \(n\) is suitably large. Here Lef means the global trace, naive.loc means the naive local term (as an important property this term vanishes if the fiber of \(K\) is zero), and Fr means the geometric Frobenius over \(\mathbb{F}_q\). This result follows by upgrading in stages the Lefschetz-Verdier trace formula as found in Sémin. Géom. Algebr. 1965-66, SGA5, Lect. Notes Math. 589 [Exposé III, 73-137 (1977; Zbl 0355.14004) by \textit{A. Grothendieck}, and Exposé III B, 138-203 (1977; Zbl 0354.14006) by \textit{L. Illusie}]. For a proper statement of (1), it is necessary to use rigid geometry. Lefschetz-Verdier trace formula; Deligne conjecture; rigid geometry; proper scheme; constructible cohomology sheaves; Frobenius map; positive characteristic Fujiwara, K., \textit{rigid geometry, Lefschetz-verdier trace formula and deligne's conjecture}, Invent. Math., 127, 489-533, (1997) Étale and other Grothendieck topologies and (co)homologies, Local ground fields in algebraic geometry, Finite ground fields in algebraic geometry Rigid geometry, Lefschetz-Verdier trace formula and Deligne's conjecture | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(M\) be the complement of a hypersurface \(V\) in the \(n\)-dimensional the projective space \(\mathbb{P}^n\). Let \(V_1,\dots,V_m\) be the irreducible components of \(V\). If \(L\) is a generic rank one local system on \(M\), it is well known that \(H^k(M,L)=0\) for all \(k \neq n\). In this paper the author gives some sufficient conditions on the components \(V_j\) and on the local system \(L\) such that two twisted cohomology groups are non-zero, namely \(H^k(M,L) \neq 0\) for \(k=n-1\) and \(k=n\). Explicit non-trivial cohomology classes are constructed using rational differential forms and some geometric examples are given. local systems; twisted cohomology; hypersurface arrangements de Rham cohomology and algebraic geometry, Divisors, linear systems, invertible sheaves, Relations with arrangements of hyperplanes Non-vanishing of the twisted cohomology on the complement of hypersurfaces | 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 Many deformation-theoretic problems and results in algebraic and complex geometry can be formulated in terms of differential graded Lie algebras. These include the Kodaira-Spencer theory of deformations of complex structures. Nevertheless, there are fundamental deformation deformation problems in geometry for which no Lie theoretic formulation is known, e.g. deformation theory of submanifolds in a fixed ambient manifold, which is the local theory of the Hilbert scheme in algebraic geometry or the Douady space in complex-analytic geometry. A principal purpose of the present article is to remedy this situation. The author works with a generalization of Lie atoms; an object that is a pair of Lie algebras \(\mathfrak g,\mathfrak h^+\), a Lie homomorphism \(\mathfrak g\rightarrow\mathfrak h^+\), and a \(\mathfrak g\)-module \(\mathfrak h\subset\mathfrak h^+\). Geometrically, a Lie atom can be used to control situations where a geometric point is deformed while some aspects of the geometry stays constant (in some trivial manner). A typical application to the deformation theory of Lie atoms is the local theory of Hilbert schemes or Douady space of submanifolds of a space \(X\).
The author points out that a Lie atom possesses some of the formal properties of Lie algebras. There is a deformation theory for Lie atoms that generalizes the case of Lie algebras and which in addition allows to treat some classical deformation problems. These include, on the one hand, the Hilbert scheme, and on the other hand heat-equation deformations. In this article the author present a systematic development of the deformation theory of Lie atoms, which are closely analogous to differential graded Lie algebras.
The \textit{Jacobi-Bernoulli complex} is a comultiplicative complex whose zeroth cohomology, dualized, yield the \textit{deformation ring} of the atom. The proof of the vanishing of the square of the differential of this complex needs a discussion of Lie identities and Bernoulli number identities. Basic algebra on Lie atoms is developed, and the Kodaira-Spencer formalism is introduced. Finally the author constructs universal deformations under suitable hypothesis of finiteness and automorphism-paucity.
The definition of Lie atoms and their deformations are explicitly given and easy to understand. The examples are important and interesting. The atomic deformation theory associated to a sheaf of Lie atoms is naturally enough harder to read, but even more interesting. The author deduces the deformation from geometric definitions rather than defining it formally by the Kodaira-Spencer class, then showing that the two points of view coincides. formal deformation theory; differential graded Lie algebra; Hilbert scheme; Bernoulli number; Lie atom; Lie pair; Jacobi-Bernoulli complex Ran Z., Lie atoms and their deformations, Geom. Funct. Anal., 2008, 18(1), 184--221 Structure of families (Picard-Lefschetz, monodromy, etc.), Deformations of complex structures, Cohomology of Lie (super)algebras Lie atoms and their deformations | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The main purpose of the article is to complete the result of \textit{E. W. Howe, E. Nart} and the reviewer [Ann. Inst. Fourier 59, No. 1, 239--289 (2009; Zbl 1236.11058)] by giving the Weil polynomials for supersingular genus \(2\) curves over the finite fields \(\mathbb{F}_{q}\) with \(q=3^d\). The result is the following: if \(d\) is odd the Weil polynomial belongs to the following list
1. \((x^2+q)(x^2-sx+q)\) for all \(s \in \{\pm \sqrt{3q}\}\);
2. \((x^2+q)^2\), if \(q>3\);
3. \(x^4+q\);
4. \(x^4+qx^2+q^2\);
5. \(x^4-2qx^2+q^2\) if \(q>3\).
And if \(d\) is even it belongs to the list
1) \((x^2-2sx+q)(x^2+sx+q)\) for all \(s \in \{\pm \sqrt{q}\}\);
2) \((x^2-sx+q)^2\) for all \(s \in \{0,\pm \sqrt{q}\}\);
3) \((x^2-2sx+q)^2\) for all \(s \in \{\pm \sqrt{q}\}\), if \(q>9\);
4) \(x^4+q^2\);
5) \(x^4-sx^3+qx^2-sqx+q^2\) for all \(s\in \{\pm \sqrt{q}\}\).
The Weil polynomials of abelian surfaces which cannot be obtained are treated using various arguments. For positive results, note that for each class, an explicit curve can be constructed. This is done by using the first part of the paper which gives a beautiful analysis of the coarse moduli space \(\mathcal{A}\) of triples \((C,E,\phi)\) where \(C\) is a supersingular genus \(2\) curve over a field of characteristic \(3\), \(E\) the elliptic curve with \(j\)-invariant \(0\) and \(\phi :C \to E\) a degree \(3\) map. The author shows (Th. 2.1) that \(C\) is supersingular if and only if its Igusa invariants \(J_2,J_4\) and \(J_8\) are zero and shows that \(C : y^2=x^6+A x^3+Bx+A^2\) represents the invariant \([0:0:A:0:B]\). Two other nice properties are shown : if \(C\) is a genus \(2\) triple cover of a supersingular elliptic curve in characteristic \(3\) then \(C\) is supersingular (Cor.3.3) ; if \(C : y^2=f(x)\) with \(f\) a sextic polynomial is a supersingular genus \(2\) curve such that \(f\) can be written as the product of two cubic factors then \(C\) is a triple cover of a supersingular elliptic curve (Th.3.4).
Finally the author shows that \(\mathcal{A}\) is isomorphic to the affine line with one point removed and is a degree \(20\) cover of the moduli space of supersingular genus \(2\) curves. On the other hand, the author shows that \(\mathcal{A}\) is also isomorphic to the moduli space of pairs \((C,G)\) where \(C\) is supersingular and \(G\) is an order \(4\) subgroup of \(\mathrm{Jac}(C)[2]\) that is not isotropic with respect to the Weil pairing. curve; Jacobian; supersingular; abelian surface; zeta function; Weil polynomial; Weil number Curves over finite and local fields, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Finite ground fields in algebraic geometry Supersingular genus-2 curves over fields of characteristic 3 | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The main results with proofs of the author's notes devoted to the adèle approach in soliton theory and in the theory of Kac-Moody algebras are stated. The two definitions of the \(\tau\)-function as a spherical function (''matrix element'') and as modular co-form (''infinite determinant'') are given. The key result (variant of the Frobenius duality theorem) is its coincidence. adèle approach in soliton theory; Kac-Moody algebras; \(\tau \) - function; spherical function; matrix element; infinite determinant I. V. Cherednik, Funct. Anal. Appl., 19, 193--206 (1985). Infinite-dimensional Lie (super)algebras, Classical groups (algebro-geometric aspects), Infinite-dimensional Lie groups and their Lie algebras: general properties, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Partial differential equations of mathematical physics and other areas of application Functional realization of basic representations of factorizable Lie groups and algebras | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems The author connects continued fractions with elliptic curves. This was first done by \textit{W. W. Adams} and \textit{M. J. Razar} [Proc. Lond. Math. Soc., III. Ser. 41, 481--498 (1980; Zbl 0403.14002)]. The arguments of these authors are made explicit in the present paper. Later on \textit{D. Zagier} [Problems posed at the St. Andrews Colloquium, 1996, Solutions, 5th day] has found out that the integrality of a certain sequence \((B_h)_{h\in\mathbb{Z}}\) has to do with the multiples of a certain point on an elliptic curve.
Anyway, the author of this paper considers the continued fraction expansion of the square root \(Y\) of a quartic polynomial \(D\in\mathbb{F}[X]\) over a field \(\mathbb{F}\) of characteristic \(n\neq 2,3\), where \(D\) is not a square (of course) and is supposed to be monic and have zero trace. Hence, we obtain the quartic elliptic curve \(C: Y^2= D(X)\) with two points at infinity \(S\) and \(O\), where \(O\) is taken as the zero of the group law. Moreover, \(D\) may be assumed to have the form
\[
D(X)= (X^2+ f)^2+ 4v(X- w).
\]
The continued fraction expansion of \(Y\) leads to a sequence
\[
Y_h= {Y+ A+ 2e_h\over v_h(X- w_h)}\qquad (h\in \mathbb{Z}\geq 0)
\]
with \(A= X^2+ f\) and a fortiori to so-called ``elliptic'' sequences \((A_h)\), \((W_h)\) and so on, which satisfy certain identities (see \textit{D. Zagier} [loc. cit.]). The results are related to two dissertations, especially to \textit{C. Swart} [Elliptic curves and related sequences, Ph.D. Thesis, Royal Holloway and Bedford New College, University of London (2003)] but also to [\textit{R. Shipsey}, Elliptic divisibility sequences, Ph.D. Thesis, Goldsmiths College, University of London (2000)].
The paper is somewhat difficult to read, in particular for readers not so acquainted with the details of the theory of continued fractions.
Since the publication of \textit{M. Ward} [Memoir on elliptic divisibility sequences, Am. J. Math. 70, 31--74 (1948; Zbl 0035.03702)] is referred to and the characteristics 2 and 3 are excluded, we mention also the multiplication formulae in [\textit{J. W. S. Cassels}, A note on the division values of \(gs(u)\), Proc. Camb. Philos. Soc. 45, 167--172 (1949; Zbl 0032.26103)] for characteristic \(\neq 2,3\), [\textit{H. G. Zimmer}, Computational aspects of the theory of elliptic curves, Number Theory and Applications, R. A. Mollin (ed.), 279--324 (1989; Zbl 0702.14029)] for characteristic \(\neq 2\) and [\textit{N. Koblitz}, Constructing elliptic curve cryptosystems in characteristic 2, Lect. Notes Comput. Sci. 537, 156--167 (1991; Zbl 0788.94012)].
The continued fraction expansion of quadratic irrationalities in function fields was introduced by Artin. The method of Artin can be turned into efficient algorithms [\textit{B. Weis} and \textit{H. G. Zimmer}, Artins Theorie der quadratischen Kongruenzfunktionenkörper und ihre Anwendung auf die Berechnung der Einheiten- und Klassengruppen, Mitt. Math. Ges. Hamb. 12, No. 2, 261--286 (1991; Zbl 0757.11046)]. We recall this citation also to supplement \S6 of the paper under consideration because it contains an exposition of the theory of continued fractions in quadratic function fields. elliptic curves; elliptic sequences; continued fractions; singular case; periodicity; function field Van Der Poorten, A.: Elliptic curves and continued fractions. Journal of integer sequences 8, No. 2, 1-19 (2005) Continued fractions, Elliptic curves over global fields, Algebraic functions and function fields in algebraic geometry, Elliptic curves, Special sequences and polynomials, Special algebraic curves and curves of low genus Elliptic curves and continued fractions | 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 Consider the following system of PDEs:
\[
\frac{\partial{\Psi}}{\partial{t_i}}=P_i(z, u_1,\dots, u_n),\quad i=1,\dots,N,
\]
\noindent where \(t_1, \dots, t_N\) are independent variables, \(u_1,\dots, u_n\) are dependent variables and \(P_i(z,u_1,\dots,u_n), i=1,\dots,N\) are some functions. The compatibility conditions of this system are called Whitham type hierarchy and have the form:
\[
\sum_{l=1}^{n}\Bigg(\bigg(\frac{\partial{P_i}}{\partial{z}}\frac{\partial{P_j}}{\partial{u_l}}-\frac{\partial{P_j}}{\partial{z}}\frac{\partial{P_i}}{\partial{u_l}}\bigg)\frac{\partial{u_l}}{\partial{z}}+\bigg(\frac{\partial{P_j}}{\partial{z}}\frac{\partial{P_k}}{\partial{u_l}}-\frac{\partial{P_k}}{\partial{z}}\frac{\partial{P_j}}{\partial{u_l}}\bigg)\frac{\partial{u_l}}{\partial{t_i}}
\]
\[
+\bigg(\frac{\partial{P_k}}{\partial{z}}\frac{\partial{P_i}}{\partial{u_l}}-\frac{\partial{P_i}}{\partial{z}}\frac{\partial{P_k}}{\partial{u_l}}\bigg)\frac{\partial{u_l}}{\partial{t_j}}\Bigg)=0.
\]
\noindent Under some conditions on the functions \(P_i(z,u_1,\dots,u_n)\) the last system is equivalent to a hydrodynamic type system of the following form:
\[
\sum_{l=1}^{n}\Bigg(a_{rl}(u_1,\dots,u_n)\frac{\partial{u_l}}{\partial{t_i}}+b_{rl}(u_1,\dots,u_n)\frac{\partial{u_l}}{\partial{t_j}}+c_{rl}(u_1,\dots,u_n)\frac{\partial{u_l}}{\partial{t_k}}\Bigg)=0.
\]
\noindent The present paper is devoted to the construction of such hydrodynamic type systems associated with compact Riemann surfaces of arbitrary genus. Potentials of these hierarchies are written explicitly as integrals of hypergeometric type. integrable hierarchies; hypergeometric functions; tau-function Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions, Relationships between algebraic curves and integrable systems A simple construction of integrable Whitham type hierarchies | 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 Consider a matrix polynomial of the form
\[
W(z) = B^0 z^m + \cdots + B^m, \quad B^i \in \mathrm{Mat}_n(\mathbb{C}).
\]
Let \(C \to \mathbb{CP}^1\) be the associated spectral curve whose affine part is given by
\[
\{(z,w)\in \mathbb{C}^2 ~|~\det\left(w\cdot \mathbf{1}_n - W(z)\right) = 0\}.
\]
If \(W\) is generic, then \(C\) is smooth and irreducible with \(n\) pairwise distinct points \(P_1, \dots, P_n\) over \(\infty\). Note that the spectral curve \(C\) comes with a natural line bundle \(\mathcal{L}\) defined by the eigenvectors of \(W\). Finally, we denote by \(\theta\) the Riemann theta function of \(C\) associated with a fixed canonical basis of \(H_1(C,\mathbb{Z})\).
The present article studies \(N\)-differentials \(\nu_{N, \mathbb{Q}}\) on \(C\) for \(N\geq 2\) and \(N\)-tuples \(\mathbb{Q}\) of points on \(C\). The \(\nu_{N, \mathbb{Q}}\) are defined by the \(N\)-th logarithmic differential of \(\theta\) at the point in the Jacobian corresponding to \(\mathcal{L}\) (after tensoring with an appropriate divisor) and the \(N\)-tuples \(\mathbb{Q}\). The surprising main result of this articles expresses each \(\nu_{N, \mathbb{Q}}\) explicitly in terms of \(W\) and \(\mathbb{Q}\) only. If \(W\) has rational coefficients, i.e. \(B^i \in \mathrm{Mat}_n(\mathbb{Q})\), then it is further shown as a corollary that \(\nu_{N, \mathbb{Q}}\) has only rational coefficients when expanded around \(\infty\).
The key of proving these statements (for \(N \geq 3\); for \(N=2\) the methods are related but more direct) is the relationship to the \(n\)-wave (or AKNS-D) hierarchy defined by
\[
[L_{a, k}, L_{b,l}] = 0,\quad L_{a, k} = \frac{\partial}{\partial t^a_k} - U_{a, k}(\mathbf{t}; z).
\]
Here \(U_{a, k}(\mathbf{t}; z)\) are any \(n \times n\)-matrix-valued polynomials in \(z\) of degree \(k+1\) of a special form. Now given a spectral curve \(C\) as above and \(N\geq 0\), a solution to the \(n\)-wave hierarchy is constructed (Proposition 2.20) following Krichever's approach and using a vector-valued Baker-Akhiezer function. We note that the \(U_{a,k}\) of the hierarchy are in fact constructed from the spectral curve \(C\). A key result (Proposition 2.24) is that the tau-function attached to such a solution (reviewed in Appendix A) can be expressed as
\[
\tau(\mathbf{t}) = a(\mathbf{t})\cdot \theta(V(\mathbf{t}) - \mathbf{u}_0).
\]
Here \(\theta\) is the Riemann theta function of \(C\) as above and \(a(\mathbf{t})\), \(V(\mathbf{t})\) are certain scalar-/vector-valued functions.
From the previous formula it follows that the \(N\)-differentials \(\nu_{N, \mathbb{Q}}\) are equivalently defined as certain \(N\)-th logarithmic differentials of the tau-function \(\tau(\mathbf{t})\). Applying results on such differentials for more general solutions of the \(n\)-wave hierarchy (proven in Appendix A), it is possible to express them in terms of \(W(z)\) for the solutions attached to \(C\). Thereby the main theorem follows (see end of Section 2.3).
Despite being technical, this article is well written and organized. It contains several interesting results along the way. For example, results on the divisor of normalized eigenvectors of \(W\) (see Section 2) and its relation to the relative Jacobian \(J(C; P_1, \dots, P_n)\) (which can be considered as the Jacobi variety of the singular curve obtained from \(C\) by identifying the points \(P_i\) over \(\infty\)). Finally, explicit examples of the main results are provided as well as an appendix on tau-functions of solutions to the \(n\)-wave hierarchy. Riemann surfaces; Riemann theta function; tau-functions; integrable hierarchies Relationships between algebraic curves and integrable systems, Theta functions and abelian varieties, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) Algebraic spectral curves over \(\mathbb{Q}\) and their tau-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 In this text, the authors study the Jacobian \(J\) of the smooth projective curve \(C\) of genus \(r-1\) with affine model \[y^r=x^{r-1}(x+1)(x+t),\] over the function field \(\mathbb{F}_p(t)\), when \(p\) is prime and \(r\geq2\) is an integer prime to \(p\). When \(q\) is a power of \(p\) and \(d\) is a positive integer, they compute the \(L\)-function of \(J\) over \(\mathbb{F}_q(t^{1/d})\) and show that the Birch and Swinnerton-Dyer conjecture holds for \(J\) over \(\mathbb{F}_q(t^{1/d})\). When \(d\) is divisible by \(r\) and of the form \(p^\nu+1\), and \(K_d:=\mathbb{F}_p(\mu_d,t^{1/d})\), they write down explicit points in \(J(K_d)\), show that they generate a subgroup \(V\) of rank \((r-1)(d-2)\) whose index in \(J(K_d)\) is finite and a power of \(p\), and show that the order of the Tate-Shafarevich group of \(J\) over \(K_d\) is \([J(K_d):V]^2\). When \(r>2\), the authors prove that the new part of \(J\) is isogenous over \(\overline{F_p(t)}\) to the square of a simple abelian variety of dimension \(\phi(r)/2\) with endomorphism algebra \(\mathbb{Z}[\mu_r] ^+\). For a prime \(l\) with \(l\nmid pr\), they prove that \(J[l](L)=\{0\}\) for any abelian extension \(L\) of \(\overline{F}_p(t)\). This monograph is organized as follows. Chapter 1, deals with the curve, explicit divisors, and relations. The authors give basic information about the curve \(C\) and Jacobian \(J\) they are studying. They write down explicit divisors in the case \(d=p^\nu+1\), and we find relations satisfied by the classes of these divisors in J. These relations turn out to be the only ones, but that is not proved in general until much later in the paper. Chapter 2, deals with descent calculations. In this chapter, the authors assume that \(r\) is prime and use descent arguments to bound the rank of \(J\) from below in the case when \(d=p^\nu+1\). Chapter 3, deals with minimal regular model, local invariants, and domination by a product of curves. The authors construct the minimal, regular, proper model \(\aleph\longrightarrow\mathbb{P}^1\) of \(C/\mathbb{F}_q(u)\) for any values of \(d\) and \(r\). In particular, they compute the singular fibers of \(\aleph\longrightarrow\mathbb{P}^1\). This allows them to compute the component groups of the Néron model of \(J\). They also give a precise connection between the model \(X\) and a product of curves. Chapter 4, deals with heights and the visible subgroup. The authors consider the case where \(d=p^\nu+1\) and \(r|d\), and they compute the heights of the explicit divisors introduced in Chapter 1. This allows them to compute the rank of the explicit subgroup \(V\) and its structure over the group ring \(\mathbb{Z}[\mu_r\times\mu_d]\). Chapter 5, deals with the \(L\)-function and the \(BSD\) conjecture. The authors give an elementary calculation of the \(L\)-function of \(J\) over \(\mathbb{F}_q(u)\) (for any \(d\) and \(r\)) in terms of Jacobi sums. They also show that the \(BSD\) conjecture holds for \(J\), and we give an elementary calculation of the rank of \(J(\mathbb{F}_q(u))\) for any \(d\) and \(r\) and all sufficiently large \(q\). Chapter 6, deals with analysis of \(J[p]\) and \(NS(\aleph_d)tor\) and Chapter 7, with index of the visible subgroup and the Tate-Shafarevich group. In these technical chapters, the authors prove several results about the surface \(X\) that allow them to deduce that the index of \(V\) in \(J(K_d)\) is a power of \(p\) when \(d=p^\nu+1\) and \(r\) divides \(d\). They also use the \(BSD\) formula to relate this index to the order of the Tate-Shafarevich group. Chapter 8, deals with monodromy of \(l\)-torsion and decomposition of the Jacobian. The authors prove strong results on the monodromy of the \(l\)-torsion of \(J\) for prime to \(pr\). This gives precise statements about torsion points on \(J\) over abelian or solvable extensions of \(\mathbb{F}_p(t)\) and about the decomposition of \(J\) up to isogeny into simple abelian varieties. The paper is supported by an appendix on an additional hyperelliptic family. curve; function field; Jacobian; abelian variety; finite field; Mordell-Weil group; torsion; rank; \(L\)-function; Birch and Swinnerton-Dyer conjecture; Tate-Shafarevich group; Tamagawa number; endomorphism algebra; descent; height; Néron model; Kodaira-Spencer map; monodromy Research exposition (monographs, survey articles) pertaining to number theory, Elliptic curves over global fields, \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture, Curves of arbitrary genus or genus \(\ne 1\) over global fields, Algebraic functions and function fields in algebraic geometry Explicit arithmetic of Jacobians of generalized Legendre curves over global 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 under review is concerned with the problem of counting points belonging to an algebraic variety \(X\) defined over a number field \(K\), weighing these points \(\xi \in X(\overline{K})\) by a function depending on their multiplicity \(\mu_\xi(X) \in \mathbb{Z}_{\geq 1}\).
More precisely, the main result of the paper (Theorem 4.5) gives an explicit upper bound for the sum
\[ \sum_{\xi \in S(X;D,B)} \mu_\xi(X) \cdot (\mu_\xi(X) - 1)^{\mathrm{dim}(X) - \mathrm{dim}(X^\text{sing})} \]
where \(X^\text{sing} \hookrightarrow X\) is the singular locus of \(X\) and
\[ S(X;D,B) := \{ \xi \in X(\overline{K}) \ \mid \ [K(\xi) \colon K] = D, H(\xi) \leq B \} \]
is a set of points of bounded height and fixed degree. The paper assumes that the height \(H\) is induced from the Weil height on \(\mathbb{P}^n_K\), via an embedding \(X \hookrightarrow \mathbb{P}^n_K\) which identifies \(X\) with a reduced hypersurface. The upper bound proved in the paper under review then depends polynomially on the degree \(\delta\) of this hypersurface.
This generalizes work of Laumon, who treated the case \(\mathrm{dim}(X^\text{sing}) = 0\) (see [\textit{G. Laumon}, Bull. Soc. Math. Fr. 104, 51--63 (1976; Zbl 0343.14014)]).
The proof of Theorem 4.5 builds on two fundamental results: on the one hand, the authors of the paper under review construct intersection trees associated to the algebraic variety \(X\), which control the multiplicities of singular points. On the other hand, Theorem 3.3 of the paper under review gives a uniform estimate for the number of points of bounded height defined on \(X\), which has the advantage of depending only on \(\mathrm{dim}(X)\) and on \(\delta\).
The paper under review, generalizing analogous results proved by its second author for varieties defined over finite fields (see [\textit{C. Liu}, ``Comptage des multiplicités dans une hypersurface sur un corps fini'', Preprint, \url{arXiv:1606.09337}]), sheds some light on the problem of counting points with respect to their multiplicities. In particular, this allows one to account for the singularities of the variety \(X\), which are often neglected in height counting problems, such as the ones related to the conjectures of \textit{B. Lehmann} and \textit{S. Tanimoto} [Res. Math. Sci. 6, No. 1, Paper No. 12, 41 p. (2019; Zbl 1431.14019)] for a recent survey). algebraic point; counting multiplicities; height function over projective space; intersection tree; rational point Counting solutions of Diophantine equations, Heights, Rational points, Singularities of surfaces or higher-dimensional varieties, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Counting multiplicities in a hypersurface over number fields | 0 |
The authors identify iterated function systems \(\Phi\) on a metric space \(X\) and regular Borel measures \(\mu\) such that the matrix subdivision process relative to a finite family \(\mathcal A\) converges. They give a characterization of the convergence of the subdivision process to a function in \(L_p(\Omega,\mu)\) in terms of the spectral properties of \(\mathcal A\). First, they give a description of subdivision schemes and the definition of convergence with observations on convergent subdivision schemes. Then, they define and discuss the concept of \(\mu\)-uniformity of a family of contractions for any \(\mu\)-measurable set. Finally, they prove that the subdivision scheme converges exponentially fast and its limit function enjoys Hölder regularity. spectral radius; iterated function systems; subdivision scheme --------, Subdivision schemes for iterated function systems , Proc. Amer. Math. Soc. 129 (2001), 1861-1872. JSTOR: Algebraic properties of function spaces in general topology, Coverings in algebraic geometry, Rings and algebras of continuous, differentiable or analytic functions, Modular representations and characters Subdivision schemes for iterated function systems Let \(f(x)\) be a polynomial on the finite dimensional vector space \(V_K\) over the \(p\)-adic field \(K\). Consider the \(p\)-adic integral \(Z(s):=\int_{V_{O_K}} \vert f(x)\vert^s\vert \,dx \vert\), where \(V_{O_K}\) is the compact set of \(p\)-adic integers in \(V_K\) and \(|dx|\) is the Haar measure on \(V_K\). It is proved that, for any polynomial \(f(x)\), \(Z(s)\) is a rational function in \(q^{-s}\) where \(q\) is the number of the elements of the residue field of \(K\). Moreover, it is conjectured that all possible poles appear in the zeros of the \(b\)-function of \(f(x)\) when \(f(x)\) is a relative invariant of a prehomogeneous vector space.
The authors of this article succeeded in proving the conjecture for some fundamental prehomogeneous vector spaces. poles of complex powers of polynomial functions; p-adic integral; zeros of the b-function; prehomogeneous vector space DOI: 10.2307/2374750 Prehomogeneous vector spaces, Hypersurfaces and algebraic geometry, Homogeneous spaces and generalizations On the poles of \(p\)-adic complex powers and the \(b\)-functions of prehomogeneous vector spaces | 0 |
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