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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give a bijection between certain colored partitions and the elements in the quotient of an affine Weyl group modulo its Weyl group. By Bott's formula these colored partitions give rise to some partition identities. In certain types, these identities have previously appeared in the work of \textit{M. Bousquet-Mélou} et al. [Ramanujan J. 1, No. 1, 101--111 (1997; Zbl 0909.05008)], \textit{H. Eriksson} and \textit{K. Eriksson} [Electron. J. Comb. 5, Research paper R18, 32 p. (1998; Zbl 0889.20002); printed version J. Comb. 5, 231--262 (1998)] and \textit{V. Reiner} [Electron. J. Comb. 2, Research paper R25, 21 p. (1995; Zbl 0849.20032); printed version J. Comb. 2, 403--422 (1995)]. In other types the identities appear to be new. For type \(A_n\), the affine colored partitions form another family of combinatorial objects in bijection with \((n + 1)\)-core partitions and \(n\)-bounded partitions. Our main application is to characterize the rationally smooth Schubert varieties in the affine Grassmannians in terms of affine partitions and a generalization of Young's lattice which refines weak order and is a subposet of Bruhat order. Several of the proofs are computer assisted. rationally smooth Schubert varieties; Young's lattice Billey, S., and Mitchell, S., Affine partitions and affine Grassmannians, preprint 46pp: arXiv:0712.2871 (2007), to appear in the Electronic Journal of Combinatorics. Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds Affine partitions and affine Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(\Bbbk\) be an algebraically closed field. We prove that a polynomial \(\Bbbk\)-derivation \(D\) in two variables is locally nilpotent if and only if the subgroup of polynomial \(\Bbbk\)-automorphisms which commute with \(D\) admits elements whose degree is arbitrary big. locally nilpotent derivations; polynomials Derivations and commutative rings, Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) A characterization of local nilpotence for dimension two polynomial derivations | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The Schubert intersection problem is the problem of counting \(r\)-dimensional subspaces of \(\mathbb{C}^n\) having certain adjacency properties with given subspaces (flags) of \(\mathbb{C}^n\). In certain situations such a Schubert intersection problem can be reduced to another Schubert intersection problem with smaller \(n\). The paper under review extends an earlier such reduction result of Thompson-Therianos to a large family of reduction results. In fact, the reductions presented in this paper are sufficient for the complete solution of a special class of intersection problems, when the Littlewood-Richardson coefficient is 1. The reductions presented are in connection with known multiplicative properties of Littlewood-Richardson coefficients. The method of the proof is a very careful analysis of a combinatorial rule (involving `measures', similar to `puzzles') for Littlewood-Richardson coefficients. Schubert variety; Littlewood-Richardson rule; puzzle; measure Bercovici, H.; Li, W. S.; Timotin, D.: A family of reductions for Schubert intersection problems, J. algebraic combin. 33, 609-649 (2011) Grassmannians, Schubert varieties, flag manifolds A family of reductions for Schubert intersection problems | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials See the preview in Zbl 0539.14008. decomposition of birational morphism; smooth curves on surfaces; with rational double points; blow-up of a smooth point; Gorenstein; threefold singularities D. Morrison, ''The biratioanl geometry of surfaces with ratiional double points,''Math. Ann.,271, 415--438 (1985). Global theory and resolution of singularities (algebro-geometric aspects), Rational and birational maps, Singularities of surfaces or higher-dimensional varieties, \(3\)-folds, Singularities in algebraic geometry The birational geometry of surfaces with rational double points | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials \noindent Let \(k\) be a commutative ring and \(F\) a field of coefficients. In [\textit{G. Tabuada}, Int. Math. Res. Not. 2005, No. 53, 3309--3339 (2005; Zbl 1094.18006)] the second author constructed the category \({\text{{NNum}}}(k)_F\) of non-commutative Chow motives over a field \(k\) and with coefficients in \(F\). The objects of this are smooth and proper \(dg\)-categories in the sense of Kontsevich whereas morphisms (correspondences) are given by the \(F\)-linearized Grothendieck group \(K_{0}({\mathcal A}^{\text{op}}{\otimes}^{\mathbb L}_{k}{\mathcal B})_{F}\). The authors prove that this category is abelian semi-simple and that for \(F={\mathbb Q}\) the category \({\text{Num}}(k)_{\mathbb Q}\) embeds in \({\text{NNum}}(k)_{\mathbb Q}\) after factorization by the action of the Tate object. These to results yield an alternative ``non-commutative'' semi-simplicity result of \textit{U. Jannsen} [Invent. Math. 107, No. 3, 447--452 (1992; Zbl 0762.14003)]. non-commutative Chow motive; Grothendieck group; semi-simplicity M. Marcolli and G. Tabuada, Noncommutative motives, numerical equivalence, and semi-simplicity, Amer. J. Math. 136 (2014), no. 1, 59-75. Derived categories, triangulated categories, Closed categories (closed monoidal and Cartesian closed categories, etc.), Generalizations (algebraic spaces, stacks) Noncommutative motives, numerical equivalence, and semi-simplicity | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We study some new theories of characteristic homology classes of singular complex algebraic (or compactifiable analytic) spaces.
We introduce a motivic Chern class transformation \(mC_{y}: K_{0}\)(var/\(X\)) \(\rightarrow G_{0}(X) \otimes \mathbb Z[y]\), which generalizes the total \(\lambda \)-class \(\lambda _{y}(T^*X)\) of the cotangent bundle to singular spaces. Here \(K_{0}\)(var/\(X\)) is the relative Grothendieck group of complex algebraic varieties over \(X\) as introduced and studied by Looijenga and Bittner in relation to motivic integration, and \(G_{0}(X)\) is the Grothendieck group of coherent sheaves of \(\mathcal O_X\)-modules. A first construction of \(mC_{y}\) is based on resolution of singularities and a suitable ``blow-up'' relation, following the work of Du Bois, Guillén, Navarro Aznar, Looijenga and Bittner. A second more functorial construction of mC \(_{y}\) is based on some results from the theory of algebraic mixed Hodge modules due to M. Saito. We define a natural transformation \(T_{y^*} : K_{0}\)(var/\(X\)) \(\rightarrow H_{*}(X) \otimes \mathbb Q[y]\) commuting with proper pushdown, which generalizes the corresponding Hirzebruch characteristic. \(T_{y^*}\) is a homology class version of the motivic measure corresponding to a suitable specialization of the well-known Hodge polynomial. This transformation unifies the Chern class transformation of MacPherson and Schwartz (for \(y = -1\)), the Todd class transformation in the singular Riemann-Roch theorem of Baum-Fulton-MacPherson (for \(y = 0\)) and the L-class transformation of Cappell-Shaneson (for \(y = 1\)). We also explain the relation among the ``stringy version'' of our characteristic classes, the elliptic class of Borisov-Libgober and the stringy Chern classes of Aluffi and De Fernex-Lupercio-Nevins-Uribe. All our results can be extended to varieties over a base field \(k\) of characteristic 0. characteristic classes; characteristic number; genus; singular space; motivic; additivity; Riemann-Roch; Grothendieck group; cobordism group; Hodge structure; mixed Hodge module Brasselet, J-P; Schürmann, J.; Yokura, S., Hirzebruch classes and motivic Chern classes for singular spaces, J. Topol. Anal., 2, 1-55, (2010) Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Riemann-Roch theorems, Global theory and resolution of singularities (algebro-geometric aspects), Global theory of complex singularities; cohomological properties, Mixed Hodge theory of singular varieties (complex-analytic aspects) Hirzebruch classes and motivic Chern classes for singular spaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In spite of much effort, algebraic surfaces of general type with \(p_g=q=0\) are still far from being well understood and every progress requires subtle reasonings. In particular the construction of new examples is always difficult.
The present paper focuses on minimal surfaces of general type with\(p_g=0\) and \(K^2=5\). The main result is the effective construction of new examples of such surfaces as bidouble covers of the projective plane, with branching divisors of degrees 2,4 and 6 and suitable singularities. By the reviewer's results [Arch. Math. 69, No.5, 435--440 (1997; Zbl 0921.14024)] the degree of the bicanonical map of minimal surfaces of general type with \(p_g=0\) and \(K^2=5\) is 1, 2 or 4. It is shown that these new examples have bicanonical map of degree \(2\) and that they have a genus \(3\) hyperelliptic fibration [cf. J. Algebr. Geom. 16, No. 4, 625--669 (2007; Zbl 1132.14036)].
Previous examples due to \textit{F. Catanese} [Algebraic geometry: Hirzebruch 70. Proceedings of the algebraic geometry conference in honor of F. Hirzebruch's 70th birthday, Stefan Banach International Mathematical Center, Warszawa, Poland, May 11-16, 1998. Providence, RI: American Mathematical Society. Contemp. Math. 241, 97--120 (1999; Zbl 0964.14012)] are also studied. It is proven that their bicanonical map has degree \(4\) and that these examples can in fact be obtained by the Burniat construction, i.e. they are Burniat surfaces.
The constructions and the analysis of the bicanonical map are clearly presented. algebraic surface; bicanonical map; Du Val double plane; Burniat surface; bidouble cover C. Werner, On surfaces with pg= 0 and K2= 5, Canad. Math. Bull. 53 (2010), 746--756. Surfaces of general type On surfaces with \(p_g = 0\) and \(K^2 = 5\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let K be a compact connected Lie group, T be a maximal torus in K and T' be its normalizer in K. The flag variety \(X=K/T\) admits a cellular decomposition \(X=\cup X_ w\) with cells indexed by all elements w of the Weyl group \(W=T'/T\). The closures \(\bar X{}_ w\) of these cells determine elements in the dual space to the complex cohomology space \(H^*(X)\) that are called Schubert cycles.
The aim of the present paper is to define and to study the analogues of Schubert cycles for the equivariant cohomology \(H^*_ K(X)\). Identifying (by the Chern-Weil isomorphism) \(H^*_ K(X)\) with the space of polynomial functions on the Lie algebra \({\mathfrak t}\) of T the author gives an explicit formula for the equivariant Schubert cycles in terms of the reduced decompositions of elements of W. polynomial functions on Lie algebra; compact connected Lie group; maximal torus; flag variety; cellular decomposition; Weyl group; Schubert cycles; equivariant cohomology; reduced decompositions A. Arabia, Cycles de Schubert et cohomologie équivariante de \?/\?, Invent. Math. 85 (1986), no. 1, 39 -- 52 (French). Discrete subgroups of Lie groups, Homology with local coefficients, equivariant cohomology, Grassmannians, Schubert varieties, flag manifolds Cycles de Schubert et cohomologie équivariante de K/T. (Schubert cycles and equivariant cohomology of K/T) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials On an arbitrary toric variety, we introduce the logarithmic double complex, which is essentially the same as the algebraic de Rham complex in the nonsingular case, but which behaves much better in the singular case.
Over the field of complex numbers, we prove the toric analog of the algebraic de Rham theorem which Grothendieck formulated and proved for general nonsingular algebraic varieties re-interpreting an earlier work of Hodge-Atiyah. Namely, for a finite simplicial fan which need not be complete, the complex cohomology groups of the corresponding toric variety as an analytic space coincide with the hypercohomology groups of the single complex associated to the logarithmic double complex. They can then be described combinatorially as Ishida's cohomology groups for the fan. -- We also prove vanishing theorems for Ishida's cohomology groups. As a consequence, we deduce directly that the complex cohomology groups vanish in odd degrees for toric varieties which correspond to finite simplicial fans with full-dimensional convex support. In the particular case of complete simplicial fans, we thus have a direct proof for an earlier result of Danilov and the author. toric variety; logarithmic double complex; de Rham complex; simplicial fan; hypercohomology groups; vanishing theorems; Ishida's cohomology groups T. Oda: ''The algebraic de Rham theorem for toric varieties'',Tohoku Math. J., Vol. 2, (1993), pp. 231--247. Vanishing theorems in algebraic geometry, de Rham cohomology and algebraic geometry, Toric varieties, Newton polyhedra, Okounkov bodies The algebraic de Rham theorem for toric varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We prove in this paper the following results. Let G be a semisimple algebraic group over an algebraically closed field k and Q a parabolic subgroup containing a Borel subgroup B. Let X be a Schubert variety (i.e. the closure of a B orbit) in G/Q. Then (a) If L is a line bundle on G/Q such that \(H^ 0(G/Q,L)\neq 0\) then \(H^ i(X,L)=0\) for \(i>0\) and the restriction map \(H^ 0(G/Q,L)\to H^ 0(X,L)\) is surjective; (b) X is normal; (c) X is projectively normal in any embedding given by an ample line bundle on G/Q. - If we prove the results for fields of positive characteristic they follow for fields of characteristic zero by semicontinuity. When char k\(=0\) we have the absolute Frobenius morphism \(F:X\to X\) defined by raising functions on X to the p-th power. In the preprint ''Frobenius splitting and cohomology vanishing for Schubert varieties'' by \textit{V. B. Mehta} and \textit{A. Ramanathan} it was shown using duality for the Frobenius morphism of the Bott-Samelson-Demazure variety (constructed in the paper of Demazure cited below) that the p-th power map \(0_ X\to F_*0_ X\) admits a section. This quickly gives (a) for ample line bundles L. In this paper we extend this method, by a closer examination of the splitting, to the general case of \(H^ 0(G/Q,L)\neq 0\). We then deduce (b) from (a) by an inductive argument involving the \({\mathbb{P}}^ 1\)-fibrations \(G/B\to G/P\) for suitable minimal parabolic subgroups P containing B.
These results prove the conjectures of \textit{M. Demazure} in his paper in Ann. Sci. Éc. Norm. Supér., IV. Sér. 7, 53-88 (1974; Zbl 0312.14009). In particular his character formula for \(H^ 0(X,L)\) for fields of arbitrary characteristic also follows. Incidentally our results uphold the main claims in Demazure's paper in spite of the falsity of proposition 11, {\S}2 of that paper. projective normality; vanishing cohomology groups; Schubert variety; line bundle; Frobenius morphism [RR]Ramanan, S. \&Ramanathan, A., Projective normality of flag varieties and Schubert varieties.Invent. Math., 79 (1985), 217--224. Grassmannians, Schubert varieties, flag manifolds, Group actions on varieties or schemes (quotients) Projective normality of flag varieties and Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We discuss the so-called secant conjecture in real algebraic geometry, and show that it follows from another interesting conjecture, about disconjugacy of vector spaces of real polynomials in one variable. disconjugacy; Wronskian; Schubert calculus Real algebraic and real-analytic geometry, Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations, Real polynomials: location of zeros Disconjugacy and the secant conjecture | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials 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 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a compact connected Lie group and \(P\subset G\) be the centralizer of a one-parameter subgroup in \(G\). We explain a program that reduces integration along a Schubert variety in the flag manifold \(G/P\) to the Cartan matrix of \(G\).As applications of the program, we complete the project of explicit computation of the degree and Chern number of an arbitrary Schubert variety started in \textit{X. Zhao} and \textit{H. Duan} [J. Symb. Comput. 33, No. 4, 507--517 (2002; Zbl 1046.14027)]. Cartan matrix; flag manifolds; Schubert calculus DOI: 10.1016/j.jsc.2004.03.005 Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Symbolic computation and algebraic computation The Cartan matrix and enumerative calculus | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Motivated by questions about partial differential equations the author introduces the notion of a coercive form. This is a homogeneous polynomial \(f\) that is a sum of squares in the polynomial ring \(\mathbb R[X_1,\dots, X_n]\) and has at least one representation \(f= \sum^k_{i=1} p^2_i\) such that \(\{z\in\mathbb C^n\mid p_1(z)=\cdots= p_k(z)= 0\}= \{0\}\). The author asks whether there exist positive non-coercive forms with \(n\) variables and degree \(2\cdot d\).
The answer is related to the existence of positive definite forms that are not sums of squares. It is shown that there are non-coercive forms of degree 4 if \(n\geq 6\) and of degree 6 if \(n\geq 4\). The constructions use the method of Gram matrices, which is familiar from representations of sums of squares of polynomials. positive definite form; sum of squares; Gram matrix; zero set of polynomials Gregory C. Verchota, Noncoercive sums of squares in \Bbb R[\?\(_{1}\),\ldots ,\?_{\?}], J. Pure Appl. Algebra 214 (2010), no. 3, 236 -- 250. Fields related with sums of squares (formally real fields, Pythagorean fields, etc.), Sums of squares and representations by other particular quadratic forms, Real algebra, Forms over real fields, Real algebraic and real-analytic geometry, Higher-order elliptic equations, Boundary value problems for higher-order elliptic equations Noncoercive sums of squares in \(\mathbb R[x_1,\ldots ,x_n]\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials 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 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This book appears at the time Robert Langland is given the Abel prize. It covers much more, but the Langlands program is included.
The book consists of three parts, the first preparatory, and is called Basics. It starts by explaining model examples, that is the simplest functors arising in algebraic geometry, number theory and topology. The functors take values in \(C^\ast\)-algebras, and are the noncommutative tori. The noncommutative torus \(\mathcal A_\theta\) is an algebra over \(\mathbb C\) on a pair of generators \(u,v\) satisfying \(vu=e^{2\pi i\theta}\) where \(\theta\) is a real constant. There are several more or less equivalent definitions of noncommutative tori. The original is the geometric, which involves a deformation of the commutative algebra \(C^\infty(T^2)\) of smooth complex valued functions on a two-dimensional torus \(T^2\). After this comes the analytic definition, and finally the algebraic definition. The different definitions give a good illustration of the connections, and the reason for the more abstract algebraic definition of a torus. Some basic facts about the torus \(\mathcal A_\theta\) are considered. The author gives the identity for when \(\mathcal A_\theta\) is Morita equivalent to \(\mathcal A_{\theta^\prime}\) in the geometric and analytic case, which says when the to tori are equivalent in the setting of noncommutative algebraic geometry. Also, complex and real multiplication, the relation to elliptic curves with Weiserstrass uniformation and Jacobi normal forms are considered. The functor \(F:\mathcal E_\tau\rightarrow\mathcal A_\tau\) is explained, and this makes the definition of the Sklyanin algebra \(S(\alpha,\beta,\gamma)\) natural. The functor \(F:\mathcal E_\tau\rightarrow\mathcal A_\theta\) is intertwined with the arithmetic of elliptic curves, and the ranks of elliptic curves are related to an invariant of algebras \(\mathcal A_{RM}\). The first chapter ends with a classification of surface automorphisms.
Via the basics of category theory, the book contains a thorough definition of \(C^\ast\)-algebras: A \(C^\ast\)-algebra is an algebra \(A\) over \(\mathbb C\) with a norm \(a\mapsto\parallel a\parallel\) and an involution satisfying \(\parallel ab\parallel\leq \parallel a\parallel\parallel b\parallel\) and \(\parallel a^\ast a\parallel=\parallel a\parallel^2\) and such that \(A\) is complete with respect to the norm. The properties of \(C^\ast\)-algebras are recalled, and the \(K\)-theory of \(C^\ast\)-algebras is given. After this it is possible to define the noncommutative tori and the almost finite (AF) algebras introduced by Bratteli, which are classified by their Bratteli diagrams. Particular cases of these are generic AF algebras, stationary AF-algebras, and the original case: Uniformly hyper-finite \(C^\ast\)-algebras (UHF-algebras). Also, the \(K\)-groups of the Cuntz-Krieger algebras are computed.
The second part of the book studies noncommutative invariants, first in the case of topology. This is done by constructing functors arising in the topology of surface automorphisms, fibre bundles, knots, links, etc. The functors have their images in the category of AF-algebras, Cuntz-Krieger algebras, cluster \(C^\ast\)-algebras and so forth, defining a set of homotopy invariants of the the corresponding topological space. Some invariants are new, and some are known: Torsion in fibre bundles, Jones and HOMFLY polyniomials, and more. For the classification of surface automorphisms of compact oriented surfaces of genus \(g\geq 1\), the text considers Pseudo-Anasov automorphisms of a surface, the Jacobians of measured foliations, Anosov maps of the torus and its numerical invariants, etc.
In the study of torsion in the torus bundles \(M_\alpha\), the study includes the Cuntz-Krieger functor and specific noncommutative invariants of torus bundles.
In the study of the obstruction theory for Anosov's bundles, a functor \(F\) from the category of mapping tori of the Anosov diffeomorphisms \(\phi:M\rightarrow M\) of a smooth manifold \(M\), the Anosov bundles, to a category of stable homomorphisms between corresponding AF-algebras is constructed. This is used to introduce an obstruction theory for continuous maps between Anosov's bundles built on the noncommutative invariants derived from the Handleman triple \((\Lambda,[I],K)\) attached to a stationary AF-algebra. Specific examples are given in dimension 2, 3 and 4.
Now follows the definition, properties and applications of cluster \(C^\ast\)-algebras and knot polynomials in the topological setting. The author gives a representation of the braid groups in the cluster \(C^\ast\)-algebra associated to a triangulation of the Riemann surface \(S\) with one or two cusps, and it is proved that the Laurent polynomials coming from the \(K\)-theory of such an algebra are topological invariants of the closure of braids. Jones and HOMFLY polynomials are special cases of the construction corresponding to the \(S\) being a sphere with two cusps and a torus with one cusp respectively. One should mention that the text covers the Birman-Hilden theorem and a lot of explicit examples.
The book turns over to noncommutative invariants in algebraic geometry: The setup is to look at \(\mathsf{CRng}\) as the category of coordinate rings of projective varieties, and to consider a functor \(F:\mathsf{CRng}\overset{\text{GL}_n} {\rightarrow}\mathsf{Grp}\hookrightarrow\mathsf{Grp-Rng}\). It is proved in the text that when \(\mathsf{CRng}\) are the coordinate rings of elliptic curves, the category \(\mathsf{Grp-Rng}\) are the noncommutative tori. Also, if \(\mathsf{CRng}\) are the rings of algebraic curves of genus \(g\geq 1\), then \(\mathsf{Grp-Rng}\) are the toric AF-algebras. If \(\mathsf{CRng}\) are coordinate rings of projective varieties of dimension \(n\geq 1\), then \(\mathsf{Grp-Rng}\) consists of the Serre \(C^\ast\)-algebras. Notice also that elliptic curves over the field of \(p\)-adic numbers are considered, and in this case \(\mathsf{Grp-Rng}\) consists of the UHF-algebras. Finally in this chapter on noncommutative invariants in algebraic geometry, it is proved that the mapping class group of genus \(g\geq 2\) are linear: They admit a faithful representation into the matrix group \(\text{GL}_{6g-6}(\mathbb Z)\). This chapter includes Elliptic curves, algebraic curves of genus \(g\geq 1\), Tate curves and UHF-algebras and the mapping class group. It should be mentioned that the link between topology and algebraic geometry is explored.
In number theory, noncommutative invariants appear via a restriction of a functor \(F:\mathsf{CRng}\rightarrow\mathsf{Grp-Rng}\) to the arithmetic schemes. An important example is when \(\mathcal E_{\text{CM}}\) is an elliptic curve with complex multiplication. Then \(F(\mathcal E_{\text{CM}})=\mathcal A_{\text{RM}}\) where \(\mathcal A_{\text{RM}}\) is a noncomutative torus with real multiplication. This is used to relate the rank of \(\mathcal E_{\text{CM}}\) to an invariant of \(\mathcal A_{\text{RM}}\). This invariant is called arithmetic complexity, and is used to prove that the complex number \(e^{2\pi i\theta+\log\log\varepsilon}\) is algebraic whenever \(\theta\) and \(\varepsilon\) are algebraic numbers in a real quadratic field. Also, the invariant is used to find generators of the abelian extension of a real quadratic number field. The text includes the definition of an \(L\)-function \(L(\mathcal A_{\text{RM}},s)\) of \(\mathcal A_{\text{RM}}\) and proves that this coincides with the Hasse-Weil function \(L(\mathcal E_{\text{CM}},s)\) of \(\mathcal E_{\text{CM}}\). This localization functor tells that the crossed products is an analogue of the prime ideals used in algebraic geometry. The function \(L(\mathcal A_{\text{RM}},s)\) is extended to the even-dimensional noncommutative tori \(\mathcal A_{\text{RM}}^{2n}\), and an analogue of the Langlands conjecture for such tori is sketched. The number of points of a projective variety \(V(\mathbb F_q)\) over a finite field \(\mathbb F_q\) is computed in terms of the invariants of the Serre \(C^\ast\)-algebra \(F(V_{\mathbb C})\) of the complex projective variety \(V_{\mathbb C}\). Also, this chapter consider isogenies of elliptic curves, symmetry of complex and real multiplication, ranks of elliptic curves, transcendental number theory, class field theory, noncommutative reciprocity, Langlands conjecture for the \(\mathcal A_{\text{RM}}^{2n}\), and finally, projective varieties over finite fields.
The final part of the book gives a survey of Noncommutative algebaic geometry (NCG). The start is the finite geometries and the axioms of projective geometry, including Desargues and Pappus axioms. Then continuous geometries in the weak geometry, von Neumann geometry. In particular, Connes' geometries is considered more deeply, including classification of type III factors, Connes' invariants, noncommutative differential geometry and Connes' index theorem. Chapter 10 contains a survey of Index theory: The Atiyah-Singer theorem and Fredholm operators, the index theorem, \(K\)-homology and Atiyah's realization, Kasparov's \(KK\)-theory. The applications of the index theory include the Novikov conjecture, Baum-Connes conhecture, positive scalar curvature and finally the coarse geometry. After this chapter follows a treatment of Jones polynomials, Braids and the trace invariant. The next to final chapter in the book is about quantum groups,and this includes examples of Hopf algebras already in the introduction to this chapter. Also Manin's quantum plane, Hopf algebras in general, operator algebras and quantum groups are treated. The final chapter of the book is a survey of noncommutative algebraic geometry. This chapter includes the most common theories of Artin, Van den Bergh (turning slightly into derived schemes). It includes the Serre isomorphism, twisted homogeneous coordinate rings, the Sklyanin algebras, and it even mentions the noncommutative algebraic geometry of O.A. Laudal.
Ok, so there is one more chapter, 14, indicating some trends in NCG, but this is most likely separated into its own chapter just because of tridecrafobia.
The book is a good survey of noncommutative geometry, and is an excellent starting point for doing good research in the field. Each section of the book ends with a list of references for going deeply into each subject, and so this book gives a framework for the field of noncommutative geometry. noncommutative tori; Anasov automorphisms; C*-algebras; K-theory; cluster C*-algebras; Sklyanin algebras; AF-algebras; UHF-algebras; Hecke eigenform; continuous geometries; Connes geometries; index theory; Kasparov KK-theory; Jones polynomials; quantum groups; Hopf algebra; noncommutative algebraic geometry; deformation quantization Research exposition (monographs, survey articles) pertaining to algebraic geometry, Noncommutative algebraic geometry, Rings arising from noncommutative algebraic geometry, Categories in geometry and topology Noncommutative geometry. A functorial approach | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Following a method used by H. Maier and G. Tenenbaum, we prove that almost all polynomials in \({\mathbb{F}}_ q[X]\) have two different divisors of the same degree. In fact, we prove the following theorem: Let h(n) be the number of polynomials of degree n in \({\mathbb{F}}_ q[X]\) which have no pair of different divisors of the same degree. Then, for every \(a\in]1/\log (3),1[\) we have
\[
h(n)=O(q^ n[\log (\log n)(\log n)^{- 1/4}]^{g(a)}),\quad where\quad g(a)=1-a+a \log (a),
\]
the constants contained in the O depending on q and a. We deduce of this theorem a result about partitions of integers in pseudo-geometric sequences. polynomials; divisors; partitions of integers; pseudo-geometric sequences Polynomials over finite fields, Algebraic functions and function fields in algebraic geometry Un problème de diviseurs dans \({\mathbb F}_q[X]\). (A divisor problem in \({\mathbb F}_q[X])\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors classify surfaces X with an ample and spanned line bundle \({\mathcal L}\) whose hyperelliptic locus \({\mathcal H}\) is large enough, i.e. dim(\({\mathcal H})\geq 2\), where \({\mathcal H}\) is defined as the closure of the set \(\{C\in | {\mathcal L}|\;| \;C\) smooth hyperelliptic\}. They use an argument by Castelnuovo to put together the \(g^ 1_ 2\)'s of all elements of an irreducible 2-dimensional component \({\mathcal H}_ 0\) of \({\mathcal H}\) and to express X as a double cover. If the degree \(d={\mathcal L}.{\mathcal L}\) is \(\geq 5\), the line bundle \(K_ X\otimes {\mathcal L}\) is spanned unless (X,\({\mathcal L})\) is a scroll. In the first part the authors study the case \(K_ X\otimes {\mathcal L}\) spanned and they consider the adjunction mapping \(\Phi\) defined by \(| K_ X\otimes {\mathcal L}|.\)
They get the classification:
(X,\({\mathcal L})\) is a conic bundle, or:
There exists a morphism of degree two \(\pi: X\to Y\) on a normal surface Y, \(L=\pi^*M\) and they give the possibilities for (Y,M).
To have this result the authors choose a component \({\mathcal H}_ 0\) of dimension \(\geq 2\) and for a point p of X they consider the image \(\gamma\) (p) on X of the closure \(\Gamma\) (p) of \(\{(p',C)\in X\times {\mathcal H}_ 0(p)| i_ C(p)=p'\},\) where \({\mathcal H}_ 0(p)={\mathcal H}_ 0\cap | {\mathcal L}-p|\) and \(i_ C\) is the hyperelliptic involution of C.
In the second part the authors look at the case \(d\leq 4\), they use the classification of polarized surfaces of \(\Delta\)-genus \(\leq 1\) and a case by case analysis of the possibilities for the map f associated to \(| {\mathcal L}|.\)
In the third part they study the case (X,\({\mathcal L})\), a conic bundle with \({\mathcal H}\neq \emptyset\), then the image \(\Phi (X)\simeq {\mathbb{P}}^ 1\) and by looking at the Stein factorization \(X\to^{\alpha}B\to^{\beta}\Phi (X)\) they get:
\(\beta\) is an embedding, then X is rational, or:
\(\beta\) is not an embedding, then B is elliptic or hyperelliptic and \(\beta: B\to {\mathbb{P}}^ 1\) has degree 2.
In the last part the authors use these results to classify the pairs (X,E) where E is an ample and spanned rank-2 vector bundle on the surface X such that \(c_ 2(E)=2\). hyperelliptic locus of line bundle; double cover; polarized surfaces of \(\Delta\)-genus Aldo Biancofiore, Maria Lucia Fania, and Antonio Lanteri, Polarized surfaces with hyperelliptic sections, Pacific J. Math. 143 (1990), no. 1, 9 -- 24. Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Families, moduli, classification: algebraic theory, Divisors, linear systems, invertible sheaves, Elliptic curves, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Polarized surfaces with hyperelliptic sections | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This book shaped article, which is now a title of the prestigious collection of the \textsl{Memoirs of the American Mathematical Society}, is addressed to experts and/or working people in combinatorics and/or cohomology of homogeneous varieties. As the emphasis is put on combinatorics, the book can be also read with no preliminary knowledge of the cohomological theories of homogeneous spaces, at the obvious cost of losing most of its tasty flavor. It is about the descripton of the product structure of the affine Grassmannian, which is an infinite dimensional kind of flag manifold.
Just not to lose those readers not familiar with the subject, it is worth recalling that homogeneous varieties are ubiquitous in mathematics. The most popular examples are projective spaces or, more generally, Grassmannian varieties \(G(k,V)\) parameterizing \(k\)-dimensional subspaces (in the following called \(k\)-\textsl{planes}) of a fixed complex vector space \(V\). The algebraic group \(Gl(V)\) acts transitively on the set of all \(k\)-planes, and the Grassmannian \(G(k,V)\) can be viewed as the quotient \(Gl(V)/H\), where \(H\) is the isotropy subgroup of one (and hence any) \(k\)-plane.
More generally, a complex homogeneous variety is the quotient \(G/P\) of a complex algebraic group \(G\) modulo a parabolic subgroup \(P\). The ``parabolicity'' required here is to ensure that the quotient space is a complete variety indeed. Homogeneous varieties come equipped with cellular decompositions, by means of the \textsl{Bruhat cells} (see e.g. the book by [\textit{T.~A.~Springer}, Linear Algebraic Groups. 2nd ed. Progress in Mathematics (Boston, Mass.). 9. Boston, MA: Birkhäuser. (1998; Zbl 0927.20024)]). The \textsl{Schubert cycles} are the homology classes of the closure of the Bruhat cells and, by general topological facts, they generate the homology module of the homogeneous varieties. The point is that Schubert cycles are usually parametrized by combinatorial objects, such as \textsl{partitions} (like in the case of the finite Grassmannian) or \textsl{permutations} (as in the case of complete flag varieties). Since a homogeneous variety is smooth (if there is a smooth point, all the points are smooth), Schubert cycles can be viewed, via Poincaré duality, as elements of a \({\mathbb Z}\)-module basis for the cohomology of \(G/P\). Schubert calculus says how to express the product of two Schubert cycles as an integral linear combination of Schubert cycles.
The product structure of the cohomology of Grassmannians, Flag Varieties and other kinds of homogeneous varieties, can be dealt with combinatorial techniques, based on Young Diagrams, Young Tableaux (suitable filling of Young Diagrams by integers) and other devices masterly described in the popular Fulton's book [\textit{W. Fulton}, Young Tableaux. With applications to representation theory and geometry. London Mathematical Society Student Texts. 35. Cambridge: Cambridge University Press. (1997; Zbl 0878.14034)]. The latter pays a special attention to an important \textsl{insertion algorithm} due to Robinson-Schensted-Knuth (RSK) for semistandard Young tableaux. The reader is invited to look at the Fulton's book for the basic vocabulary. In fact, purely combinatorially speaking, the paper--book under review aims to generalize the RSK algorithm to deduce a Pieri's rule for the so called \textsl{affine Grassmannian} associated to the group \(G:=Sl(n,{\mathbb C})\). It is the main character of the paper, an infinite dimensional object, defined as follows. Let \({\mathbb C}[[t]]\) be the \({\mathbb C}\)-algebra of formal power series in one indeterminate \(t\) and let \({\mathbb C}((t)):={\mathbb C}[[t]][t^{-1}]\) be the corresponding \({\mathbb C}\)-algebra of formal Laurent series. Then the affine Grassmannian \(Gr\), is the quotient \(G({\mathbb C}((t)))/G({\mathbb C}[[t]])\), where for a \({\mathbb C}\)-algebra \(A\), one lets \(G(A)\) being the set of all the \(A\)-valued points of \(G\), i.e. the set of all \({\mathbb C}\)-algebra homomorphisms \(G\rightarrow A\). It turns out that \(Gr\) is a homogeneous space in the same sense explained above because it is a quotient \({\mathcal G}/{\mathcal P}\), where \({\mathcal G}\) is a Kac-Moody group and \({\mathcal P}\) is a parabolic subgroup of it. In this case the elements of the Schubert basis for both homology and cohomology are parameterized by the subset \(\widetilde{S}^0_n\) of the elements of minimal length in their cosets \(\widetilde{S}_n/S_n\), called \textsl{Grassmannian elements}. They are analogous of the Grassmannian permutations which can be alternatively used, instead of partitions, to parameterize Schubert bases for the cohomology of the ordinary Grassmannian. The authors then introduces \textsl{dual \(k\)-Schur functions}, which are generating functions of certain \(k\)-tableaux [\textit{L. Lapointe} and \textit{J. Morse}, Trans. Am. Math. Soc. 360, No. 4, 2021--2040 (2008; Zbl 1132.05060)]. The notion of tableau itself needs to be refined within this infinite dimensional framework, which leads to distinguish \textsl{weak tableaux} from \textsl{strong tableaux}, a classification already introduced in some authors' previous work-- see e.g. [\textit{T. Lam}, Am. J. Math. 128, No. 6, 1553--1586 (2006; Zbl 1107.05095)].
The essential juice of the paper is Theorem 4.2. It establishes an amazing bijection between the set of all nonnegative integers matrices with row sums less than \(n\) and pairs \((P,Q)\) with \(P\) a strong tableau and \(Q\) a weak tableau with the same shape. The proof is based on the \textsl{Fomin's growth diagrams} [\textit{S. Fomin}, J. Algebr. Comb. 4, No.1, 5--45 (1995; Zbl 0817.05077)]. Affine insertion makes evident a duality between the weak and strong orders of the affine symmetric group which have not appeared in any previous literature. Such a bijection reduces to the usual RSK row insertion algorithm as \(n\) tends to \(\infty\). Thanks to the algorithm itself, the authors are able to establish Pieri rules for the Schubert bases of \(H^*(Gr)\) and \(H_*(Gr)\), expressing the product of a special Schubert class and an arbitrary Schubert class as a \({\mathbb Z}\)-linear combination of Schubert classes. In addition, the new combinatorial definition for \(k\)-Schur functions introduced in the paper permits to represent Schubert bases of \(H_*(Gr)\). The corresponding cap product pairing between cohomology and homology, well known in classical finite dimensional Grassmannians, is interpreted in a elegant pure combinatorial fashion. A natural question arises. To which extent the methods and results exposed in the book can be extended to the more complicated case of affine flag varieties? The authors offer a partial answer proposing the conjectural extension of their Pieri's rule to such finer framework.
The organization of the book is as follows. After a comprehensive introductions, listing problems, questions, motivations, results and conjectures, Chapter 1 aims to smooth the learning path of the reader by collecting a list of preliminary facts one needs to read the subsequents chapters (such as, e.g., a review about the affine symmetric group). Strong and Weak Tableaux are introduced and discussed in Chapter 2 and 3. The core of the book is Chapter 4, where the affine insertion algorithm is stated and proven. The last two chapters explain affine insertions using the combinatorial language of \textsl{cores}, while chapter 8 cares of the translation of the picture in terms of partitions. Chapters between 5 and 7 investigate the consequences of the insertion algorithm and, in particular, Chapter 7 contains the proof of the bijectivity of the affine insertion.
In conclusion, this is a very well structured paper, cut around the shape of the reader wishing to learn the subject in a systematic way. It is not self contained, but the final bibliography is able to supply references to find all the necessary prerequisites to begin such an exciting mathematical adventure. Robinson-Schensted insertion; Schubert calculus; Pieri formula; affine Grassmannian Lam, T., Lapointe, L., Morse, J., Shimozono, M.: Affine insertion and Pieri rules for the affine Grassmannian. Mem. Am. Math. Soc. \textbf{208}, 977 (2010) Research exposition (monographs, survey articles) pertaining to algebraic geometry, Classical problems, Schubert calculus, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Affine insertion and Pieri rules for the affine Grassmannian | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(\mathcal O_q(M_{m,n}(k))\) be a generic coordinate \(k\)-algebra on rectangular matrices of size \(m\times n\). It is assumed that \(m\leqslant n\). If \(I=\{i_1<\cdots<i_u\}\), \(K=\{k_1<\cdots<k_v\}\subset M=\{1,\dots,m\}\) and \(J=\{j_1<\cdots<j_u\}\), \(L=\{l_1<\cdots<l_v\}\subset\{1,\dots,n\}\), then \((I,J)\leqslant_{st}(K,L)\) if and only if \(u\leqslant v\) and \(i_s\leqslant k_s\), \(j_s\leqslant l_s\) for all possible indices \(s\). Denote by \(\Pi_{m,n}\) the set of all pairs of indices \((I,J)\) in which \(u=m\).
The quantum Grassmannian \(\mathcal O_q(G_{m,n}(k))\) is the subalgebra in \(\mathcal O_q(M_{m,n}(k))\) generated by all \(m\times m\) quantum minors. In the authors' previous paper [J. Algebra 301, No. 2, 670-702 (2006; Zbl 1108.16026)], it is shown that there is a vector basis in \(\mathcal O_q(G_{m,n}(k))\) consisting of monomials \([I_1\mid M]\cdots [I_t\mid M]\) such that \((I_1,M)\leqslant_{st}\cdots\leqslant_{st}(I_t,M)\).
Take \(\gamma\in\Pi_{m,n}\) and denote by \(\Pi_{m,n}^\gamma\) the set of all \(\alpha\in\Pi_{m,n}\) such that \(\alpha\ngeqslant_{st}\gamma\). The quantum Schubert variety \(\mathcal O_q(G_{m,n}(k))_\gamma\) associated with \(\gamma\) is the algebra \(\mathcal O_q(G_{m,n}(k))\) factorized by the ideal generated by \(\Pi_{m,n}^\gamma\). It is shown that
\[
\mathcal O_q(G_{m,n}(k))[Y^{\pm 1};\varphi]\simeq\mathcal O_q(G_{m,n}(k))_\gamma[\gamma^{-1}].
\]
It is given a criterion under which a quantum Schubert variety has left and right finite injective dimensions. Let \(I_t\) be the ideal in \(\mathcal O_q(M_{m,n}(k))\) generated by all minors of a fixed size \(t\). It is shown that \(\mathcal O_q(M_{m,n}(k))/I_t\) is a normal domain. rings arising in quantum group theory; generic coordinate algebras; quantum Grassmannians; quantum minors; quantum Schubert varieties; normal domains T. H. Lenagan and L. Rigal, Quantum analogues of Schubert varieties in the Grassmannian, Glasg. Math. J. 50 (2008), no. 1, 55 -- 70. , Grassmannians, Schubert varieties, flag manifolds, Divisibility, noncommutative UFDs, Rings arising from noncommutative algebraic geometry, Noetherian rings and modules (associative rings and algebras) Quantum analogues of Schubert varieties in the Grassmannian. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials It has been shown recently that the normalized median Genocchi numbers are equal to the Euler characteristics of the degenerate flag varieties. The \(q\)-analogues of the Genocchi numbers can be naturally defined as the Poincaré polynomials of degenerate flag varieties.
We prove that the generating function of the Poincaré polynomials can be written as a simple continued fraction. As an application we prove that the Poincaré polynomials coincide with the \(q\)-version of the normalized median Genocchi numbers introduced by Han and Zeng. Poincaré polynomials of degenerate flag varieties Feigin, E, The Median Genocchi numbers, \(q\)-analogues and continued fractions, Eur. J. Comb., 33, 1913-1918, (2012) Factorials, binomial coefficients, combinatorial functions, \(q\)-calculus and related topics, Grassmannians, Schubert varieties, flag manifolds, Bernoulli and Euler numbers and polynomials The median Genocchi numbers, \(q\)-analogues and continued fractions | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials 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 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials An index calculus attack for the elliptic curve discrete logarithm over extension fields requires the determination of relations between random points and the points of the factor base. For this task, \textit{P. Gaudry} [J. Symb. Comput. 44, No. 12, 1690--1702 (2009; Zbl 1177.94148)] gave a decomposition algorithm based on Samaev's summation polynomials. On the other hand, \textit{K. E. Stange} [Lect. Notes Comput. Sci. 4575, 329--348 (2007; Zbl 1151.94570)] gave a decision method for the decomposition problem using multi-variable elliptic functions. In this paper, the authors present some relations between the two methods and prove that the corresponding index calculus attacks are equivalent. Samaev's summation polynomials; Stange's elliptic nets; index calculus attack Applications to coding theory and cryptography of arithmetic geometry, Elliptic curves Some relations between Semaev's summation polynomials and Stange's elliptic nets | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The book under review offers a comprehensive discussion of the classical theories of algebraic curves and surfaces extended to fields of positive characteristics, and that from both the modern and the classical viewpoint.
The first part of the book is devoted to the study, classification, and description of the forms of the affine line. In particular, the automorphism group of the forms, their divisor class group, and the birational model of hyperelliptic forms of the affine line are explicitly described.
The second part of the books deals with the study of surfaces, their fibrations, and resolution of singularities. It begins with a chapter on vector fields where a correspondence between the vector fields and the action of infinitesimal groups is built. It continues with the study of (affine) Zariski surfaces and the data of their resolution of singularities. Moreover, the (quasi) elliptic fibrations and the characterization of the birational model of a unirational surface equipped with such fibrations are treated. In particular, the classification of the reducible fibers of rational (quasi) elliptic surfaces is provided. As the main core of this part, the geometry of the Artin-Schreier coverings of a smooth projective surface is studied.
In the third part, the authors give a detailed account of the theory of rational double points, their classification via the correspondence between the Du Val graphs and normal forms. The classification of simple singularities, an algorithm to detect the rational double points, and a description of the equisingular locus of all types of rational double points are provided.
Before the publication of the book under review, most of its material was dispersed among numerous research articles, and it is very convenient to get it collected together. One of the most attractive features of the book is the exhibition of elucidating examples, remarks, and open problems at the end of chapters. The authors have provided detailed proofs and explicit calculations over fields of different characteristics. algebraic curves; algebraic surfaces; forms; automorphisms; fibrations; coverings; vector fields; singularities; rational double points Research exposition (monographs, survey articles) pertaining to algebraic geometry, Special surfaces, Positive characteristic ground fields in algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Automorphisms of surfaces and higher-dimensional varieties Algebraic surfaces in positive characteristics. Purely inseparable phenomena in curves and surfaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let B denote a Borel subgroup of a semi-simple group G. \(P^-\) stands for the set of B-modules of dimension 1 such that \(H^ 0(G/B,{\mathcal L}(\lambda))\neq 0\). \({\mathcal S}\) denotes the class of B-modules, having a Schubert filtration. The main result is as follows: For every union S of Schubert varieties, every antidominant weight \(\lambda\) and every \(\mu \in P^-\), \(M:=H^ 0(S,\lambda)\otimes \mu\) is a B-module, having a Schubert filtration that allows an explicit description. Furthermore, if S is irreducible, M allows an excellent filtration (meaning that M allows a composition series, with quotients isomorphic to \(H^ 0(X_ w,\lambda)\). This work, - though independently developed, - extends the work of Mathieu (characteristic zero). Borel subgroup; semi-simple group; Schubert filtration; Schubert varieties; antidominant weight; excellent filtration Polo, P, Modules associés aux variétés de Schubert, C. R. Acad. Sci. Paris Sér. I Math., 308, 123-126, (1989) Cohomology theory for linear algebraic groups, Representation theory for linear algebraic groups, Grassmannians, Schubert varieties, flag manifolds Modules associés aux variétés de Schubert. (Modules associated with Schubert varieties) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(X\) be a smooth projective curve and \(L\in \text{Pic} (X)\). The author studies the injectivity the symmetric multiplication map \(\mu_L: S^2(H^0(X,L))\to H^0(X, L^{\otimes 2})\) for certain double coverings of high genus curves. double covering of curves Coverings of curves, fundamental group, Divisors, linear systems, invertible sheaves, Special divisors on curves (gonality, Brill-Noether theory) On the injectivity of the symmetric multiplication map for line bundles on smooth projective curves. II. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In [ibid. 45, No. 1, 25--74 (2009; Zbl 1225.11151)], \textit{A. Abbes} and \textit{T. Saito} define a measure of wild ramification of \(\ell\)-adique sheaves at a generic point of a complete trait of characteristic \(p\), with \(p\neq \ell\). Adapting their construction to differential modules of characteristic zero, we show for such a module \(\mathcal{M}\) a formula which expresses that geometric invariant in terms of differential forms appearing in the Levelt-Turrittin decomposition of \(\mathcal{M}\). differential modules; Levelt-Turrittin polynomials; nearby cycles Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Sheaves of differential operators and their modules, \(D\)-modules The construction of Abbes and Saito for meromorphic connections: formal aspects of dimension 1 | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Consider the Grassmannian \(\mathcal G_K(k,n)\) over any field \(K\), some flags \(F_i\), \(i=1,\dots,m\), and consider the corresponding sets of conditions \(\Omega(F_i)\) on the intersection of \(k\)-planes with the elements of each \(F_i\). This datum defines a Schubert problem \(\omega=\{\Omega(F_i)\}\). The author studies the case where the expected set of solutions is finite. The problem \(\omega\) is enumerative over \(K\) when one finds the expected number \(\deg(\{\Omega(F_i)\})\) of solutions, formed by distinct \(k\)-planes, all of them defined over \(K\).
The author proves that any Schubert problem is enumerative over \(\mathbb R\), as well as over algebraically closed field of any characteristic. Moreover, he finds that Schubert problems over finite fields are enumerative in a set of positive density. One tool for proving the results is an extension of the Bertini-Kleiman smoothness condition. The author proves that the map from the set of solutions in the universal Grassmannian, to the product of \(m\) copies of the flag variety, is generically smooth.
The author finally studies the Galois group of an enumerative problem, for the Grassmannians of lines and planes. Schubert cycles R. Vakil, Schubert induction. Ann. Math. 164, 489--512 (2006) Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Enumerative problems (combinatorial problems) in algebraic geometry Schubert induction | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author formulates an approach to supergravity by using the idea of Penrose transformation. Recall that results of Ogevetsky, Sokatchev and Schwarz imply a ''compensation'' of the hyperbolicity of Einstein equations by the non-commutativity of the superextension so that supergravity may be described by means of complex variables. In the author's formulation the main structure is a complex supermanifold, i.e. a sheaf of analytic Grassmann algebras. The examples are superspaces of flags of complex superdimension 4N. At the beginning the author presents supergeometry of the aforementioned flag spaces. Then he defines a so called Frobenius form for general supermanifolds. This form helps him to prove two formulae of decomposition of sheaves of Berezinian's into objects generated by left and right spinors. The author observes that the Lagrangian of simple supergravity is equal to the Berezinian of a transformation between two distinguished frames. In the paper he defines these frames explicitly by means of the Frobenius form, the Ogevetsky- Sokatchev prepotential and the real structure on the base supermanifold. In this way the presented theory becomes mathematically complete. The author notes that his approach leads to a contradiction with physical models of extended supergravity in cases \(N=2,4\). The reviewed paper is the first one where all main data of a supergravity were formulated in terms of advanced global analytic structures. super Schubert cells; analytic supermanifold; Frobenius form; supergravity; Penrose transformation; complex supermanifold; sheaf of analytic Grassmann algebras; flag spaces; Berezinian Applications of global differential geometry to the sciences, Grassmannians, Schubert varieties, flag manifolds, Sheaves and cohomology of sections of holomorphic vector bundles, general results, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Constructive quantum field theory, General relativity Geometry of supergravity and super Schubert cells | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors study equidistribution of solutions of word equations of the form \(w(x,y)=g\) in the family of finite groups \(\mathrm{SL}(2,q)\). They provide criteria for equidistribution in terms of the trace polynomial of \(w\). This allows them to get an explicit description of certain classes of words possessing the equidistribution property and show that this property is generic within these classes. word maps; finite groups of Lie type; equidistribution; trace polynomials Bandman, T.; Kunyavskii, B., Criteria for equidistribution of solutions of word equations on \(S L(2)\), J. Algebra, 382, 282-302, (2013) Algebraic geometry over groups; equations over groups, Linear algebraic groups over finite fields, Simple groups: alternating groups and groups of Lie type, Probabilistic methods in group theory, Arithmetic and combinatorial problems involving abstract finite groups, Classical groups, Rational points, Finite ground fields in algebraic geometry Criteria for equidistribution of solutions of word equations on \(\mathrm{SL}(2)\). | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [Part I: \textit{P. Berthelot}, Ann. Sci. Éc. Norm. Supér., IV. Sér. 29, No. 2, 185-272 (1996; Zbl 0886.14004)].
The paper under review is another step in the author's systematic study of rings of differential operators in crystalline cohomology. These are filtered by the niveau which measures what type of factorials appear in the denominator. The main result is that Frobenius raises the niveau by 1 and induces (up to that change of niveau) an equivalence of categories of \(D\)-modules. This is preceded by a comparison of left and right \(D\)-modules, which are exchanged by Grothendieck-Hartshorne duality. Finally it is shown that Frobenius commutes with the usual (``six'') operations, and applications to Frobenius-action on cohomology and \(F\)-\(D\)-modules are given. \(D\)-module; \(F\)-crystal; \(p\)-curvature; adjoint operator; rings of differential operators; crystalline cohomology; niveau; Grothendieck-Hartshorne duality; Frobenius-action Berthelot, Pierre, \(\mathcal{D}\)-modules arithmétiques. II. Descente par Frobenius, Mém. Soc. Math. Fr. (N.S.), 81, vi+136 pp., (2000) \(p\)-adic cohomology, crystalline cohomology, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, de Rham cohomology and algebraic geometry, Rings of differential operators (associative algebraic aspects), Sheaves of differential operators and their modules, \(D\)-modules \({\mathcal D}\)-modules arithmétiques. II: Descente par Frobenius. (Arithmetic \({\mathcal D}\)-modules. II: Frobenius descent) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This note is an announcement of the observation of an example of a mirror symmetry phenomenon [in the sense \textit{A. Strominger, S.-T. Yau} and \textit{E. Zaslow}, ``Mirror symmetry is T-duality'', Nucl. Phys. B 479, No. 1-2, 243-259 (1996; Zbl 0896.14024)]. The authors describe some hyper-Kähler varieties which are equipped with special Lagrangian torus fibrations, dual to each other. The main result of this paper claims that the stringy mixed Hodge polynomials of these varieties are equal. This provides mathematical evidence in support of mirror symmetry.
One of the hyper-Kähler manifolds is Simpson's moduli space of local systems over a smooth projective curve with structure group \(SL(n)\). Its mirror partner, which corresponds to the Langlands dual group \(PGL(n)\), is obtained as a quotient of such a moduli space. It is an orbifold. Both spaces are non-compact, which seems to be the reason for the equality of the Hodge numbers rather than their usual mirror relation. The torus fibrations arise from Hitchin systems on moduli spaces of stable Higgs bundles (such a moduli space is diffeomorphic to a moduli space of local systems). It is conjectured that similar results hold for any reductive algebraic group and its Langlands dual group. This would strongly relate mirror symmetry to Langlands duality.
At the moment this review was written, the full version, with proofs and improved results, was available as preprint only [math.AG/0205236 on arXiv.org: \textit{T. Hausel} and \textit{M. Thaddeus}: ``Mirror symmetry, Langlands duality, and the Hitchin system'']. In the full version, the meaning and function of the B-field have been clarified. Notation, background and results are formulated more clearly and proofs can be found there. Nevertheless, the paper under review might still serve as a bilingual introduction to the full version. hyperKähler manifold; orbifold; representation of fundamental group; E-polynomial; Hodge polynomial; Hitchin map; SYZ fibration; mirror partner; stringy mixed Hodge polynomials; mirror symmetry; Langlands duality Hausel, T.; Thaddeus, M.: Examples of mirror partners arising from integrable systems. C. R. Acad. sci. Paris sér. I math. 333, No. 4, 313-318 (2001) Calabi-Yau manifolds (algebro-geometric aspects), Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Vector bundles on curves and their moduli, Relationships between algebraic curves and integrable systems, Transcendental methods of algebraic geometry (complex-analytic aspects), Hyper-Kähler and quaternionic Kähler geometry, ``special'' geometry Examples of mirror partners arising from integrable systems | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(\Sigma\) be an exterior differential system and let \(B_ k\) be the fiber bundle of ordinary integral k-flags of \(\Sigma\). The main result is: If b is a fixed ordinary integral k-flag then in a neighborhood of b the set of ordinary integral k-elements of \(\Sigma\) is a Schubert manifold (a submanifold of a projective Grassmann manifold). exterior differential system; integral k-flag; Schubert manifold Exterior differential systems (Cartan theory), Grassmannians, Schubert varieties, flag manifolds, Integral geometry Schubert conditions for integral elements of an exterior differential system | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Numerical Campedelli surfaces are smooth minimal surfaces of general type with \(p_g=0\) and \(K^2=2\). Although they have been studied by several authors, their complete classification is not known. In this paper the authors classify numerical Campedelli surfaces with an involution, i.e. an automorphism of order 2. Using results on involutions on surfaces of general type with \(p_g=0\) [cf. \textit{A. Calabri, C. Ciliberto} and \textit{M. M. Lopes}, Trans. Am. Math. Soc. 359, No. 4, 1605--1632 (2007; Zbl 1124.14036)], they show that an involution \(\sigma\) on a numerical Campedelli surface \(S\) has either four or six isolated fixed points, and the bicanonical map of \(S\) is composed with the involution if and only if the involution has six isolated fixed points. In the latter case they prove that the ramification divisor \(R\) on \(S\) is not 0, and the quotient surface \(S/\sigma\) is either birational to an Enriques surface or a rational surface; if \(S/\sigma\) is rational, then there are four possible cases and each of the four cases actually occurs. If the involution has four isolated fixed points, they show that the ramification divisor \(R\) is either \(0\) or constituted by one, two or three \(-2-\)curves. In this case there are more possibilities for the quotient surface \(S/\sigma\):
1)\(S/\sigma\) is of general type (a numerical Godeaux surface) if and only if the ramification divisor \(R\) is equal to 0;
2) if \(R\) is not empty and irreducible, then \(S/\sigma\) is properly elliptic;
3) if \(R\) has two or three components then \(S/\sigma\) may be rational or birational to an Enriques surface or properly elliptic. There are examples for all cases, except when the quotient surface is a rational surface for this case. The authors also study a family of numerical Campedelli surfaces with torsion \(\mathbb{Z}_3^2\), showing that every surface in this family has two involutions, one with four isolated fixed points and one with six isolated fixed points, whose quotients are respectively birational to a numerical Go deaux surface and a rational surface. Finally, they study the involutions of numerical Campedelli surfaces with torsion \(\mathbb{Z}_2^3\), the so-called ``classical Campedelli surfaces''. Using the description of these surfaces as a \(\mathbb{Z}_2^3\)-cover of \(\mathbb{P}^2\) branched on 7 lines, they show that these involutions are all composed with the bicanonical map. Campedelli surfaces; involutions on surfaces; double covers Calabri, A., Mendes Lopes, M., Pardini, R.: Involutions on numerical Campedelli surfaces. Tohoku Math. J. 60(1), 1--22 (2008) Surfaces of general type Involutions on numerical Campedelli surfaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper gives a detailed analysis of the resolution of an isolated double point of a surface, i.e. an isolated singularity of multiplicity two of an algebraic surface. These double points are locally given by an equation \( z^2=f(x,y),\) where \(f\) is a square-free polynomial such that the curve \(f=0\) has a singular point at \((0,0)\). The paper explains and complements the papers by \textit{D. J. Dixon} [Pac. J. Math. 80, 105--117 (1979; Zbl 0422.14003)] and \textit{H. B. Laufer} [Isr. J. Math. 31, 315--334 (1978; Zbl 0415.14003)] on the same subject.
The results of the paper are: 1. relations between the canonical and the minimal resolution of a double point, 2. relations and formulas for cycles connected to the resolution process: fundamental cycle and fiber cycle, 3. computations of conditions that a double point imposes to canonical and pluricanonical systems of a surface. surface singularity; double point; resolution of singularity; fundamental cycle; fiber cycle A. Calabri - R. Ferraro, Explicit resolutions of double point singularities of surfaces, Collect. Math. 53 (2002), 99--131. Singularities of surfaces or higher-dimensional varieties, Complex surface and hypersurface singularities, Global theory and resolution of singularities (algebro-geometric aspects), Singularities in algebraic geometry Explicit resolutions of double point singularities of surfaces. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The commutative ring \(R(P(t))=\mathbb C[t^{\pm 1},u\mid u^2=P(t)]\), where \(P(t)=\sum_{i=0}^na_it^i=\prod_{k=1}^n(t-\alpha_i)\) with \(\alpha_i\in \mathbb C\) pairwise distinct, is the coordinate ring of a hyperelliptic curve when \(n>4\). The Lie algebra \(\mathcal {R}(P(t))=\operatorname{Der}(R(P(t)))\) of derivations is called the hyperelliptic Lie algebra associated to \(P(t)\) and is a particular type of multipoint Krichever-Novikov algebra. In this paper, we describe the universal central extension of \(\operatorname{Der}(R(P(t)))\) in terms of certain families of polynomials which in a particular case are associated Legendre polynomials. Moreover we describe certain families of polynomials that arise in the study of the group of units for the ring \(R(P(t)\)), where \(P(t)=t^4-2bt^2+1\). In this study, pairs of Chebyshev polynomials \((U_n,T_n)\) arise as particular cases of a pairs \((r_n,s_n)\) with \(r_n+s_n\sqrt{P(t)}\) a unit in \(R(P(t)\)). We explicitly describe these polynomial pairs as coefficients of certain generating functions and show that certain of these polynomials satisfy particular second-order linear differential equations. Krichever-Novikov algebras; automorphism groups; Pell's equation; associated Legendre polynomials; universal central extensions; superelliptic Lie algebras; superelliptic curves; DJKM algebras; Fáa di Bruno's formula; Bell polynomials Infinite-dimensional Lie (super)algebras, Riemann surfaces; Weierstrass points; gap sequences, Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) Certain families of polynomials arising in the study of hyperelliptic Lie algebras | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For the entire collection see Zbl 0655.00011.]
The author classifies quasi-smooth hypersurfaces V of degree \(h\) in a weighted projective space \({\mathbb{P}}(a,b,c,1)\) belonging to some classes described below. Let
\[
\chi (T)=T^{-h}(T^ h-T^ a)(T^ h-T^ b)(T^ h-T^ c)/(T^ a-1)(T^ b-1)(T^ c-1)=T^{m_ 1}+T^{m_ 2}+...+T^{m_ r}
\]
for some integers \(m_ 1,...,m_ r\), called the exponents. The smallest exponent \(\epsilon =a+b+c-h\) has the meaning that the canonical sheaf of V is equal to \({\mathcal O}_ V(-\epsilon)\). The author studies the following classes: (i) \(\epsilon <0\) but all other exponents are positive; (ii) \(\epsilon <0\) with some other exponents equal to 0; (iii) \(\epsilon =-2.\)- This includes the classification of such surfaces with \(\epsilon =-1.\)
As was shown earlier by the reviewer there are 31 types of such surfaces corresponding to the 31 types of Fuchsian cocompact groups whose algebra of automorphic forms is generated by three elements [Funct. Anal. Appl. 9, 149-151 (1975); translation from Funkts. Anal. Prilozh. 9, No.2, 67-68 (1975; Zbl 0321.14003) and in Group actions and vector fields, Proc. Pol.-North Am. Semin., Vancouver 1981, Lect. Notes Math. 956, 34-71 (1982; Zbl 0516.14014)]. They are all K3-surfaces with double cyclic singularities. Class (i) consists of 49 types, including 22 types corresponding to the Fuchsian groups of genus 0. The smallest exponent takes the value from the set \(\{-1,-2,-3,-4,-7\}\). There are 12 types of surfaces in class (ii), including 9 types corresponding to the Fuchsian groups of positive genus. Here \(\epsilon\in \{-1,-2,-3\}\). Finally, there are 21 types of surfaces in class (iii). There are 7 types defining K3- surfaces, 8 types corresponding to elliptic surfaces and the remaining 6 types are Horikawa's surfaces of general type. Bibliography; hypersurfaces in weighted projective space; K3-surfaces with double cyclic singularities K. Saito, Algebraic surfaces for regular systems of weights, to appear, preprint RIMS-563. Families, moduli, classification: algebraic theory, Projective techniques in algebraic geometry, Complete intersections Algebraic surfaces for regular systems of weights | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A theta divisor on an abelian variety \(A\) over a field is an effective ample divisor \(\Theta\) such that the global sections of the line bundle \(O_{A}(\Theta)\) are one-dimensional. For example, the image of the \((g-1)\)st symmetric power of a smooth projective curve \(C\) of genus \(g\) with a rational point in its Jacobian is a theta divisor. Berthelot, Bloch and Esnault proved the following
Theorem 1.1. Let \(\Theta, \Theta'\) be two divisors on an abelian variety, all defined over a finite field \({\mathbb F}_{q.}\) Then \( | \Theta ({\mathbb F}_{q})| \equiv | \Theta' ({\mathbb F}_{q})| \mod q. \)
Question 1.2. Is \(| \Theta | \equiv | \Theta' | \mod {\mathbb L} \in K_{0}(\mathrm{Var}_{k})\)?
Here \(K_{0}(\mathrm{Var}_{k})\) denotes the naive Grothendieck ring of varieties over \(k,\) i.e. the free abelian group on isomorphism classes \([X]\) of varieties over \(k\) modulo the relations \([X] = [X -Y] + [Y]\) for \(Y \subset X\) a closed subvariety. The ring structure is given by product of varieties, and \({\mathbb L}\) is the class \([A^{1}_{k}]\) of the affine line. The author gives a negative answer to the above question using the fact that products of two elliptic curves can be Jacobians. symmetric power of a smooth projective curve; naive Grothendieck ring of varieties Theta functions and abelian varieties, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Arithmetic ground fields for abelian varieties, Divisors, linear systems, invertible sheaves, Étale and other Grothendieck topologies and (co)homologies, Theta functions and curves; Schottky problem, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Elliptic curves, Finite ground fields in algebraic geometry A note on congruences for theta divisors | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Immunity against differential cryptanalysis techniques represents an important criterion when analysing the security of symmetric ciphers. In the light of Lai's results regarding higher order derivatives and their applications to differential cryptanalysis, the current paper can be seen as a natural and technically valuable follow-up of [\textit{J. F. Voloch}, in: Algebraic geometry and its applications. Dedicated to Gilles Lachaud on his 60th birthday. Proceedings of the first SAGA conference, Papeete, France, May 7--11, 2007. Hackensack, NJ: World Scientific. 135--141 (2008; Zbl 1151.14319)]. Thus, a density theorem which may be considered an extension of Voloch's main result is stated and proved. More precisely, instead of analysing the differential uniformity of a polynomial \(f \in \mathbb{F}_q[x]\), where \(q=2^n\), the authors explore the ``second order differential uniformity''.
The paper is structured in eight sections. The first section discusses introductory aspects, a very short presentation of the article's structure and establishes notations. Sections 2 to 6 cover all the technical details necessary to construct the main theorem of the paper and prove it in section 7. Section 8 provides information about an inversion mapping which is of great importance in the study of (good) S-boxes. A specific instantiation of the previously mentioned inversion mapping is used precisely in the case of AES, a block cipher which is widely adopted nowadays.
The mathematical concepts and properties discussed in each section are generally presented in a clear and accessible manner for (graduate) students and more experienced readers (especially if interested in differential cryptanalysis). The lemmas, the propositions and the theorems stated in the paper are accompanied by well written proofs.
As a side note regarding the structure of the paper (more like personal opinions rather than shortcomings), a number of things which could have added readability or completion (especially for cryptography enthusiasts) are described next:
1. The \(Introduction\) lacks a reference to section 8 (only for uniformity, as the other sections are briefly tackled).
2. Usually, papers submitted to cryptography conferences of journals include a liaison with real world applications. Nonetheless, the current article was published in the Journal of Pure and Applied Algebra and, thus, it is all right for the writing style to be rather arid (and lacking motivation). The importance of the results is not clearly underlined in real world scenarios (e.g., concepts like S-boxes and block ciphers like AES are vaguely mentioned in section 8).
3. A section including future work would have been interesting.
Given the above, we recommend the readers interested in differential cryptanalysis to attentively read this paper and, maybe, extend its results as they are of clear importance to symmetric cryptography.
In conclusion, the current article is a valuable research work for both mathematicians and cryptographers. second order derivative; differential uniformity; generic polynomials; density theorem; differential cryptanalysis; second order differential uniformity; inversion mapping Applications to coding theory and cryptography of arithmetic geometry, Algebraic coding theory; cryptography (number-theoretic aspects), Cryptography Differential uniformity and second order derivatives for generic polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In the theory of abelian functions on Jacobians, the key role is played by entire functions that satisfy the Riemann vanishing theorem. Here we introduce polynomials that satisfy an analog of this theorem and show that these polynomials are completely characterized by this property. By rational analogs of abelian functions we mean logarithmic derivatives of orders \(\geq 2\) of these polynomials. We call the polynomials thus obtained the Schur-Weierstrass polynomials because they are constructed from classical Schur polynomials, which, however, correspond to special partitions related to Weierstrass sequences. Recently, in connection with the problem of constructing rational solutions of nonlinear integrable equations [\textit{M. Adler} and \textit{J. Moser}, Commun. Math. Phys. 61, 1--30 (1978; Zbl 0428.35067) and \textit{I. M. Krichever}, Zap. Nauchn. Sem. LOMI, 84, No. 1, 117--130 (1979; Zbl 0413.35008)], special attention was focused on Schur polynomials [\textit{J. J. Duisterman} and \textit{F. A. Gruenbaum}, Commun. Math. Phys. 103, 177--204 (1986; Zbl 0625.34007) and \textit{V. G. Kac}, Infinite Dimensional Lie Algebras, Birkhäuser (1983; Zbl 0584.17007)]. Since a Schur polynomial corresponding to an arbitrary partition leads to a rational solution of the Kadomtsev-Petviashvili hierarchy, the problem of connecting the above solutions with those defined in terms of abelian functions on Jacobians naturally arose. One results open the way toward solving this problem on the basis of the Riemann vanishing theorem. KP hierarchy; Jacobians; Riemann vanishing theorem; Schur-Weierstrass polynomials Bukhshtaber, VM; Ènol'skiĭ, VZ; Leĭkin, DV, Rational analogues of Abelian functions, Funct. Anal. Appl., 33, 83-94, (1999) Relationships between algebraic curves and integrable systems, Jacobians, Prym varieties, Riemann surfaces; Weierstrass points; gap sequences, Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions Rational analogues of Abelian functions. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper, related to Hilbert's 17th problem, the authors give a sketch of the functional-analytic proof for the \textit{B. Reznick}'s result [Math. Z. 220, 75-97 (1995; Zbl 0828.12002)]. Any positive homogeneous polynomial on \(\mathbb{R}^n\) is written as a sum of squares of polynomials, all divided by a power of \(x^2_1+x^2_2 +\cdots+x^2_n\). The proof is based on simple geometric constructions and a separation theorem of convex cones in a topological vector space. Hilbert's 17th problem; sum of squares of polynomials; convex cones M. Putinar and F.-H. Vasilescu, \textit{Positive polynomials on semialgebraic sets}, C. R. Math. Acad. Sci. Paris, 328 (1999), pp. 585--589. Semialgebraic sets and related spaces, Fields related with sums of squares (formally real fields, Pythagorean fields, etc.), General theory of locally convex spaces Polynômes positifs sur des ensembles semi-algébriques | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Consider an elliptic curve \(E\) over a number field \(K\). Suppose that \(E\) has supersingular reduction at some prime \(\mathfrak{p}\) of \(K\) lying above the rational prime \(p\). We completely classify the valuations of the \(p^n\)-torsion points of \(E\) by the valuation of a coefficient of the \(p{\text{th}}\) division polynomial. This classification corrects an error in earlier work of Lozano-Robledo. As an application, we find the minimum necessary ramification at \(\mathfrak{p}\) in order for \(E\) to have a point of exact order \(p^n\). Using this bound we show that sporadic points on the modular curve \(X_1(p^n)\) cannot correspond to supersingular elliptic curves without a canonical subgroup. We generalize our methods to \(X_1(N)\) with \(N\) composite. elliptic curves; torsion points; division fields; torsion fields; division polynomials; sporadic points; modular curves Elliptic curves over global fields, Elliptic curves Ramification in division fields and sporadic points on modular curves | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Zhegalkin zebra motives are tilings of the plane by black and white polygons representing certain \(\mathbb{F}_2\)-valued functions on \(\mathbb{R}^2\). They exhibit a rich geometric structure and provide easy to draw insightful visualizations of many topics in the physics and mathematics literature. The present paper gives some pieces of a general theory and a few explicit examples. Many more examples will be shown in the forthcoming article ``Zhegalkin zebra motives: algebra and geometry in black and white''. Zhegalkin polynomials; motives; dimer models; mirror symmetry Tilings in \(2\) dimensions (aspects of discrete geometry), Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics, Toric varieties, Newton polyhedra, Okounkov bodies Zhegalkin zebra motives digital recordings of mirror symmetry | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper deals with the number of critical points of real polynomials in two variables and reports the work done at a research seminar for undergraduate students. number of critical points of real polynomials; two variables Durfee, A., Kronenfeld, N., Munson, H., Roy, J. and Westby, I. (1993). Counting critical points of real polynomials in two variables. Amer. Math. Monthly 100 255-271. JSTOR: Real polynomials: analytic properties, etc., Polynomials in real and complex fields: location of zeros (algebraic theorems), Topology of real algebraic varieties Counting critical points of real polynomials in two variables | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In [\textit{J. Briançon}, \textit{Y. Laurent} and \textit{Ph. Maisonobe}, C. R. Acad. Sci., Paris, Sér. I 313, No. 5, 285-288 (1991; Zbl 0743.32011)] the equivalence between the existence of relative Bernstein polynomials associated with a deformation \(F\) of isolated singularities hypersurfaces and equisingularity properties was proved. After having defined a natural equisingularity notion for an analytic map \((F_1,\dots,F_p)\) defined on \(\mathbb{C}^3\), we show that the existence of relative Bernstein polynomials still characterizes the equisingularity and is also equivalent to a ``non-characteristic'' type condition. existence of relative Bernstein polynomials; equisingularity H. Biosca, Caractérisation de l'existence de polynômes de Bernstein relatifs associés à une famille d'applications analytiques, C. R. Acad. Sci. Paris Sér. I Math. 325 (1997), no. 4, 395-398. Equisingularity (topological and analytic), Deformations of complex singularities; vanishing cycles, Global theory and resolution of singularities (algebro-geometric aspects) Characterization of the existence of relative Bernstein polynomials associated to a family of analytic maps | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We introduce genomic tableaux, with applications to Schubert calculus. We report a combinatorial rule for structure coefficients in the torus-equivariant \(K\)-theory of Grassmannians for the basis of Schubert structure sheaves. This rule is positive in the sense of \textit{D. Anderson} et al. [J. Eur. Math. Soc. (JEMS) 13, No. 1, 57--84 (2011; Zbl 1213.19003)]. We thereby deduce an earlier conjecture of \textit{H. Thomas} and \textit{A. Yong} [``Equivariant Schubert calculus and jeu de taquin
'', Ann. Inst. Fourier (to appear)] for the coefficients. Moreover, our rule specializes to give a new Schubert calculus rule in the (non-equivariant) \(K\)-theory of Grassmannians. From this perspective, we also obtain a new rule for \(K\)-theoretic Schubert structure constants of maximal orthogonal Grassmannians, and give conjectural bounds on such constants for Lagrangian Grassmannians. Schubert calculus; equivariant \(K\)-theory; Grassmannians; genomic tableaux Pechenik, O., Yong, A.: Genomic tableaux and combinatorial \(K\)-theory. Discrete Math. Theor. Comput. Sci. Proc. \textbf{FPSAC'15}, 37-48 (2015) Combinatorial aspects of representation theory, Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds Genomic tableaux and combinatorial \(K\)-theory | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We construct an upgrade of the motivic volume by keeping track of dimensions in the Grothendieck ring of varieties. This produces a uniform refinement of the motivic volume and its birational version introduced by Kontsevich and Tschinkel to prove the specialization of birational types. We also provide several explicit examples of obstructions to stable rationality arising from this technique. rationality of algebraic varieties; Grothendieck ring of varieties; motivic nearby fiber Arcs and motivic integration, Applications of methods of algebraic \(K\)-theory in algebraic geometry A refinement of the motivic volume, and specialization of birational types | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper introduces schemes called ``correspondence scrolls'' and gives a general study of them. For a closed subscheme \(Z\) in \(\Pi _{i=1}^n {\mathbb A}^{a_i+1}\), defined by a multigraded ideal \(I\subset A:={\Bbbk}[x_{i,j} : 1\le i \le n, 0 \le j \le a_i]\), and for \(\mathbf{b}=(b_1, \ldots , b_n) \in \mathbb{N}_+^n\), consider the homomorphism \({\Bbbk}[z_{i,\alpha }] \rightarrow A/I\) which sends a variable \(z_{i, \alpha }\) to the monomial \(x_i^\alpha\), of degree \(b_i\). Here \(x_i^ {\alpha }\) denotes \(x_{i,0}^{\alpha _0} \cdots x_{i, a_i}^{\alpha _{a_i}}\). The kernel of the above map defines a closed projective subscheme \(C(Z, {\mathbf b}) \subset { \mathbb P}^N\) (\(N= \sum \binom{a_i+b_i}{a_i}-1)\), called \textit{correspondence scroll}. This definition includes classical correspondences as well as interesting non-classical ones: rational normal scrolls, double structures which are degenerate \(K3\) surfaces, degenerate Calabi-Yau threefolds, etc. Many invariants or properties of correspondence scrolls are studied: dimension, degree, nonsingularity, Cohen-Macaulay and Gorenstein property and others. The paper is very well written and invites to further research. rational normal scroll; Veronese embedding; join variety; multiprojective space; variety of complexes; variety of minimal degree; double structure; \(K3\) surface; Calabi-Yau scheme; Gorenstein ring; Gröbner basis \(n\)-folds (\(n>4\)), Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Linkage, complete intersections and determinantal ideals, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Rational and ruled surfaces, \(K3\) surfaces and Enriques surfaces, Calabi-Yau manifolds (algebro-geometric aspects), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal), Determinantal varieties, Rational and unirational varieties, Projective techniques in algebraic geometry Correspondence scrolls | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For the entire collection see Zbl 0563.00006.]
This is a survey note about some recent questions and results concerning the Weierstrass points (W-points), on families of curves. Let C be a smooth complete curve of genus g and \(\alpha =\alpha (C,p)\) the Schubert index of the point \(p\in C\), where \(\alpha =(\alpha_ 0,\alpha_ 1,...,\alpha_{g-1})\). The point p is a Weierstrass one if \(\alpha\) \(\neq (0,...,0)\). Let \(M_ g\) be the moduli spaces of curves and \(C_ g\) the moduli spaces of pointed curves. There is a stratification of \(C_ g\) by W-points such that for each index \(\alpha\) one has \(C_{\alpha}=\{(C,p),\alpha (C,p)=\alpha \}.\) The authors discuss some old and new problems on the subject as for instant: the determination of \(\alpha\) so that \(C_{\alpha}=\emptyset\), the codimension of \(C_{\alpha}\), the geometric relationship between \(C_{\alpha}\), the locus of the W-points in \(C_ g\), and so on. Many of these questions are still open. Weierstrass points; Schubert index; moduli spaces of curves Eisenbud, D., Harris, J.: Recent progress in the study of Weierstrass points. Conf. Proceedings, Rome 1984. Lect. Notes Math. (to appear) Families, moduli of curves (analytic), Riemann surfaces; Weierstrass points; gap sequences, History of algebraic geometry, Families, moduli of curves (algebraic), History of mathematics in the 20th century Recent progress in the study of Weierstrass points | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(E_1\) and \(E_2\) be two isogenous elliptic curves defined over a number field and \(g:E_1\to E_2\) be an isogeny between them. \textit{F. Pazuki} [Int. J. Number Theory 15, No. 3, 569--584 (2019; Zbl 1446.11126)] proved that
\[
|h(j(E_1))-h(j(E_2))|\leq 9.204+12\ln(\deg(g)),
\]
where \(h(j(E_1))\) (\(h(j(E_2))\) respectively) is the Weil height of the \(j\)-invariant of \(E_1\) (\(E_2\) respectively). In the present article, the main goal of the authors is to give a function field analogue of Pazuki's result. Before we state the main result, we introduce a few notation and definitions.
Let \(\mathbb{F}_q\) be the finite field with \(q\) elements and \(A:=\mathbb{F}_q[t]\) be the set of polynomials in \(t\) with coefficients from \(\mathbb{F}_q\). We define \(F:=\mathbb{F}_q(t)\), the fraction field of \(A\), and let \(K\) be a finite extension of \(F\). We denote the set of places of \(K\) (\(F\) respectively) by \(M_{K}\) (\(M_F\) respectively) and for each place \(v\in M_K\), we denote its absolute value by \(|\cdot|_v\) normalized in a way that \(|\alpha|_v=|\alpha|_w\) for any \(\alpha\in F\) and \(w\in M_F\) lying under \(v\). For each \(v\in M_K\), we further set \(K_v\) (\(F_v\) respectively) to be the completion of \(K\) (\(F\) respectively) at \(v\) and define \(n_v:=[K_v:F_v]\).
Let \(L\) be any, not necessarily finite, extension of \(F\). We define \(L[\tau]\) to be the non-commutative polynomial ring subject to the condition
\[
\tau c=c^q\tau, \ \ c\in L.
\]
Let \(r\geq 2\) be an integer. \textit{A Drinfeld \(A\)-module \(\phi\) of rank \(r\) defined over \(K\)} is an \(\mathbb{F}_q\)-algebra homomorphism \(\phi:A\to K[\tau]\) given by
\[
\phi_t:=\phi(t)=t+g_1\tau+\dots+g_r\tau^{r}, \ \ g_1,\dots,g_r\in K, \ \ g_r\neq 0.
\]
We further define \textit{the height \(h_G(\phi)\) corresponding to the coefficients \(g_1,\dots,g_r\) of \(\phi\)} by
\[
h_G(\phi):=\frac{1}{[K:F]}\sum_{v\in M_K}n_v\log_q(\max\{|g_1|_v^{1/(q-1)},|g_2|_v^{1/(q^2-1)},\dots,|g_r|_v^{1/(q^r-1)} \}).
\]
Let \(\overline{K}\) be a fixed algebraic closure of \(K\). Each element \(u=c_0+c_1\tau+\dots+c_s\tau^s\in \overline{K}[\tau]\) has an action on \(\overline{K}\) given by \( u(x):=c_0x+c_1x^q+\dots+c_sx^{q^s}\in \overline{K}. \) Thus we can now also define the set \(Ker(u):=\{x\in \overline{K} \ \ |u(x)=0\}\).
Let \(\phi'\) be another Drinfeld \(A\)-module of rank \(r\) defined over \(K\). \textit{An isogeny \(f:\phi\to \phi'\) of \(\phi\) and \(\phi'\)} is a non-zero element of \(\overline{K}[\tau]\) satisfying \(f\phi_t=\phi'_tf\). We also set \(\deg(f)\) to be the cardinality of \(Ker(f)\).
For a given isogeny \(f:\phi\to \phi'\), there exists an isogeny (dual isogeny) \(\hat{f}:\phi'\to \phi\) with \(\deg(\hat{f})\leq \deg(f)^{r-1}\) satisfying \(\hat{f} f=\phi_N\) and \(f\hat{f}=\phi'_N\) where \(N\in A\) is an element of minimal degree such that \(Ker(f)\subset Ker(\phi_N)\). The main result of the present paper then shows that (see Theorem 3.1)
\[
|h_G(\phi')-h_G(\phi)|\leq \Big(\frac{q}{q-1}-\frac{q^r}{q-1}\Big) + \deg_t(N).
\]
Since \(\deg_t(N)=r^{-1}(\log_q(\deg(f))+\log_q(\deg(\hat{f})))\leq \log_q(\deg(f))\), one can now obtain
\[
|h_G(\phi')-h_G(\phi)|\leq \Big(\frac{q}{q-1}-\frac{q^r}{q-1}\Big)+ \log_q(\deg(f))
\]
which can be seen as a function field analogue of Pazuki's result.
The proof of the main result mainly uses the analytic estimates achieved by \textit{E. U. Gekeler} in [``On the Drinfeld discriminant function'', Compos. Math. 106, 181--202 (1997); J. Théor. Nombres Bordx. 29, No. 3, 875--902 (2017; Zbl 1411.11041); J. Reine Angew. Math. 754, 87--141 (2019; Zbl 1444.11124)] and the work of \textit{Y. Taguchi} [J. Number Theory 44, No. 3, 292--314 (1993; Zbl 0781.11024)].
As an application of Theorem 3.1, the authors also introduce, in Proposition 6.5, an upper bound on the coefficients of Drinfeld modular polynomials in the rank 2 case which are the objects firstly defined by \textit{S. Bae} [J. Number Theory 42, No. 2, 123--133 (1992; Zbl 0754.11018)] in an analogy with the classical modular polynomials. Drinfeld modules; heights; isogenies; modular polynomials Drinfel'd modules; higher-dimensional motives, etc., Heights, Positive characteristic ground fields in algebraic geometry, Arithmetic varieties and schemes; Arakelov theory; heights Heights and isogenies of Drinfeld modules | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For the entire collection see Zbl 0741.00020.]
The paper summarizes recent results by Matsuki, Oshima, Uzawa, Brion, Vinberg on the space \(H\backslash G/P\) of double cosets of a connected real semisimple Lie group \(G\). Here \(P\) is a minimal parabolic subgroup of \(G\), \(G/P\) is the corresponding flag manifold, and \(H\) is (almost) the fixed point subgroup of an involutive automorphism of \(G\). Among the topics discussed here (often without proof) are:
--- the ``symbol'' of a double coset, when \(G\) is a complex classical group (\(G=GL(n,\mathbb{C})\), \(SO(n,\mathbb{C})\) or \(Sp(n,\mathbb{C})\)).
--- Uzawa's function and vector field on \(G/P\),
--- spherical subgroups of a complex semisimple Lie group \(G\). double cosets; connected real semisimple Lie group; parabolic subgroup; flag manifold; complex semisimple Lie group Matsuki, T.: Orbits on flag manifolds. In: Proceedings of the International Congress of Mathematicians, Kyoto 1990, Vol. II. Springer, pp. 807-813 (1991) Semisimple Lie groups and their representations, Grassmannians, Schubert varieties, flag manifolds, Differential geometry of homogeneous manifolds Orbits on flag manifolds | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Wu Wen-tsün's method of mechanical theorem proving in geometry is complete for certain elementary geometry problems involving equality only. For the corresponding algebraic geometry problem, the method is complete for problems with an algebraically closed field as the associated field. The authors present a theorem to extend some theoretical property of Wu's method from algebraically closed fields to arbitrary fields. The theorem implies a decision method for the membership of the set of all polynomials which fix a certain type of algebraic variety denoted by \(V^*\) by Wu Wen-tsün. mechanical theorem proving in geometry; membership of the set of all polynomials Computational aspects in algebraic geometry, A decision method for certain algebraic geometry problems | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The \(\mathcal{S}\)-cone provides a common framework for cones of polynomials or exponential sums which establish non-negativity upon the arithmetic-geometric inequality, in particular for sums of non-negative circuit polynomials (SONC) or sums of arithmetic-geometric exponentials (SAGE). In this paper, we study the \(\mathcal{S}\)-cone and its dual from the viewpoint of second-order representability. Extending results of \textit{G. Averkov} [SIAM J. Appl. Algebra Geom. 3, No. 1, 128--151 (2019; Zbl 1420.90043)] and of
\textit{J. Wang} and \textit{V. Magron} [``A second order cone characterization for sums of nonnegative circuits'', Preprint, \url{arXiv:1906.06179}] on the primal SONC cone, we provide explicit generalized second-order descriptions for rational \(\mathcal{S}\)-cones and their duals. positive polynomials; sums of non-negative circuit polynomials; arithmetic-geometric exponentials; dual cone; \(\mathcal{S}\)-cone; second-order cone Polynomial optimization, Semialgebraic sets and related spaces, Convex sets in \(n\) dimensions (including convex hypersurfaces), Convex programming, Semidefinite programming, Forms over real fields The \(\mathcal{S}\)-cone and a primal-dual view on second-order representability | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(K_0(\mathcal{V}_k)\) be the Grothendieck group of \(k\)-varieties. The first and third author [``Devissage and localization for the Grothendieck spectrum of varieties'', Preprint, \url{arXiv:1811.08014}] have constructed a higher algebraic \(K\)-theory spectrum \(K(\mathcal{V}_k)\) such that \(\pi_0 K(\mathcal{V}_k) = K_0(\mathcal{V}_k)\). In this paper we construct non-trivial classes in the higher homotopy groups of \(K(\mathcal{V}_k)\) when \(k\) is finite or a subfield of C. To do this we give a recipe for lifting motivic measures \(K_0(\mathcal{V}_k) \rightarrow K_0(\mathcal{E})\) to maps of spectra \(K(\mathcal{V}_k) \rightarrow K(\mathcal{E})\). We consider two special cases: the classical local zeta function, thought of as a homomorphism \(K_0(\mathcal{V}_{\mathbb{F}_q}) \rightarrow K_0(\mathrm{End}(\mathbb{Q}_\ell))\), and the compactly-supported Euler characteristic, thought of as a homomorphism \(K_0(\mathcal{V}_{\mathbb{C}}) \rightarrow K_0(\mathbb{Q})\). We use lifts of these motivic measures to prove that the Grothendieck spectrum of varieties contains nontrivial geometric information in its higher homotopy groups by showing that the map \(\mathbb{S} \rightarrow K(\mathcal{V}_k)\) is nontrivial in higher dimensions when \(k\) is finite or a subfield of C, and, moreover, that when \(k\) is finite this map is not surjective on higher homotopy groups. \(K\)-theory; zeta function; \(K\)-theory of varieties; Grothendieck group of varieties; motivic measure Applications of methods of algebraic \(K\)-theory in algebraic geometry, Zeta functions and \(L\)-functions, Motivic cohomology; motivic homotopy theory, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects), Grothendieck groups (category-theoretic aspects), Stable homotopy theory, spectra Derived \(\ell\)-adic zeta functions | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(X\) be a smooth complex projective threefold and \(\pi:X\rightarrow C\) a surjective morphism with a section onto a smooth curve \(C\) such that every geometric fibre is a union of smooth surfaces of multiplicity one with normal crossings. In this case the Leray spectral sequence \(E_{2}^{p,q}=H^{p}(C,R^{q}\pi_{\star}\mathbb{Q})\Rightarrow H^{\star}(X,\mathbb{Q})\) degenerates [\textit{S. Zucker}, Ann. Math. (2) 109, 415--476 (1979; Zbl 0446.14002)]. By a careful analysis of the associated exact sequences, the author proves the Grothendieck standard conjecture \(B(X)\) of Lefschetz type under certain natural assumptions on the Hodge structures on certain pieces \(E_{2}^{p,q}\). In particular the conjecture is shown to be true when the generic fibre of \(\pi\) is birationally equivalent to either a ruled surface, an Enriques surface, or a \(K3\)-surface. algebraic cycles; Grothendieck standard conjecture; threefolds С. Г. Танкеев, ``О стандартной гипотезе типа Лефшеца для комплексных проективных трехмерных многообразий'', Изв. РАН. Сер. матем., 74:1 (2010), 175 -- 196 Algebraic cycles, Classical real and complex (co)homology in algebraic geometry, \(3\)-folds On the standard conjecture of Lefschetz type for complex projective threefolds | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper under review is concerned with the small quantum Schubert Calculus on flag varieties, namely homogeneous spaces of the form \(G/B\), where \(G\) is a connected simply connected complex Lie group and \(B\) is a Borel subgroup of it. Each homogeneous space carries a natural cellular decomposition and so, by general results of algebraic topology, the homology classes of the closure of the affine cells provide a basis of the homology over the integers, whose elements are said to be Schubert cycles. Poincaré duality holds in \(G/B\), because of its smoothness, and so the Schubert cocycles, Poincaré dual of the Schubert cycles, form a basis of the cohomology ring of the flag variety.
Schubert calculus of \(G/B\) amounts to knowing the \textsl{Schubert structure constants}, according to the authors' terminology, i.e., the structural constants of the cohomology algebra with respect to the basis of Schubert (co)cycles. In the case when \(G=SL(n+1, {\mathbb C})\) and \(B\) is the Borel subgroup of the triangular matrices, the picture is very well understood, as one falls in the usual intersection theory of the Grassmann variety: the structure constants are traditionally known as Littlewood--Richardson coefficients.
The two main theorems concern the small quantum cohomology \(QH^*(G/B)\) of flag varieties, i.e., the determination of what the authors call the \textsl{quantum Schubert constants}. Here \(QH^*(G/B)\) is a deformation of \(H^*(G/B)\): the support is \(H^*(G/B)\otimes_{\mathbb Z}{\mathbb C}[[t]]\) while the product structure is obtained by ``correcting'' the classical product by means of the appropriate Gromow-Witten invariants, which basically count number of rational curves in \(G/B\) having suitable incidence properties with respect to some given configuration of Schubert varieties.
The first main theorem computes the quantum Schubert structural constants of \(G/B\) by means of a certain function involving some combinatorially defined quantities, which are rational functions in simple roots. Its precise and detailed explanation in a review would not be more helpful for the interested reader. The second theorem computes, through a very explicit formula, the structural constants of the Pontryagin products in \(H^T_*(\Omega K)\), which is the equivariant (Borel-Moore) homology, with respect to the action of a \(n\)-dimensional torus \(T\), of the based loop group of the maximal compact subgroup \(K\) of \(G\).
The beginning of the paper consists in a careful introduction followed by a review of general knowledges about Kac--Moody algebras and a set up of notation (Section 2). Section 3 is devoted to deduce an explicit formula for the Pontryagin product on \(H^T_* (\Omega K)\) while Section 4 uses the formula to prove the Main Theorem. The sequence of instructive examples of Section 5 give the measure of how is the formula effective. The appendix is finally devoted to provide the proofs of some properties stated in Section 4. Quantum Schubert Calculus of homogeneous spaces; Pontryagin product, loop groups, Equivariant homology and cohomology of homogeneous spaces, Gromov-Witten invariants Leung, N. C.; Li, C., Gromov-Witten invariants for \(G / B\) and Pontryagin product for {\(\omega\)}\textit{K}, Trans. Amer. Math. Soc., 5, 2567-2599, (2012) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Infinite-dimensional Lie groups and their Lie algebras: general properties Gromov-Witten invariants for \(G/B\) and Pontryagin product for \(\Omega K\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For the entire collection see Zbl 0722.00006.]
The Hartshorne-Rao module of a curve \(C\subset\mathbb{P}^ 3_ k\) is defined as \(M_ C=\sum_{n\in\mathbb{Z}}H^ 1({\mathcal I}_ C(n)))\), where \({\mathcal I}_ C\) denotes the ideal sheaf of \(C\). Let \(C_ i\), \(i=1,2\), be two curves. Is there a construction of a curve \(C\) (in terms of \(C_ i)\) such that \(M_ C\) is isomorphic to some shift of \(M_{C_ 1}\oplus M_{C_ 2}\)? This is solved by \textit{P. Schwartau} in his unpublished thesis ``Liaison addition and monomial ideals'' [Ph. D. thesis, Brandeis University 1982), see also \textit{J. Stückrad} and \textit{W. Vogel} [``Buchsbaum rings and applications. An interaction between algebra, geometry, and topology'', VEB Deutscher Verlag der Wissenschaft (Berlin 1986; Zbl 0606.13017); published simultaneous by Springer Verlag)] for a reproduction of his arguments.
In this paper the authors generalize Schwartau's result in several senses: (1) There is a liaison addition in any codimension in \(\mathbb{P}^ n_ k\), \(n\geq 3\), of schemes of mixed codimensions. --- (2) There is an addition of any number of schemes. --- (3) There is no need that the schemes are attached via a complete intersection (as in Schwartau's result).
The basic idea is the construction of a scheme \(Z\) consisting set- theoretically of the union of certain given schemes such that the cohomology of \(Z\) is related to those of the given schemes in terms of a long exact sequence. This far reaching generalization yields, in the case the underlying scheme is arithmetically Cohen-Macaulay, that the cohomology of \(Z\) is the shift of the direct sum of the involved schemes. There are a number of applications of this clever technique to the construction of arithmetically Buchsbaum schemes, the double linkage, Hilbert functions, etc. decomposition of Hartshorne-Rao module; arithmetical Cohen-Macaulay scheme; liaison addition; arithmetically Buchsbaum schemes; double linkage; Hilbert functions Linkage, Linkage, complete intersections and determinantal ideals A generalized liaison addition | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials It has been conjectured by \textit{A. Rényi} [Hung. Acta Math. 1, 30--34 (1947; Zbl 0030.11402)] and \textit{P. Erdős} [Nieuw Arch. Wiskd., II. Ser. 23, 63--65 (1949; Zbl 0032.00203)] that if a complex polynomial \(f\) has \(T\) terms, and \(f^2\) has \(t\) terms, then \(t\) tends to infinity with \(T\). This conjecture has been established by \textit{A. Schinzel} [Acta Arith. 49, No. 1, 55--70 (1987; Zbl 0632.12024)]; see also Sect. 2.6 of his book [Polynomials with special regard to reducibility. Cambridge: Cambridge University Press (2000; Zbl 0956.12001)] in a stronger form, replacing the square of \(f\) by any power \(f^s\) and obtaining the bound \(t\ge c(s)\log\log T\). Later \textit{A. Schinzel} and \textit{U. Zannier} [Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 20, No. 1, 95--98 (2009; Zbl 1221.11068)] improved this bound to \(t\ge 2+\log(T-1)/\log(4s)\) .
In this paper the authors provide an essential generalization by showing in Theorem 1.1 that if \(f(x,y)\) is a complex polynomial, monic of degree \(d\ge1\) in \(y\), having \(l\) terms in \(x\), and the polynomial \(g(x)\) satisfies \(f(x,g(x))=0\), then \(g(x)\) has at most \(B=B(d,l)\) terms. They note also that this theorem can be stated also in the following form (Theorem 1.2):
If \(f(t_1,\dots, t_l,y)\) is a non-constant complex polynomial, monic in \(y\) and of degree \(\le d\) in each variable, and if the polynomial \(g(x)\) satisfies
\[
f(x^{n_1},\dots,x^{n_l},g(x))=0
\]
with some natural numbers \(n_1,\dots,n_l\), then \(g(x)\) has at most \(B_1=B_1(d,l)\) terms.
A similar result holds also for rational complex functions \(g(x)\). In this case one does not have to assume that \(f\) is monic in \(y\), and the assertion states that \(g(x)\) is the ratio of two polynomials each having at most \(B_5=B_5(d,l)\) terms (Theorem 2.2). Other variations of Theorem 1.2, dealing with factorizations of \(f(x^{n_1},\dots,x^{n_l})\), are given by Theorems 2.1 and 2.3. There are several other consequences of the main results, one of them being an analogue of Bertini's theorem for covers of tori (Theorem 1.5). Rényi-Erdős conjecture; Bertini theorem; polynomials with few terms Polynomials in number theory, Nonstandard arithmetic (number-theoretic aspects), Polynomials in general fields (irreducibility, etc.), Rational points, Surfaces and higher-dimensional varieties On fewnomials, integral points, and a toric version of Bertini's theorem | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The main result of the paper is a characterization of those Schubert varieties for the general linear group that are local complete intersection (lci). The authors prove that a Schubert variety \(X_w\) associated to a permutation \(w \in S_m\) is lci if and only if \(w\) avoids the six patterns 53241, 52341, 52431, 35142, 42513, and 35162.
The paper is organized as follows. Section 2 presents the basic definitions and set-up. Sections 3 and 4 prove that Schubert varieties associated to permutations avoiding the given patterns are lci. The proof involves showing that the variety \(X_w\) is lci at the identity point by identifying a minimal set generators for its defining ideal. This uses the notion of the \textit{essential set} of \(w\) introduced in [\textit{W. Fulton}, Duke Math. J. 65, No. 3, 381--420 (1992; Zbl 0788.14044)] and Schubert varieties \textit{defined by inclusions} from work of \textit{V. Gasharov} and \textit{V. Reiner} [J. Lond. Math. Soc., II. Ser. 66, No. 3, 550--562 (2002; Zbl 1064.14056)].
Section 5 proves the necessity of the pattern avoidance. The strategy is to identify a collection of intervals \([u,v]\) in the Bruhat order for which the corresponding slice is not lci. The authors then show that if \(w\) contains one of the six given patterns, then \(w\) \textit{interval contains} one of the intervals \([u,v]\) above and they conclude that \(X_w\) is not lci.
Section 6 contains a number of applications, including formulas for Kostant polynomials and presentations of cohomology rings for lci Schubert varieties. Section 7 concludes with a number of open questions. Schubert varieties; local complete intersection; pattern avoidance \beginbarticle \bauthor\binitsH. \bsnmÚlfarsson and \bauthor\binitsA. \bsnmWoo, \batitleWhich Scubert varieties are local complete intersections? \bjtitleProc. Lond. Math. Soc. (3) \bvolume107 (\byear2013), page 1004-\blpage1052. \endbarticle \endbibitem Grassmannians, Schubert varieties, flag manifolds, Reflection and Coxeter groups (group-theoretic aspects), Complete intersections, Combinatorial aspects of commutative algebra, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Permutations, words, matrices Which Schubert varieties are local complete intersections? | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials ``This book'', as we can read on the back cover, ``aims to promote constructive mathematics, not by defining it or formalizing it, but by practicing it, by basing all definitions and proofs on finite algorithms''. And this is exactly what the book does. The author chose to write about the fundamental theorem of algebra, Galois theory, Gauss's composition of binary quadratic forms and the quadratic reciprocity law, the genus of algebraic curves, as well as the spectral theorem in linear algebra. He does so by sticking to Kronecker's philosophy that everything in mathematics should be reduced to calculation with natural numbers.
The first two chapters deal with polynomial rings and Galois theory; in particular, the author discusses the factorization of polynomials with integer coefficients. These two chapters are a must-read for anyone interested in how Galois theory was perceived by people like Abel, Galois and Kronecker.
In Chapter 3, the theory of binary quadratic forms is discussed. Since the author chose not to use negative integers (giving ``the theory a pleasing economy of structure'') or integrally closed rings, he bumps into all the problems that were resolved by Dedekind; in particular his class group is a semigroup. After stubbornly refusing to use negative numbers to the extent of remarking that the values of the Legendre symbol may be taken to be the residue classes \(1\) and \(3 \bmod 4\), he finally accepts them (``as Gauss did'') when starting to discuss the composition of forms.
Chapter 4 is about algebraic curves. After a few historical remarks on Abel's theorem and Euler's addition formula for elliptic integrals, he introduces the genus; this is followed by an interlude on Newton polygons, a description of an algorithm for determining the genus, and a discussion of the theorem of Riemann-Roch. This is again a beautiful introduction to the classical works of Abel and Riemann that cannot readily be found elsewhere.
The last chapter contains remarks on the fundamental theorem of algebra, an existence proof (of the Sylow theorems) by contradiction that the author claims is not ``not constructive'', and ends with a tongue-in-cheek comparison of the author's Linear Algebra (1995; Zbl 0823.15001) with \textit{S. Axler}'s Linear Algebra Done Right (1997; Zbl 0886.15001). A final essay deals with a remark often attributed to Kronecker according to which he once claimed that irrational numbers do not exist.
In the preface the author claims that he believes that the most interesting parts of mathematics can be dealt with constructively. If this is true, then why is almost all the material covered in this book more than 150 years old?
The formulation of the last two sentences
does not convey the intended meaning, which was:
1. The author's constructive approach deals
with time-honored topics; the development
of the material, on the other hand, is original.
2. The reviewer would be highly (but pleasantly)
surprised if a similar constructive approach
in the spirit of Kronecker could be successfully
applied to topics that have dominated
algebra, number theory, and algebraic
geometry in the last 50 years. (Revised version) constructivism; Galois theory; factorization of polynomials; splitting field; binary quadratic forms; composition of forms; Newton's polygon; genus; theorem of Riemann-Roch; Sylow's theorems [Edwards 2005] Edwards, H. \textit{Essays in Constructive Mathematics}. Springer-Verlag, New York. Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to field theory, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry Essays in constructive mathematics | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This article considers the relation between deformations of a rational surface singularity with a reflexive module, and deformations of a partial resolution of the singularity with the locally free strict transform of the module. The results indicate how a family of small resolutions of a 3-dimensional index one terminal singularity and its flop are obtained by blowing up in a maximal Cohen-Macaulay module and its syzygy.
The article is motivated by the work on the geometrical McKay correspondence which can be said to give a one-to-one correspondence between the isomorphism classes of indecomposable reflexive modules \(\{M_i\}\) and the prime components \(\{E_j\}\) of the exceptional divisor in the minimal resolution \(\tilde X\rightarrow X\) of a rational double point (RDP), that is \(A_n\), \(D_n\), \(E_{6 - 8}.\)
For a natural class of special reflexive modules named Wunram modules after its inventor, the correspondence holds for any rational surface singularity. \textit{M. Van den Bergh} [Duke Math. J. 122, No. 3, 423--455 (2004; Zbl 1074.14013)] used the endomorphism ring of a higher dimensional Wunram module to prove derived equivalences for flops, and this again led to attention to the \(2\)-dimensional case with interesting results by \textit{O. Iyama} and \textit{M. Wemyss} [Math. Z. 265, No. 1, 41--83 (2010; Zbl 1192.13012); Ill. J. Math. 55, No. 1, 325--341 (2011; Zbl 1258.13015)] and \textit{M. Wemyss} [Math. Ann. 350, No. 3, 631--659 (2011; Zbl 1233.14012)].
In this article, the authors prove that blowing up a rational surface singularity \(X\) in a reflexive module \(M\) gives a partial resolution \(f:Y\rightarrow X\) where \(Y\) is normal, dominated by the minimal resolution , and where the strict transform \(\mathcal M=f^\Delta(M)\) is locally free. This partial resolution is determined by the first Chern class \(c_1(\mathcal F)\) of the strict transform \(\mathcal F\) of \(M\) to \(\tilde X.\) Thus, in particular, any partial resolution dominated by the minimal resolution is given by blowing up in a Wunram module, and the authors mention the RDP-resolution obtained by contracting the \((-2)\)-curves in the minimal resolution in particular; this is given by blowing up in the canonical module \(\omega_X.\)
Consider the category of deformations \(\text{Def}_{Y,\mathcal M}\) of the pair \((Y,\mathcal M)\) blowing down to \((X,M)\). The main result in the present article says that the blowing down map \(\alpha:\text{Def}_{Y,\mathcal M}\rightarrow\text{Def}_{X,M}\) is injective and that it commutes with the forgetful map \(\beta:\text{Def}_{Y,\mathcal M}\rightarrow\text{Def}_{Y}\) and the blowing down map \(\delta:\text{Def}_Y\rightarrow\text{Def}_X.\) Furthermore, the forgetful map \(\beta\) is smooth and an isomorphism in many situations. The blowing down map \(\delta\) is a Galois covering onto the Artin component \(A\) on spaces, which for RDPs equals \(\text{Def}_X\). In general, it not injective, making the injectivity of \(\alpha\) surprising. The authors prove that \(\beta\) is an isomorphism if \(M\) is Wunram, implying that \(\delta\) factors through a closed embedding \(\alpha\beta^{-1}:\text{Def}_Y\subseteq\text{Def}_{(X,M)}\) realizing deformations of the pair as conjectured by \textit{C. Curto} and \textit{D. R. Morrison} [J. Algebr. Geom. 22, No. 4, 599--627 (2013; Zbl 1360.14053)] in the RDP case.
A deformation of the pair \((X,M)\) in the geometric image of \(\text{Def}_{(Y,\mathcal M)}\) lifts to a deformation of \((Y,\mathcal M)\) without any base change. In general, \(\text{Def}_{X,M}\) is not dominated by \(\text{Def}_{(Y,\mathcal M)}\) even for RDPs. A main ingredient in Wahl's proof that the covering \(\text{Def}_{\tilde X}\rightarrow A\) has Galois action by a product of Weyl groups is the injectivity of \(\delta\) in the case that \(Y\) is the RDP resolution. This follows directly from the authors result because \(\text{Def}_{X,\omega_X}\cong\text{Def}_{X}\). The results indicate that there are interesting relations to \(\text{Def}_{X}\), for instance the component structure.
The main application of the result above is a generalization of three conjectures of Curto and Morrison [loc. cit.] concerning the nature of small partial resolutions of \(3\)-dimensional index one terminal singularities and their flops. When \(g:W\rightarrow Z\) is a small partial resolution and \(X\subseteq Z\) is a sufficiently generic hyperplane section with strict transform \(f:Y\rightarrow X\), a result of Reid states that \(f\) is a partial resolution of an RDP. Thus \(g\) is a \(1\)-parameter deformation of \(f\) and so an element in \(\text{Def}_Y\). The authors prove that \(Y\) then is the blowing up of \(X\) in a reflexive module \(M\). As a consequence, \(\alpha\beta^{-1}\) takes this \(g\) to a \(1\)-parameter deformation of \((Z,N)\) of the pair \((X,M).\) Verbatim:
Corollary. There is a maximal Cohen-Macaulay \(\mathcal O_Z\)-module \(N\) such that (i) The small partial resoltuion \(W\rightarrow Z\) is given by blowing up \(Z\) in \(N.\) (ii) Blowing up \(Z\) in the syzygy module \(N^+\) of \(N\) gives the unique flop \(W^+\rightarrow Z\). (iii) The length of the flop equals the rank of \(N\) if the flop is simple.
A version of this statement is given for flat families of small partial resolutions and flops. It is proved that there is a family of pairs \((\mathbf{X},\mathbf{M})\) in \(\text{Def}_{(X,M)}\) such that the blowing up of \(\mathbf{X}\) in \(\mathbf{M}\) and in the syzygy \(\mathbf{X}^+\) give two simultaneous partial resolutions \(\mathbf{Y}\rightarrow\mathbf{X}\leftarrow\mathbf{Y}^+\) inducing any local family of flops \(g\) by pullback, for any \(g\) with hyperplane section \(f\). In fact, \(\mathbf{X}\) equals \(A_1, D_4, E_6, E_7, E_8\) for \(l=1,2,3,4,5,6\) respectively, so that the result above proves that there is in each case a unique reflexive module \(M\) of rank \(l\) such that any simple flop of length \(l\) is obtained by pullback from the \(\mathbf{Y}\rightarrow\mathbf{X}\leftarrow\mathbf{Y}^+.\) This gives the universal simple flop of length \(l\) realized as blowing-ups in families of reflexive modules (as suggested by Curto and Morrison [loc. cit.]).
The RDPs are hypersurfaces, and any maximal Cohen-Macaulay module is given by a matrix factorization. The conjectures of Curto and Morrison [loc. cit.] are stated by matrix factorizations, and they are verified for \(A_n\) and \(D_n\) by brute force computations. The present argument is conceptual, coordinate-free, and makes the conjectures transparent.
The singularities considered in this work will all be henselisations of finite algebras and the results will therefore have finite type representations locally in the étale topology.
Work by Donovan, Wemyss et.al. links properties of various noncommutative algebras to flops. This involves quiver algebras, mutations, tilting theory and GIT-constructions with endomorphism algebras as input. This work offers a direct proof of the original Curto-Morrison conjectures using deformation theory where the blowing up ideal for the small, partial resolution is obtained directly from the \(2\)-dimensional Wunram module. Any flop with fixed RDP hyperplane section and Dynkin diagram is a pullback from a pair of universal blowing ups.
The article is impressing. The deformation functors are studied conceptually, more as fibred categories than as functors, and natural transformations are transformed into maps between the categories of resolutions of singularities. The article gives a lot of techniques that that can be used in the study of more general contractions, and shows a brilliant use of deformation theory in general. Also, the article is self contained with respect to the deformation theory, and contains all preliminaries needed for understanding the importance of the results. flatifying blowing-up; maximal Cohen Macaulay module; simultaneous partial resolution; small resolution; rational double point; RDP; matrix factorization; deformation of algebras; deformation of rational singularities; deformations of exceptional module; partial resolution; domination of resolution; contracting curves; strict transform; Wunram module; blowing up Deformations of singularities, Minimal model program (Mori theory, extremal rays), Stacks and moduli problems, McKay correspondence Deformations of rational surface singularities and reflexive modules with an application to flops | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(k\) be a number field with algebraic closure \(\overline{k}\), and let \(S\) be a finite set of primes of \(k\), containing all the infinite ones. Consider a Chebyshev dynamical system on \({\mathbb P}^2\). Fix the effective divisor \(D\) of \({\mathbb P}^2\) that is equal to a line nondegenerate on \([-2,2]^2\). Then we prove that the set of preperiodic points on \({\mathbb P}^2(\overline{k})\) which are \(S\)-integral relative to \(D\) is not Zariski dense in \({\mathbb P}^2 \). canonical measures; Chebyshev polynomials; equidistribution; height; integral points; preperiodic points doi:10.1016/j.jnt.2010.10.003 Dynamical systems over global ground fields, Elliptic curves over global fields, Varieties over global fields, Distribution modulo one, Rational points, Height functions; Green functions; invariant measures in arithmetic and non-Archimedean dynamical systems, Arithmetic properties of periodic points A nondensity property of preperiodic points on Chebyshev dynamical systems | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials ``Let \(A=\begin{pmatrix} 2&-2\\-2&2 \end{pmatrix}\), \({\mathfrak g}(A)\) the associated Kac-Moody Lie-algebra and G(A) \((=\hat SL_ 2\)) the associated Kac-Moody group. Let P be a (maximal) parabolic subgroup of G(A). Let W (resp. \(W_ P)\) be the Weyl group of \(G(A)\) (resp. P). For \(\tau \in W/W_ P\), let \(X(\tau)\) be the Schubert variety in G(A)/P associated to \(\tau\). We construct explicit bases for \(H^ 0(X(\tau),L^ m)\), \(m\in {\mathbb{Z}}^+\), in terms of ``standard monomials'' where L denotes the tautological line bundle on \(P^ N\) [as well as its restriction to \(X(\tau)\)] for some canonical projective embedding \(X(\tau)\hookrightarrow P^ N\). As a consequence, we obtain similar results for Schubert varieties in \(\hat SL_ 2/B\). Kac-Moody Lie-algebra; ŜL\({}_ 2\); Kac-Moody group; parabolic subgroup; Weyl group; Schubert variety; standard monomials; tautological line bundle V. Lakshmibai and C. S. Seshadri, ?Thèorie monomiale standard pour \(\widehat{\mathfrak{s}\mathfrak{l}}_2 \) ,? C. R. Acad. Sci. Paris Ser. I,305, 183-185 (1987). Infinite-dimensional Lie groups and their Lie algebras: general properties, Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Grassmannians, Schubert varieties, flag manifolds, Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.) Théorie monomiale standard pour \(\hat SL_ 2\). (Standard monomial theory for \(\hat SL_ 2)\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Double covers of \(\mathbb P^3\) with at most nodes, so-called double solids, were studied by \textit{C. H. Clemens} [Adv. Math. 47, 107--230 (1983; Zbl 0509.14045)] focusing particularly on quartic double solids, i.e. double covers of \(\mathbb P^3\) ramified along quartic nodal surfaces. All double solids are irrational when their ramification surfaces are of degree greater than six, but if the ramification surfaces are of lower degree, then the problem is not simple. For example, smooth quartic double solids are irrational, but nodal ones may be rational or irrational. In general, the question of the rationality of nodal quartic double solids can be very delicate and must be handled using the intermediate Jacobians technique.
In the paper under review, the authors consider the irrationality issue for sextic double solids, where the intermediate Jacobians technique seems very hard to implement; this issue has yet to be clarified even for the smooth case, except for the irrationality of a sufficiently general smooth sextic double solid obtained using a degeneration technique cf. \textit{A. Beauville} [Ann. Sci. Éc. Norm. Supér. (4) 10, 309--391 (1977; Zbl 0368.14018)], \textit{A. N. Tyurin} [Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 12, 5--57 (1979; Zbl 0415.14023)].
In addition to the irrationality of sextic double solids, other properties, such as \(\mathbb Q\)-factoriality, potential density and elliptic fibration structures, are studied too. The authors also consider some relevant problems over fields of positive characteristic.
The introduction provides historical notes, with complete references, concerning the rationality problem and other related problems of algebraic varieties. double solids; irrationality problems; \(\mathbb Q\)-factoriality; potential density; elliptic fibration structures Cheltsov, Ivan; Park, Jihun, Sextic double solids. Cohomological and geometric approaches to rationality problems, Progr. Math. 282, 75-132, (2010), Birkh\H{a}user Boston, Inc., Boston, MA Rational and birational maps, Rationality questions in algebraic geometry, Rational points, Positive characteristic ground fields in algebraic geometry Sextic double solids | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this article a unified description of the structure of the small cohomology rings for all projective homogeneous spaces \(SL_n(\mathbb C)/P\) (with \(P\) a parabolic subgroup) is given. First the results on the classical cohomology rings are recalled. Then the algebraic structure of the quantum cohomology ring is studied. Important results are the general quantum versions of the Giambelli and Pieri formulas of the classical cohomology (classical Schubert calculus). They are obtained via geometric computations of certain Gromov-Witten invariants, which are realized as intersection numbers on hyperquot schemes. quantum cohomology; Gromov-Witten invariants; Schubert calculus; small cohomology rings; homogeneous spaces; Pieri formulas Ionuţ Ciocan-Fontanine, On quantum cohomology rings of partial flag varieties, Duke Math. J. 98 (1999), no. 3, 485 -- 524. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Homogeneous spaces and generalizations, Gromov-Witten invariants, quantum cohomology, Frobenius manifolds On quantum cohomology rings of partial flag varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Consider the cohomology or Chow ring of a homogeneous space $G/P$. The classes of Schubert varieties of $G/P$ form a ``geometric'' basis. In Schubert calculus one studies the structure constants of the ring with respect to this basis -- c.f. Littlewood-Richardson coefficients. Much is known about cohomological (Chow) Schubert calculus, and even about its $K$-theory generalization. \par The paper under review studies an even more general cohomology theory, in fact the universal oriented algebraic cohomology theory: algebraic cobordism. In such generality the classes of Schubert varieties are not well defined, they depend on choices. The authors make their choices (Bott-Samelson resolution), and prove a formula for the product of the class of a smooth Schubert variety with an arbitrary Bott-Samelson class -- for type A Grassmannians. The last sections of the paper also establish some results on polynomials (``generalized Schubert polynomials'') representing Schubert varieties for hyperbolic formal group laws. Schubert calculus; cobordism; Grassmannian; generalized cohomology Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Connective \(K\)-theory, cobordism, Bordism and cobordism theories and formal group laws in algebraic topology Smooth Schubert varieties and generalized Schubert polynomials in algebraic cobordism of Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper, the discriminant scheme of homogeneous polynomials is studied in two particular cases: the case of a single homogeneous polynomial and the case of a collection of \(n-1\) homogeneous polynomials in \(n\geqslant 2\) variables. In both situations, a normalized discriminant polynomial is defined over an arbitrary commutative ring of coefficients by means of the resultant theory. An extensive formalism for this discriminant is then developed, including many new properties and computational rules. Finally, it is shown that this discriminant polynomial is faithful to the geometry: it is a defining equation of the discriminant scheme over a general coefficient ring \(k\), typically a domain, if \(2\neq 0\) in \(k\). The case where \(2=0\) in \(k\) is also analyzed in detail. elimination theory; discriminant of homogeneous polynomials; resultant of homogeneous polynomials; inertia forms Busé, L; Jouanolou, JP, On the discriminant scheme of homogeneous polynomials, Math. Comput. Sci., 8, 175-234, (2014) Solving polynomial systems; resultants, Computational aspects in algebraic geometry, Determinantal varieties, Polynomial rings and ideals; rings of integer-valued polynomials On the discriminant scheme of homogeneous polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials \textit{V. V. Deodhar} [Invent. Math. 79, 499--511 (1985; Zbl 0563.14023)] defined a family of decompositions of the flag variety \(G/B\). This decomposition was generalized by \textit{B. Webster} and \textit{M. Yakimov} [Transform. Groups 12, No. 4, 769--785 (2007; Zbl 1144.14044)] to double flag variety \(G/B\times G/B^-\), where \(B^-\) is the opposite Borel subgroup of \(G\). The main motivation of the author of this paper is to better understand the cell decomposition on the double flag variety by Webster-Yakimov. The author constructs cell decompositions on the double Bott-Samelson variety \(Z_{u,v}\), where \(u,v\) are elements in the Weyl group. He also introduces coordinates on each cell of the decomposition and identifies open subsets of the cell where the coordinates are well-defined as regular functions. Another motivation of the paper is to study the Poisson structure on the double Bott-Samelson variety \(Z_{u,v}\). cell decomposition; double Bott-Samelson varieties Mouquin, V.: Cell decompositions of double Bott-Samelson varieties. Int. math. Res. not. 18, 8372-8410 (2015) Homogeneous spaces and generalizations, Classical groups (algebro-geometric aspects) Cell decompositions of double Bott-Samelson varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(X\subset {\mathbb{P}}^ M\) be a projective variety and \(L\subset L'\subset {\mathbb{P}}^ M\) be two linear subspaces such that \(X\cap L'=0\). Let \(\pi_ L: X\to {\mathbb{P}}^ N\) and \(\pi_{L'}: X\to {\mathbb{P}}^{N'}\) denote the linear projections with center L and L' respectively. This note investigates the relations between the double point and ramification schemes or rational equivalence classes associated to \(\pi_ L\) and \(\pi_{L'}\) (most of the time assuming dim L'\(=\dim L+1)\). linear projections; double point; ramification schemes Ramification problems in algebraic geometry Double points of compositions of projections | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This book is neither a graduate text nor a research monograph, but somewhere in between. It expounds how to compute topological invariants of algebraic varieties (mostly hypersurfaces) and their complements --- homology groups, fundamental groups, Alexander polynomials, and (eventually mixed) Hodge structures, in both global and local cases, with some emphasis on the relation between the two. Inevitably this requires rather an extensive array of prerequisites. This is dealt with in 3 ways: first, there are three introductory chapters explaining respectively Whitney stratifications, the structure of plane curve and normal surface singularities, and the Milnor fibration and lattice for an isolated hypersurface singularity. As well as giving background material on these topics, numerous examples are included, and calculations of topological invariants giving a foundation for subsequent calculations. Secondly, three appendices give digests of information on integral bilinear forms, weighted projective varieties and mixed Hodge structures respectively. Thirdly, the author is always ready to quote results from other sources and refer the reader to them for further information; modulo this, the book is reasonably self-contained.
The three main chapters of the book describe methods of computation of fundamental groups of hypersurface complements, of cohomology of complete intersections (smooth or with isolated singularities), and of de Rham cohomology of complements of hypersurfaces (with critical locus of dimension \(\leq 1\)). The book is well written: the explanations are firmly rooted in detailed treatments of examples, which indeed occupy most of the text.
This is not a book to skip over: without following how the calculations are done, one loses the whole point. There is a steady progression to more sophisticated ideas, and the final chapter culminates with some very nice results of the author calculating Alexander polynomials, where a crucial ingredient is the defect of some linear system.
It is unusual to find a book devoted to explaining how to make calculations. This is not a book to suit everyone, but for those who want to understand how to calculate topological invariants in local and global complex algebraic geometry it is unrivalled. topological invariants of algebraic varieties; homology groups; fundamental groups; Alexander polynomials; Hodge structures; Whitney stratifications; plane curve and normal surface singularities; Milnor fibration; lattice for an isolated hypersurface singularity; integral bilinear forms; weighted projective varieties; mixed Hodge structures; hypersurface complements; cohomology of complete intersections A. Dimca, ''Singularities and Topology of Hypersurfaces'', Universitext, Springer-Verlag, New York, 1992. DOI: 10.1007/978-1-4612-4404-2 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes, Research exposition (monographs, survey articles) pertaining to algebraic topology, Algebraic topology on manifolds and differential topology, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces, Deformations of complex singularities; vanishing cycles, Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc., Classical real and complex (co)homology in algebraic geometry, Global theory and resolution of singularities (algebro-geometric aspects), de Rham cohomology and algebraic geometry, Mixed Hodge theory of singular varieties (complex-analytic aspects), Stratifications; constructible sheaves; intersection cohomology (complex-analytic aspects), Milnor fibration; relations with knot theory, Stratifications in topological manifolds, Complex surface and hypersurface singularities Singularities and topology of hypersurfaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We classify coherent modules on \(k [x, y]\) of length at most 4 and supported at the origin. We compare our calculation with the motivic class of the moduli stack parametrizing such modules, extracted from the Feit-Fine formula. We observe that the natural torus action on this stack has finitely many fixed points, corresponding to connected skew Ferrers diagrams. coherent sheaves; finite length modules; Grothendieck ring of varieties; Hilbert scheme of points; torus actions Stacks and moduli problems, Parametrization (Chow and Hilbert schemes) On coherent sheaves of small length on the affine plane | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials cubic surfaces; double line; projections; branch loci; resolvent surfaces; fundamental groups; projective plane Faenzi, D.: A remark on Pfaffian surfaces and aCM bundles. In: Vector Bundles and Low Codimensional Subvarieties: State of the Art and Recent Developments, vol. 21 of Quad. Mat. pp. 209-217. Dept. Math., Seconda Univ. Napoli, Caserta (2007) Surfaces and higher-dimensional varieties Cubic surfaces with a double line | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Four various ``ansatzes'' of the Krichever curves for the periodic-in-\(t\) solutions of the nonlinear Schrödinger equation are considered. An example is given. double-periodic-in-\(t\) solutions; finite-gap solutions; nonlinear Schrödinger equation A. O. Smirnov, Solutions of the nonlinear Schrödinger equation that are elliptic in \?, Teoret. Mat. Fiz. 107 (1996), no. 2, 188 -- 200 (Russian, with English and Russian summaries); English transl., Theoret. and Math. Phys. 107 (1996), no. 2, 568 -- 578. NLS equations (nonlinear Schrödinger equations), Elliptic curves The elliptic-in-\(t\) solutions of the nonlinear Schrödinger equation | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(p\) be real polynomial of degree \(d\) in the variables \(x=(x_1,x_2,\dots,x_n)\) such that \(p(x)=1\) when \(s(x):=\sum_{j=1}^nx_j\). Let \(N(p)\) denote the number of distinct monomials of \(p\). In general, there is no function \(C\) such that
\[
d\leq C(n,N(p)).
\]
Nevertheless, under the additional assumption on the positivity of all coefficients of \(p\) (a condition that arises naturally in CR geometry), \textit{J.~P.~D'Angelo, Š.~Kos} and \textit{E.~Riehl} [J. Geom. Anal. 13, No. 4, 581--593 (2003; Zbl 1052.26016)] proved that for \(n=2\) we have the following sharp estimate
\[
d\leq 2N(p)-3.
\]
J.~P.~D'Angelo conjectured, that for polynomials with positive coefficients and \(n\geq3\) the best possible bound is
\[
C(n,N(p))=\frac{N(p)-1}{n-1}.
\]
\textit{J.~P.~D'Angelo, J.~Lebl} and \textit{H.~Peters} [Mich. Math. J. 55, No. 3, 693--713 (2007; Zbl 1165.32005)] proved that
(a) if \(n\geq3\) then
\[
d\leq\frac43\frac{2N(p)-3}{2n-3};
\]
(b) if \(n\) is sufficiently larger than \(d\) then we have the sharp bound
\[
d\leq\frac{N(p)-1}{n-1}.
\]
In the paper under review the authors extend the results above by proving that the D'Angelo conjecture holds for \(n=3\) and obtaining similar estimastes under weaker assumption of indecomposability (instead of positivity of coefficients). They prove the following:
1. if \(n=2\) and \(p\) is indecomposable then we have the sharp estimate
\[
d\leq2N(p)-3;
\]
2. if \(n=3\) and all coefficients of \(p\) are positive then we have the sharp estimate
\[
d\leq\frac{N(p)-1}{2};
\]
3. if \(n=3\), \(p\) is indecomposable, and \(p\) satisfies additional mild assumption of \textit{no overhang} then we have the sharp estimate
\[
d\leq\frac{N(p)-1}{2};
\]
4. if \(n\geq2\) and \(p\) is indecomposable then
\[
d\leq\frac43\frac{2N(p)-3}{2n-3}.
\]
The authors also study the connection with monomial CR maps of hyperquadrics and prove in that case the similar bounds. polynomials constant on a hyperplane; CR mappings of spheres and hyperquadrics; monomial mappings; degree estimates; Newton diagram Lebl, J., Peters, H.: Polynomials constant on a hyperplane and CR maps of hyperquadrics. arXiv: 0910.2673 Real algebraic and real-analytic geometry, Combinatorial inequalities, Proper holomorphic mappings, finiteness theorems, Polynomials in number theory Polynomials constant on a hyperplane and CR maps of hyperquadrics | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(S\) be an hyperelliptic surface of general type, \(f:S\to C\) a genus \(g\) hyperelliptic fibration. In this paper, we prove that if the maximal \(\mathbb Z_2\)-quotient rank of the vertical part of the fundamental group of \(S\) is \(r\), then its slope
\[
\lambda(f) \geq \begin{cases} 4+ \frac{4r-8}{g(r+1)-r^2},\quad &\text{if \(r\) is even},\\ 4+\frac{4r-8}{g(r+3)-(r+1)^2},\quad &\text{if \(r\) is odd},\end{cases}
\]
with equality only if the branch locus \(R\) of the double cover induced by the hyperelliptic involution on \(S\) has \((r+1\to r+1)\) type singularities (if \(r\) is even), or \((r+2\to r+2)\) type singularities (if \(r\) is odd). double cover; hyperelliptic surface of general type; hyperelliptic fibration; fundamental group; slope; singularities Surfaces of general type, Coverings in algebraic geometry, Fibrations, degenerations in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry, Singularities of surfaces or higher-dimensional varieties The slopes of hyperelliptic surfaces with \(Z_2\)-quotient ranks | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors develop Plücker like formuluae for certain plane curves relating the number of same side and opposite side bitangencies with the number of double points and points of inflection. For that purpose they define a function \(F\) for a given curve such that it vanishes at double points, inflection points, and bitangencies. The close inspection of the function \(F\) and its index allows to prove the main result. In this way results of \textit{B. Halpern} [Bull. Am. Math. Soc. 76, 96--100 (1970; Zbl 0192.58502)] are extended. plane curve; inflection point; bitangency; double point; triple point F. S. Dias and L. F. Mello,Geometry of plane curves. Bull. Sci. Math. 135 (2011),333--344. DOI: 10.1016/j.bulsci.2011.03.007 Plane and space curves, Curves in Euclidean and related spaces Geometry of plane curves | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(f\in \mathbb C[x_1,\ldots,x_n]\) be a polynomial which determines a central hyperplane arrangement (not necessarily reduced) and let \(A_n(\mathbb C) = \mathbb C[x_1,\ldots,x_n, \partial_1, \ldots, \partial_n]\) be the Weyl algebra. Given a factorization \(f = f_1 \cdots f_r\) (not necessarily into linear terms) and two divisors \(f^\prime\) and \(g\) of \(f\), the author considers an analogue of Bernstein-Sato ideal in \(\mathbb C[s_1,\ldots,s_r]\) consisting of the polynomials \(B(s)\) satisfying the functional equation \(B(s)f^\prime f_1^{s_1} \cdots f_r^{s_r} \in A_n(\mathbb C)[s_1, \cdots, s_r]gf^\prime f_1^{s_1+1} f_r^{s_r+1}\) (cf. [\textit{P. Maisonobe}, ``Idéal de Bernstein d'un arrangement central générique d'hyperplans'', Preprint, \url{arXiv:1610.03357}]). First he computes the zero locus of the usual (when \(f^\prime = 1\)) Bernstein-Sato ideal in the sense of \textit{N. Budur} [Ann. Inst. Fourier 65, No. 2, 549--603 (2015; Zbl 1332.32038)] for any factorization of a free and reduced arrangement and for certain factorizations of a non-reduced \(f\). Then he gives an explicite expression for the roots of the Bernstein-Sato polynomial for any power of a free and reduced arrangement, generalizes a duality formula from [\textit{L. Narváez Macarro}, Adv. Math. 281, 1242--1273 (2015; Zbl 1327.14090)], and so on. If \(f\) is tame, the author obtains a combinatorial formula for the roots containing in \([-1, 0)\). For \(f^\prime\neq 1\) and any factorization of a line arrangement, he computes the zero locus of the Bernstein-Sato ideal, and so on. As an interesting application he also shows that ``small roots of the Bernstein-Sato polynomial can force lower bounds for the minimal number of hyperplanes one must add to a tame \(f\) so that the resulting arrangement is free''; this question is closely related to the paper [\textit{D. Mond} and \textit{M. Schulze}, J. Singul. 7, 253--274 (2013; Zbl 1292.32014)]. b-function; D-modules; Weyl algebra; Bernstein-Sato polynomials; hyperplane arrangements; tame arrangements; free divisors; Euler-homogeneous divisors; Saito-holonomic divisors; logarithmic differential forms; logarithmic vector fields; Spencer complex Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Local complex singularities, Relations with arrangements of hyperplanes, Sheaves of differential operators and their modules, \(D\)-modules Combinatorially determined zeroes of Bernstein-Sato ideals for tame and free arrangements | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(\mathcal{\bar{M}}_{g,n}\) denote the moduli stack over \(\mathbb{Z}\) classifying the stable \(n\)-poined curves of genus \(g\), and let \(\mathcal{M} _{g,n}\) be its open substack classifying smooth \(n\)-pointed curves. The algebraic fundamental group of \(\mathcal{M}_{g,n}\otimes _{\mathbb{Z}} \mathbb{\bar{Q}}\) is isomorphic to the profinite completion of the Teichmüller modular group. Thus the Galois action of the absolute Galois group (over the rationals) on this completion can be considered by taking a certain base point in \(\mathcal{M}_{g,n}.\) Grothendieck conjectured that the Teichmüller groupoid, i.e. the fundamental groupoid for \(\mathcal{M} _{g,n}( \mathbb{C}) \) whose base points are the points at infinity, endowed with this Galois action will have a description in terms of ``basic'' groupoids induced from \(\mathcal{M}_{0,4}\) and \(\mathcal{M} _{1,1}.\) (This conjecture is what's known as a ``game of Lego-Teichmüller''.)
Here, the author studies Grothendieck's conjecture using a theory on Schottky-Mumford uniformized universal deformations of degenerate curves. First these deformations enable them to construct a \(( 3g+n-3)\)-dimensional real orbifold which consists of simple and fusing (i.e. associativity) moves. It is possible to view these moves as a base set for the fundamental groupoid described above. Next, they show that the Galois action on these moves can be described by basic Galois action induced from \( \mathcal{M}_{0,4}\) and \(\mathcal{M}_{1,1}.\)
These results describe the Galois action on profinite Teichmüller groupoids of all types, as opposed to previous works by \textit{H. Nakamura} and \textit{L. Schneps} [Invent. Math. 141, 503--560 (2000; Zbl 1077.14030)] which describe the action on profinite modular groups with respect to tangential base points of restricted types. Teichmüller groupoids; Galois action; Grothendieck conjecture Ichikawa, Teichmüller groupoids and Galois action, J. Reine Angew. Math. 559 pp 95-- (2003) Group actions on varieties or schemes (quotients), Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Families, moduli of curves (algebraic) Teichmüller groupoids and Galois action | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials For an elliptic curve \(E\) over a field \(F\) (supposed to have a nontrivial discrete valuation \(v\), valuation ring \(\mathcal O\) and perfect residue field \(k\)) with a finite subgroup scheme \(P\subset E\) defined over \(F\), and for any integer \(n\geq 1\), one has the Eisenstein symbol map
\[
\mathcal E^ n_ P: \mathbb Q[P]^ 0\to H_{\mathcal M}^{n+1}(E^ n,\mathbb Q(n+1))_{\text{sgn}},
\]
where \(H^ i_{\mathcal M}(-,\mathbb Q(j))\) is motivic cohomology, \(\mathbb Q[P]^ 0\) is the \(\mathbb Q\)-vector space of \(\text{Gal}(\bar F/F)\)-invariant functions \(\beta: P(\bar F)\to \mathbb Q\) satisfying \(\sum_{x\in P(\bar F)}\beta (x)=0\), \(E^ n\) is identified with the kernel of the sum map \(E^{n+1}\to E\) (thus giving an action of the symmetric group \(\mathcal S_{n+1}\) on \(E^ n)\), and where the subscript `sgn' denotes the image under the projector
\[
\prod_{\text{sgn}}=\frac{1}{(n+1)!}\sum_{\sigma \in\mathcal S_{n+1}}\text{sgn}(\sigma)\cdot \sigma.
\]
Write \(E/k\) for the special fibre of the minimal regular model \(E/{\mathcal O}\) of \(E\) and suppose that \(E/k\) is a Néron \(N\)-gon for some \(N\geq 1\). Furthermore suppose that \(P\) extends to a finite flat subgroup scheme \(P/\mathcal O\) of the Néron model of \(E\) over \(\mathcal O\).
Also, let \(\overset \circ E\) denote the connected component of the Néron model of \(E\) over \(\mathcal O\). An isomorphism \(\overset \circ E/k\overset \sim \rightarrow\mathbb G_ m\) induces a bijection between \(\mathbb Z/N\mathbb Z\) and the set of components \(C_{\nu}\) of \(E/k\). Thus \(E/k=\cup_{\nu \in\mathbb Z/N\mathbb Z}C_{\nu}\). For \(\beta\in\mathbb Q[P]^ 0\) let \(d_{\beta}(\nu)\) be the degree of the restriction of the flat extension of \(\beta\) to \(C_{\nu}\). The localization sequence for the pair (\(\overset \circ E^ n/{\mathcal O},\overset \circ E^ n/k)\) gives a boundary map
\[
\partial^ n: H_{{\mathcal M}}^{n+1}(E^ n,\mathbb Q(n+1))_{\text{sgn}}\to H^ n_{\mathcal M}(\overset \circ E^ n/k,\mathbb Q(n))_{\text{sgn}}.
\]
The target space is a 1-dimensional \(\mathbb Q\)-vector space generated by an element of the form \(\Phi^ n_ n=\prod_{\text{sgn}}(y_ 0\cup...\cup y_ n)\), where \(y_ 0y_ 1...y_ n=1\), and \(y_ i\), \(1\leq i\leq n\), is a coordinate on the ith copy of \(\mathbb G_{m/k}\). The main result of the paper is following theorem:
\[
\partial^ n\circ{\mathcal E}^ n_ P(\beta)=C^ n_{P,N}\left(\sum_{\nu\in\mathbb Z/N\mathbb Z}d_{\beta}(\nu)B_{n+2}\left(\langle\frac{\nu}{N}\rangle\right)\right)\cdot \Phi^ n_ n,
\]
where \(C^ n_{P,N}\) is an explicit nonzero constant, \(B_ k(X)\) is the \(k\)th Bernoulli polynomial, and \(0\leq \langle x\rangle<1\) is a representative of \(x\in\mathbb Q/\mathbb Z\).
For the proof one may restrict to the situation where \(E/k\) is an untwisted Néron \(N\)-gon with \(N\geq 3\), \(P=\mu_ n\times\mathbb Z/N\mathbb Z\subset E(F)\) is a level \(N\) structure on \(E\), and \(P/k\) gives the standard level \(N\) structure on \((E/k)^{\text{smooth}}=\mathbb G_ m\times\mathbb Z/N\mathbb Z\). Then \(C^ n_{P,N}\) turns out to be \(\pm N^ n(n+1)/(n+2)!\).
The theorem is shown to follow from an explicit formula for the boundary map
\[
\partial^ n_ v: H_{{\mathcal M}}^{n+1}(U^{n'}/F,\mathbb Q(n+1))^{P^ n}_{\text{sgn}}\to H^ n_{{\mathcal M}}(U^{n'}/k,\mathbb Q(n))^{P^ n}_{\text{sgn}},
\]
where \(H^{\bullet}_{{\mathcal M}}(U^{n'},{\mathbb{Q}}(*))^{P^ n}_{\text{sgn}}\) is a suitable \(P(\bar F)^ n\)-invariant sgn-part of the motivic cohomology of \(U^{n'}=\{(x_ 1,...,x_ n)\in E^ n| x_ i\not\in P\), \(\forall i,0\leq i\leq n\}\subset E^ n\), with \(x_ 0=-x_ 1-...-x_ n\). One defines a map
\[
\Theta^ n_ P: \mathbb Q[P]^{0\otimes (n+1)}\to H_{{\mathcal M}}^{n+1}(U^{n'},\mathbb Q(n+1))^{P^ n}_{\text{sgn}}
\]
and then the formula for \(\partial_{\nu}\circ \Theta^ n_ P(\otimes \beta_ i)\) involves, among other things, a sum of expressions containing \(\zeta\in \mu N\), \(\zeta\neq 1\), and this leads, on account of their distributional property, to the Bernoulli polynomials. The explicit calculation uses the fact that the boundary maps in Milnor and Quillen \(K\)-theory agree. Then the theorem is verified for the case \(n=1\) and \(F\) a number field.
The general case consists in the ``weight decomposition'' of \(H^{\bullet}_{{\mathcal M}}(U^{n'}/F\), \(\mathbb Q(*))^{P^ n}_{\text{sgn}}\) under the ``\(L^{-1}\)''-multiplication. Actually, this ``\(L^{-1}\)''-multiplication (\(L\geq 1\) an integer) induces a Galois covering \([\times L]: \tilde U^{n'}\to U^{n'}\) and a homomorphism on (motivic) cohomology that plays a role throughout. The main step is a result, due to Beilinson and Deninger, which identifies \(H^{\bullet}_{{\mathcal M}}(E^ n,\mathbb Q(*))_{\text{sgn}}\) with the \(L^{-n}\)-eigenspace (for a certain endomorphism) of \(H^{\bullet}_{{\mathcal M}}(U^{n'},\mathbb Q(*))^{P^ n}_{\text{sgn}}\). The Eisenstein symbol \({\mathcal E}^ n_ P(\beta)\) is then defined as the projection of \(\Theta^ n_ P(\beta \otimes \alpha^{\otimes n})\), \(\alpha= \sum_{x\in P(\bar F)}(0)-(x)\), into the \(L^{-n}\)-eigenspace, viewed as an element of \(H_{{\mathcal M}}^{n+1}(E^ n,\mathbb Q(n+1))\). If \(F\) is a number field and \(v\) is a place of bad reduction of \(E\) one obtains a description of the `integral' cohomology
\[
H_{{\mathcal M}}^{n+1}(E^ n/F,\mathbb Q(n+1))_{\mathbb Z}\subset H_{{\mathcal M}}^{n+1}(E^ n/F,\mathbb Q(n+1)).
\]
Also, in the modular case, one obtains a new proof of a result of Beilinson which says that the boundary map
\[
\partial: H_{{\mathcal M}}^{n+1}(E^ n,\mathbb Q(n+1))_{\text{sgn}} \rightarrow \{f: \text{GL}_ 2(\mathbb Z/N\mathbb Z)\to \mathbb Q\mid f(g\begin{pmatrix} *&*\\0&1 \end{pmatrix})= f(g)=(-1)^ nf(-g)\},
\]
where \(E\) is the universal elliptic curve with level \(N\) structure, defined over the function field of the modular curve of level \(N\), \(N\geq 3\), is an isomorphism on the image of the Eisenstein symbol. Eisenstein symbol map; motivic cohomology; Néron model; Bernoulli polynomials; boundary maps; K-theory; place of bad reduction; elliptic curve; modular curve Schappacher, N.; Scholl, A. J., \textit{the boundary of the Eisenstein symbol}, Math. Ann., 290, 303-321, (1991) Elliptic curves over global fields, \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols, Symbols and arithmetic (\(K\)-theoretic aspects), Higher symbols, Milnor \(K\)-theory The boundary of the Eisenstein symbol | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give a simple proof of a criterion for the existence of global sections for line bundles on Schubert varieties. Unlike previous proofs, which use the nontrivial fact that Schubert varieties are normal, our proof is based on an elementary geometric property of root systems and their Weyl groups. global sections; line bundles; Schubert varieties; root systems; Weyl groups R. Dabrowski, A simple proof of a necessary and sufficient condition for the existence of nontrivial global sections of a line bundle on a Schubert variety, in: Kazhdan-Lusztig Theory and Related Topics (Chicago, IL, 1989), Contemp. Math., Vol. 139, Amer. Math. Soc., Providence, RI, 1992, pp. 113-120. Linear algebraic groups over arbitrary fields, Grassmannians, Schubert varieties, flag manifolds, Simple, semisimple, reductive (super)algebras A simple proof of a necessary and sufficient condition for the existence of nontrivial global sections of a line bundle on a Schubert variety | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Investigations on determinantal ideals and varieties are an interesting part of commutative ring theory and algebraic geometry. The authors avoid geometric methods and develop a purely algebraic approach to determinantal rings using mainly the theory of algebras with straightening law (Hodge algebras) on posets of minors. This is done simultaneously with the treatment of the homogeneous coordinate rings of the Schubert varieties of Grassmannians (so called Schubert cycles), where the combinatorial structure is simpler. [Algebraically every determinantal ring may be considered as a dehomogenization of a Schubert cycle.]
The subjects of this book include results on height and grade, the Cohen- Macaulay property and normality of determinantal rings and Schubert cycles and the computation of their singular locus. Moreover the divisor class groups of Schubert cycles and determinantal rings over a normal ground ring of coefficients are considered. In section \(12\) the authors also discuss Hochster-Eagon's proof of the perfection of determinantal ideals where principal radical systems instead of standard monomials are used. Finally Kähler differentials and representation-theoretical aspects of determinantal rings are described in a systematic way. The dominating example for illustrating properties of determinantal ideals is the ideal generated by the maximal minors.
The book is self-contained and includes most of the details needed for beginners. determinantal ideals; algebras with straightening law; Hodge algebras; Schubert cycles; divisor class groups W. Bruns, U. Vetter, \(Determinantal Rings\). Lecture Notes in Mathematics, vol. 1327 (Springer, New York, 1988) Theory of modules and ideals in commutative rings, Research exposition (monographs, survey articles) pertaining to commutative algebra, Determinantal varieties, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Ideals and multiplicative ideal theory in commutative rings, Relevant commutative algebra Determinantal rings | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give a permutation pattern avoidance criteria for determining when the projection map from the flag variety to a Grassmannian induces a fiber bundle structure on a Schubert variety. In particular, we show that a Schubert variety has such a fiber bundle structure if and only if the corresponding permutation avoids the split patterns \(3|12\) and \(23|1\). We also show that a Schubert variety is an iterated fiber bundle of Grassmannian Schubert varieties if and only if the corresponding permutation avoids (non-split) patterns 3412, 52341, and 635241. permutation pattern avoidance; Schubert varieties Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Classical problems, Schubert calculus Pattern avoidance and fiber bundle structures on Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Duality for complete discrete valuation fields with perfect residue field with coefficients in (possibly \(p\)-torsion) finite flat group schemes was obtained by Bégueri, Bester, and Kato. In this paper, we give another formulation and proof of this result. We use the category of fields and a Grothendieck topology on it. This simplifies the formulation and proof and reduces the duality to classical results on Galois cohomology. A key point is that the resulting site correctly captures extension groups between algebraic groups. category of fields; duality for local fields; Grothendieck topologies Étale and other Grothendieck topologies and (co)homologies, (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.), Geometric class field theory Duality for local fields and sheaves on the category of fields | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(E\) be an elliptic curve defined over a field \(K\) and, for any \(n\in \mathbb{N}-\{0\}\), let \([n](z)\in K[[z]]\) be the power series defining the multiplication by \(n\) on the formal group of \(E\). Let \(F_n\) be the \(n\)-th division polynomial, i.e. the element in the function field of \(E\) which has divisor \(E[n]-n^2O\) and verifies \(\displaystyle{ \left(\frac{z^{n^2}F_n}{[n](z)}\right)(O)=1}\) (where \(E[n]\) is the \(n\)-torsion of \(E\) and \(O\) is its point at infinity). Then the coordinates of \([n]P\) are \(\displaystyle{ \left(\frac{\gamma^2G_n(P)}{F_n^2(P)},\frac{\gamma^3H_n(P)}{F_n^3(P)}\right)}\) for some \(\gamma\in \mathbb{C}^*\) and division polynomials \(G_n\), \(H_n\) which have explicit expressions in terms of \(F_n\) and of the coefficients of the Weierstrass equation of \(E\). Papers of \textit{M. Ward} [Am. J. Math. 70, 31--74 (1948; Zbl 0035.03702)], \textit{M. Ayad} [Ann. Inst. Fourier 43, No. 3, 585--618 (1993; Zbl 0781.11007)] and \textit{J.H. Silverman} [Math. Ann. 332, No. 2, 443--471, addendum 473--474 (2005; Zbl 1066.11024)] proved periodicity of the sequences \(\{F_n(P)\}_{n\in\mathbb{N}-\{0\}}\) (when \(K\) is a finite field) and of the sequence \(\{F_n(P) \pmod{p^\ell} \}_{n\in\mathbb{N}-\{0\}}\) (when \(F=\mathbb{Q}\), \(P\) has infinite order and \(p\) is a prime such that \(P\) modulo \(p\) is nonsingular), giving also explicit formulas for the periods and for the coefficients appearing in the relations between \(F_{n+r}\) and \(F_n\) (when avaliable).
The paper under review generalizes these results providing similar formulas for the \(G_n\) and the \(H_n\) via their relation with \(F_n\) and explicit computations. With the same techniques the authors generalize also another result of Silverman [loc. cit.] on the \(p\)-adic convergence of subsequences of \(F_n(P)\) to subsequences of \(G_n(P)\) and of \(H_n(P)\). elliptic curves; division polynomials; elliptic divisibility sequences Elliptic curves, Elliptic curves over local fields, Local ground fields in algebraic geometry, Recurrences Sequences generated by elliptic curves | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We investigate the sine-square deformation (SSD) of free fermions in one-dimensional continuous space. On the basis of supersymmetric quantum mechanics, we prove the correspondence between the many-body ground state of the system with SSD and that of the uniform system with periodic boundary conditions. We also discuss the connection between the SSD in the continuous space and its lattice version, where the geometric correction due to the real-space deformation plays an important role in relating the eigenstates of the lattice SSD with those of the continuous SSD. sine square deformation; SUSY quantum mechanics; orthogonal polynomials K. Okunishi and H. Katsura, Sine-square deformation and supersymmetric quantum mechanics, . Supersymmetry and quantum mechanics, Many-body theory; quantum Hall effect, Discrete version of topics in analysis, Formal methods and deformations in algebraic geometry Sine-square deformation and supersymmetric quantum mechanics | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We point out an explicit connection between graphs drawn on compact Riemann surfaces defined over the field \(\overline{\mathbb Q}\) of algebraic numbers -- the so-called Grothendieck's dessins d'enfants -- and a wealth of distinguished point-line configurations. These include simplices, cross-polytopes, several notable projective configurations, a number of multipartite graphs and some ''exotic'' geometries. Among them, remarkably, we find not only those underlying Mermin's magic square and magic pentagram, but also those related to the geometry of two- and three-qubit Pauli groups. Of particular interest is the occurrence of all the three types of slim generalized quadrangles, namely GQ(2,1), GQ(2,2) and GQ(2,4), and a couple of closely related graphs, namely the Schläfli and Clebsch ones. These findings seem to indicate that dessins d'enfants may provide us with a new powerful tool for gaining deeper insight into the nature of finite-dimensional Hilbert spaces and their associated groups, with a special emphasis on contextuality. Grothendieck's dessins d'enfants; quantum contextuality; finite geometries M. Planat, A. Giorgetti, F. Holweck and M. Saniga, Quantum contextual finite geometries from dessins d'enfants, Int. J. Geom. Methods Mod. Phys. 12 (2015), 1550067, 10.1142/ s021988781550067x. Arithmetic aspects of dessins d'enfants, Belyĭ theory, Contextuality in quantum theory, Quantum information, communication, networks (quantum-theoretic aspects), Dessins d'enfants theory, Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices Quantum contextual finite geometries from dessins d'enfants | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper connects two degenerations related to the manifold \(F_n\) of complete flags in \({\mathbb C}^n\). \textit{N. Gonciulea} and \textit{V. Lakshmibai} [Transform.~Groups 1, No.~3, 215--248 (1996; Zbl 0909.14028)] used standard monomial theory to construct a flat sagbi degeneration of \(F_n\) into the toric variety of the Gelfand-Tsetlin polytope, and recently \textit{A. Knutson} and \textit{E. Miller} [Ann.~Math. (2) 161, No.~3, 215--248 (2005; Zbl 1089.14007)] constructed Gröbner degenerations of matrix Schubert varieties into linear spaces corresponding to monomials in double Schubert polynmials. The flag variety is the geometric invariant theory (GIT) quotient of the space \(M_n\) of \(n\) by \(n\) matrices by the Borel group \(B\) of lower triangular matrices. A matrix Schubert variety is an inverse image of a Schubert variety under this quotient. The main result in the paper under review is that this GIT quotient extends to the degenerations. The sagbi degeneration is a GIT quotient of the Gröbner degeneration.\smallskip The nature of this GIT quotient is quite interesting. The authors exhibit an action of the Borel group \(B\) on the product \(M_n\times{\mathbb C}\) of \(M_n\) with the complex line, so that the GIT quotient \(B\backslash\backslash(M_n\times{\mathbb C})\) remains fibred over \({\mathbb C}\) and is the total space of the sagbi degeneration. In this GIT quotient, the total space of the Gröbner degeneration (as in Knutson and Miller) of a matrix Schubert variety in \(M_n\times{\mathbb C}\) covers the total space of the Lakshmibai-Gonciulea degeneration of the corresponding Schubert variety. At the degenerate point, the matrix Schubert variety has become a union of coordinate planes, each of which covers a component of the sagbi degeneration of the Schubert variety indexed by a face of the Gelfand-Tsetlin polytope.
The authors use this to identify which faces of the Gelfand-Tsetlin polytope occur in a given degenerate Schubert variety, and to give a simple explanation of the classical Gelfand-Tsetlin decomposition of an irreducible polynomial representation of \(\text{GL}_n\) into one-dimensional weight spaces; in the degeneration, sections of a line bundle over \(F_n\) become sections of the defining line bundle on the toric variety for the Gelfand-Tsetlin polytope. Flag variety; Schubert variety; sagbi basis; Gelfand-Tsetlin pattern; Borel-Weil theorem Kogan, M., Miller, E.: Toric degeneration of Schubert varieties and Gelfand--Tsetlin polytopes. Adv. Math. 193(1), 1--17 (2005) Grassmannians, Schubert varieties, flag manifolds, Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) Toric degeneration of Schubert varieties and Gelfand--Tsetlin polytopes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a simple and simply connected complex algebraic group, \(\text{Gr}_G\) be its affine Grassmanian, and \(P\subset G\) a parabolic subgroup. The authors prove that the quantum cohomology ring of a flag manifold \(QH^*(G/P)\) is a quotient of \(H^*(\text{Gr}_G)\) after localization, and give the quotient map explicitly in terms of Schubert classes. This result was stated without a proof by Dale Peterson in 1997.
The authors' proof also extends to the equivariant setting. Partial manifestations of this correspondence for \(G/B\), where \(B\) is the Borel subgroup, appear in the work of \textit{R. Bezrukavnikov}, \textit{M. Finkelberg} and \textit{I. Mirković} [Compos. Math. 141, No. 3, 746--768 (2005; Zbl 1065.19004)] and \textit{B. Kim} [Ann. Math. (2) 149, No. 1, 129--148 (1999; Zbl 1054.14533)]. But even for \(P=B\), the ring property of the quotient map is new. For \(G=SL_{k+1}(\mathbb{C})\), a closely related ring homomorphism was studied by \textit{L. Lapointe} and \textit{J. Morse} [J. Comb. Theory, Ser. A 112, No. 1, 44--81 (2005; Zbl 1120.05093)] in terms of \(k\)-Schur functions. The authors note that it looks promising to also compare other structures, such as mirror symmetry on \(QH^*(G/P)\) and Hopf algebra structure with nil-Hecke action on \(H^*(\text{Gr}_G)\).
For \(P=B\), the proof relies on the replationship between the quantum Bruhat graph and the Bruhat order on the elements of the affine Weyl group with a large translation component. It also utilizes algebraic properties of \(QH^*(G/B)\) including the \(T\)-equivariant quantum Chevalley formula of Peterson, proved by \textit{L. C. Mihalcea} [Duke Math. J. 140, No. 2, 321--350 (2007; Zbl 1135.14042)] (\(T\subset G\) is a maximal torus). A byproduct of the proof are expressions for affine Schubert classes in terms of generating functions over paths in the quantum Bruhat graph.
For \(P\neq B\), the authors utilize the Coxeter combinatorics of the affinization of the Weyl group of the Levy factor of \(P\). It allows them to use the comparison formula of Woodward to relate quantum Chevalley formulas of \(QH^*(G/P)\) and \(QH^*(G/B)\). A formula for quantum multiplication by a Schubert class labeled by the reflection in the highest root is then deduced. It turns out that the ring homomorphism of Lapointe and Morse differs from the one here by the strange duality of \(QH^*(G/P)\) discovered by \textit{P.-E. Chaput}, \textit{L. Manivel} and \textit{N. Perrin} [Int. Math. Res. Not. 2007, No. 22, Article ID rnm107, 29 p. (2007; Zbl 1142.14033)]. flag manifold; quantum cohomology ring; Schubert class; affine Grassmanian; quantum Bruhat graph; quantum Chevalley formula; strange duality of \(QH^*(G/P)\) Lam, Thomas; Shimozono, Mark, Quantum cohomology of \(G/P\) and homology of affine Grassmannian, Acta Math., 204, 1, 49-90, (2010) Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Quantum cohomology of \(G/P\) and homology of affine Grassmannian | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The extensive framework of étale topology, sheaf theory, and cohomology was both initiated and largely developed by A. Grothendieck, M. Artin, J.-L. Verdier, and others in the early 1960s. Based on previous ideas due to J.-P. Serre, this algebro-geometric instrument was primarily created as a novel approach to tackle the famous Weil conjectures (from 1949) on varieties over finite fields and their zeta functions. In fact, the étale approach proved to be highly successful in arithmetic geometry ever since, especially when P. Deligne finally verified the Weil conjectures in this context around 1973. The fundamental concepts, methods, techniques, and results of étale topology and cohomology theory where first published in the volumes SGA 1, SGA 4, SGA \(4\frac12\) and SGA 5 of the series ``Séminaire de Géometrie Algébrique'' (SGA) by A. Grothendieck and his collaborators between 1960 and 1966, originally as mimeographed notes, but in the 1970s also as Lecture Notes in Mathematics by Springer Verlag. Among the classical standard textbooks on the subject are the popular primers [\textit{J. S. Milne}, Étale cohomology. Princeton, New Jersey: Princeton University Press (1980; Zbl 0433.14012)] and [\textit{E. Freitag} and \textit{R. Kiehl}, Étale cohomology and the Weil conjecture. With a historical introduction by J. A. Dieudonné. Berlin etc.: Springer-Verlag (1988; Zbl 0643.14012)], apart from the concise course notes [\textit{G. Tamme}, Introduction to étale cohomology. Translated by Manfred Kolster. Berlin: Springer-Verlag (1994; Zbl 0815.14012)].
The book under review is the second, revised edition of another, more recent introduction to the subject of étale cohomology theory. The original edition appeared in 2011 and has been very briefly reviewed back then [Zbl 1228.14001], but here we would like to be a little more precise concerning both its contents and its features. First of all, the book was written as a complete, basically self-contained preparation for a series of talks on Deligne's proof of the Weil conjectures, with the main focus on both the conceptual and technical toolkit of étale cohomology theory, and without any reference to the Weil conjectures themselves as for motivation or application of the machinery developed in the text. In this respect, the present book differs quite a bit form the above-mentioned classics by Milne and Freitag-Kiehl, respectively, but on the other hand it offers much more methodological completeness, detailedness of proofs, and technical rigor in presenting the foundational material as covered in the basic SGA volumes cited before. Accordingly, the current book may serve as a perfect companion to the venerable classics on étale cohomology, and as a profound introduction to the study of the original research works in the field, too. As for the precise contents, the book comprises ten chapters, each of which consists of several thematic sections. Each chapter begins with a list of references related to the respective contents, with strong emphasis on the relevant volumes of the SGA series as for further, more advanced and thorough reading.
Chapter 1 gives an introduction to descent theory, including the basics on flat modules and morphisms, descent properties of sheaves, morphisms and schemes, quasi-finite morphism, and passage-to-limit properties. Chapter 2 discusses sheaves of relative differentials, étale morphisms and smooth morphisms of schemes, infinitesimal liftings of morphisms, the concept of Henselization as well as direct and inverse limits in categories. Chapter 3 deals with the concept of étale fundamental group of a scheme and its functorial properties, whereas Chapter 4 briefly explains group cohomology, profinite groups and their cohomology, cohomological dimension, and Galois cohomology. Chapter 5 turns to the main subject of the book and provides the basic concepts of étale cohomology, thereby introducing Čech cohomology, étale sheaves, Grothendieck topologies, the notion of étale cohomology, calculation methods for étale cohomology, constructible sheaves, and the according passage-to-limit results. Chapter 6 deals with triangulated categories, derived categories and functors, together with their effective use in algebraic geometry, while Chapter 7 is devoted to various base change theorems in algebraic geometry, together with applications to particular cohomology theories and higher direct image sheaves. Chapter 8 treats duality theory, including extensions of Henselian discrete valuation rings, trace morphisms, duality for curves, the functor \(Rf^!\), Poincaré duality (à la SGA 4), and cohomology classes corresponding to algebraic cycles. Chapter 9 presents the finiteness theorems for constructible sheaves on Noetherian schemes and the so-called biduality theorem, complemented by sections on nearby cycles and vanishing cycles, sheaves with group actions, and generic local acyclicity, respectively, whereas Chapter 10 finally turns to the fundamentals of \(l\)-adic cohomology, especially regarding the adic formalism in general, the Grothendieck-Ogg-Shafarevich formula, Frobenius correspondences, the Lefschetz trace formula, and Grothendieck's formula for \(L\)-functions on compactifiable schemes over finite fields.
As for the prerequisites for reading this utmost comprehensive, however rather functional primer on étale cohomology theory, the reader is assumed to have a profound background knowledge of both basic commutative algebra and advanced modern algebraic geometry, the latter perhaps through R. Hartshorne's standard textbook or -- even better -- through the volumes EGA I--IV by A. Grothendieck and J. Dieudonné.
Summing up, this book is highly useful and valuable for any seasoned reader looking for a thorough introduction to the toolkit of étale cohomology with a view toward further study of its applications in both algebraic and arithmetic geometry. étale cohomology; \(l\)-adic cohomology; theory of descent; étale fundamental group; group cohomology; Galois cohomology; duality theory; Grothendieck topology; derived categories Fu, L., Etale cohomology theory, pp., (2015), World Scientific, Hackensack, NJ Research exposition (monographs, survey articles) pertaining to algebraic geometry, Étale and other Grothendieck topologies and (co)homologies, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Homotopy theory and fundamental groups in algebraic geometry, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Derived categories, triangulated categories Étale cohomology theory | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(X \subset \mathbb{P}^N\) be a smooth nondegenerate projective variety of dimension \(n\) (\(\geq 2\)), codimension \(e\) and degree \(d\) (over an algebraically closed field of characteristic zero). It is known that the linear system \(|{\mathcal O}_X(d-n-2)\otimes \omega_X^\vee|\) is base point free, as their elements can be seen as the double point divisors of linear projections onto hypersurfaces of \(\mathbb{P}^{n+1}\). Moreover, it has been proved (see the Introduction of the paper under review and references therein) that they separate points unless \(X \subset \mathbb{P}^N\) is of a particular type (a projection of a so called \textit{Roth variety}). The main purpose of this paper is to prove further positivity results on these linear systems. To be precise: It is shown that the base locus of \(|{\mathcal O}_X(d-n-e-1)\otimes \omega_X^\vee|\) is a finite set except if \(X \subset \mathbb{P}^N\) belongs to a finite list of projective varieties (completely stated, see Thm. 4). For the study of these exceptions, varieties whose generic inner projections have exceptional divisors are classified.
Some applications of these results are also provided: inequalities for the delta and sectional genera, property \((N_{k-d+e})\) for \({\mathcal O}_X(k)\) and new evidences of the regularity conjecture, in fact that \({\mathcal O}_X\) is \((d-e)\)-regular. double point divisors; base locus; linear projections; regularity Noma, A., Generic inner projections of projective varieties and an application to the positivity of double point divisors, Trans. Amer. Math. Soc., 336, 4603-4623, (2014) Classical problems, Schubert calculus, Projective techniques in algebraic geometry Generic inner projections of projective varieties and an application to the positivity of double point divisors | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The \({\mathbb{Z}}^r\)-covers of a finite CW-complex \(X\) are parametrized by the Grassmann variety of \(r\)-dimensional subspaces in the cohomology group \(H^1(X, {\mathbb{Q}})\). The subspaces corresponding to coverings having the first \(i\) Betti numbers finite form a subset \(\Omega_r^i(X)\) in this Grassmannian. The author studies these sets using the jumping loci for the homology of \(X\) with coefficients in rank one local systems. Special attention is given to the case when \(X\) is a smooth quasi-projective or projective variety. free abelian covers; Betti numbers; characteristic varieties; Grassmannians; Schubert varieties Suciu, AI, Characteristic varieties and Betti numbers of free abelian covers, Int. Math. Res. Not. IMRN, 2014, 1063-1124, (2014) Covering spaces and low-dimensional topology, Homology with local coefficients, equivariant cohomology, Homotopy theory and fundamental groups in algebraic geometry, Coverings of curves, fundamental group Characteristic varieties and Betti numbers of free abelian covers | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The birational rigidity of multidimensional algebraic varieties with a pencil of double quadrics of index 1 sufficiently twisted with respect to the base is established. Fano variety; birational automorphisms; ample divisors; birationally superrigid Fano fibration; pencil of double quadrics Rational and birational maps, Automorphisms of surfaces and higher-dimensional varieties, Fano varieties Birational automorphisms of algebraic varieties with a pencil of double quadrics | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Following \textit{S. Lichtenbaum} [Invent. Math. 88, 183-215 (1987; Zbl 0615.14004)], one defines for arithmetic schemes X the motivic, i.e. universal, cohomology groups denoted by \(H^{i+2}(K,{\mathbb{Z}}(i))\), \(i\geq -1\), where K is the function field of X. For instance, \(H^ 3(K,{\mathbb{Z}}(1))\) is defined as \(H^ 2(K,{\mathbb{G}}_ m)\), the Brauer group of K. A wide range of known as well as new facts are obtained in a unified way with a general reciprocity homomorphism, which, if it is an isomorphism, establishes a duality between motivic cohomology group and an idele class group of X.
For \(i=0\), this duality amounts to higher dimensional class field theory; for \(i=-1\), it is a reciprocity uniqueness theorem. If X is regular of dimension 2, for \(i=2\) one has a certain Hasse principle for the existence of O-cycles on X of degree 1 and information about the kernel and cokernel of the reciprocity homomorphism for \(i=1\) is related to the Brauer-Grothendieck group of X. reciprocity; duality; motivic cohomology group; higher dimensional class field theory; Hasse principle; Brauer-Grothendieck group S. Saito, ''Some observations on motivic cohomology of arithmetic schemes,'' Invent. math., vol. 98, iss. 2, pp. 371-404, 1989. (Co)homology theory in algebraic geometry, Generalizations (algebraic spaces, stacks), Global ground fields in algebraic geometry Some observations on motivic cohomology of arithmetic schemes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Schubert polynomials \(\mathfrak{S}_w\) represent cohomology classes of Schubert cycles in the full flag variety. Their coefficients are nonnegative integers. There exists a number of combinatorial formulas for computing these coefficients. This paper is devoted to the following question: when are all the coefficients of a Schubert polynomial equal to \(0\) or \(1\)? Such polynomials are called zero-one Schubert polynomials.
To answer this question, the authors first make the following observation: if a permutation \(\sigma\in S_m\) is a pattern of \(w\in S_n\), then the Schubert polynomial \(\mathfrak{S}_w\) equals a monomial times \(\mathfrak{S}_\sigma\) plus a polynomial with nonnegative coefficients. Hence the set of 0-1 Schubert polynomials is closed under pattern containment. Using Magyar's orthodontia, an inductive algorithm for computing \(\mathfrak{S}_w\) in terms of the Rothe diagram of \(w\), they describe the set of twelve avoided patterns, and also formulate equivalent confitions for a Schubert polynomial to be 0-1 in terms of Rothe diagrams and orthodontic sequences of permutations.
According to the recent result of the same authors about the supports of Schubert polynomials (see [\textit{A. Fink} et al., Adv. Math. 332, 465--475 (2018; Zbl 1443.05179)]), this implies that each 0-1 Schubert polynomial is equal to the integer transform of a generalized permutahedron. Schubert polynomial; pattern avoidance; Rothe diagram Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of algebraic geometry, Permutations, words, matrices Zero-one Schubert polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this short communication, the author announces that the Grothendieck standard conjecture of Lefschetz type holds true for a certain type of algebraic threefold. The conjecture is known to hold for all smooth complex projective curves, surfaces, abelian varieties, and threefolds of Kodaira dimension \(\kappa(X)<3\). The present article contributes to give an important example of threefold with \(\kappa(X)=3\) for which the conjecture holds.
A precise statement is as follows: Let \(\pi:X\rightarrow S\) be a projective morphism of a smooth threefold onto a smooth projective surface. Assume that \(\pi\) has a section and \(\mathrm{End}(\mathrm{Pic}^0(X_s))=\mathbb{Z}\) for some closed smooth fiber \(X_s\). If, for some nonempty open subset \(U\subset S\), the morphism \(\pi|_{\pi^{-1}(U)}:\pi^{-1}(U)\rightarrow U\) is smooth and the rank of the Kodaira-Spencer map of the tangent space of the surface at any point \(u\in U\) equals \(1\), then the conjecture is true. Grothendieck standaard conjecture of Lefschetz type; threefold Algebraic cycles, Transcendental methods, Hodge theory (algebro-geometric aspects), Variation of Hodge structures (algebro-geometric aspects) On the standard conjecture for a 3-dimensional variety fibered over a surface | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper intends to give an introduction to and an overview of Voevodsky's \(h\)- and \(qfh\)-topologies and Suslin's homology in the study of algebraic cycles. An introductory section on topological and algebraic \(K\)-theory, Chow groups and the Grothendieck-Riemann-Roch theorem, and Bloch's construction of the higher Chow groups as motivic cohomology having the localization property, is presented as historical background material. Then the Quillen-Lichtenbaum conjecture for complex varieties, stating an isomorphism between algebraic and topological \(K\)-theory with \({\mathbb Z}/n\)-coefficients, and also the more general situation for arbitrary base fields, then stating an isomorphism between algebraic and étale \(K\)-theory with \({\mathbb Z}/n\)-coefficients, is briefly discussed. One hopes for natural isomorphisms between Bloch's motivic cohomology and étale cohomology with \({\mathbb Z}/n\)-coefficients. And indeed, by a homological version of Bloch's cycle complex, introduced by Suslin in 1988, and the interpretation of the mod \(n\) dual of Suslin's homology as a cohomology theory of sheaves on Voevodsky's \(h\)- and \(qfh\)-sites in 1992, Suslin and Voevodsky were able to prove the existence of such isomorphisms, at least for algebraically closed base fields.
Suslin's homology is defined as follows: Let \(X\) and \(S\) be \(k\)-schemes, with \(S\) smooth and irreducible. Define the group \(C_0(S;X)\) as the free abelian group on the subvarieties \(W\) of \(X\times S\) such that the projection \(W\rightarrow S\) is finite and surjective. Set \(C_n(S;X)=C_0(S\times\Delta^n;X)\), where \(\Delta^n\) is the \(n\)-cosimplex given by the linear equation \(\sum_{i=0}^nt_i=0\) in \(n+1\)-space. Suslin's homology \(H_p^{\text{sing}}(S;X)\) is defined as \(H_p(C_0(S\times\Delta^*;X))\). It is covariant in \(X\) and contravariant in \(S\). Voevodsky's topologies are defined on \({\mathcal S}ch/k\). The \(h\)-topology on \({\mathcal S}ch/k\) is the Grothendieck topology for which an \(h\)-cover of a \(K\)-scheme \(Y\) is a universal topological epimorphism \(X\rightarrow Y\), i.e., for each map of schemes \(Z\rightarrow Y\), the projection \(Z\times_YX\rightarrow Z\) is surjective on points and where \(Z\) has the quotient topology. The \(qfh\)-topology on \({\mathcal S}ch/k\) has as covers of \(Y\) the quasi-finite universal topological epimorphisms \(X\rightarrow Y\). Then the \(h\)-topology is finer than the \(qfh\)-topology, and the \(qfh\)-topology is finer than the étale topology.
One has the following comparison isomorphisms: Let \({\mathcal F}\) be an étale sheaf and let \({\mathcal G}\) be a \(qfh\)-sheaf on \({\mathcal S}ch/k\), then
\[
\text{Ext}^*_{\acute{e}t}({\mathcal F},{\mathbb Z}/n)=\text{Ext}^*_{qfh}(\beta^*{\mathcal F},{\mathbb Z}/n), \text{and Ext}^* _{qfh}({\mathcal G},{\mathbb Z}/n)=\text{Ext}^*_{h}(\alpha^*{\mathcal G},{\mathbb Z}/n),
\]
where \({\mathcal S}ch/k_{h}{\buildrel\alpha\over\rightarrow}{\mathcal S}ch/k_{qfh}{\buildrel\beta\over\rightarrow}{\mathcal S}ch/k_{\acute{e}t}\) are the comparisons between the various topologies. Let \(X\) be in \({\mathcal S}ch/k\) and let \({\mathcal F}\) be a presheaf on \({\mathcal S}ch/k\). One defines the homological complex \({\mathcal F}_*(X)\) by \({\mathcal F}_n(X)={\mathcal F}(X\times\Delta^n)\). For an abelian group \(A\), one sets \((i)\) \(C_*({\mathcal F})={\mathcal F}_*(\text{Spec}(k))={\mathcal F}(\Delta^*)\), \((ii)\) \(H_*^{\text{sing}}({\mathcal F},A)=H_*(C_*({\mathcal F})\otimes^{\mathbb L}A)\), and \((iii)\) \(H^*_{\text{sing}}({\mathcal F},A)=H^*(\text{RHom}(C_*({\mathcal F}),A)\). For any presheaf \({\mathcal F}\) on \({\mathcal S}ch/k\), let \(\widetilde{\mathcal F}_{qfh}\) be the associated \(qfh\)-sheaf. Then, for algebraically closed \(k\), and \(n>0\) prime to \(\text{char}(k)\) and \({\mathcal F}\) admitting transfers, one has a canonical isomorphism \(H^*_{\text{sing}}({\mathcal F},{\mathbb Z}/n)\simeq\text{Ext}^*_{qfh}(\widetilde{\mathcal F}_{qfh},{\mathbb Z}/n)\).
As a corollary one obtains a natural isomorphism \(H^*_{\text{sing}}(X,{\mathbb Z}/n)\simeq H^*_{\acute{e}t}(X,{\mathbb Z}/n)\) for a scheme of finite type over \(k\). One can also relate Suslin homology to Bloch's higher Chow groups to obtain an isomorphism between \(H_c^*(X,{\mathbb Z}/n)\) (étale cohomology with compact supports) and \(\text{Ext}^*_h(\widetilde{z}_0(X)_h,{\mathbb Z}/n)\), where \(z_0(X)\) is a certain \(qfh\)-sheaf on \({\mathcal S}ch/k\) coming from a functor on the category of smooth \(k\)-schemes of finite type. This gives a natural isomorphism \(CH^q(X,p;{\mathbb Z}/n)\simeq H_c^{2(d-q)+p}(X,{\mathbb Z}/n(d-q))^{\vee}\), where \(X\) is an affine variety and where \(q>d=\dim(X)\). Using Poincaré duality, one obtains the isomorphism
\[
CH^q(X,p;{\mathbb Z}/n)\simeq H^{2d-p}_{\acute{e}t}(X,{\mathbb Z}/n(q))
\]
for \(X\) smooth affine. These results extend to quasi-projective (resp.\ smooth quasi-projective) \(X\).
Two fundamental theorems are discussed: \((i)\) The representation theorem, relating families of \(0\)-cycles and \(qfh\)-topology, and \((ii)\) the rigidity theorem which says that there is a canonical isomorphism \(\text{Ext}^*_h(\widetilde{\mathcal F},{\mathbb Z}/n)\rightarrow\text{Ext}^*_{{\mathcal A}b}({\mathcal F}(\text{Spec} k),{\mathbb Z}/n)\) for a homotopy invariant presheaf \({\mathcal F}\) on \({\mathcal S}ch/k\) which admits transfers.
The paper ends with an epilogue with the remark that Suslin and Voevodsky have been able to reduce the Quillen-Lichtenbaum conjectures for motivic cohomology to the Bloch-Kato conjecture, and that Voevodsky has proved the mod 2 Bloch-Kato conjecture. Many questions remain unanswered but one may be sure that the new ideas of Suslin, Voevodsky and others will lead to further breakthroughs. \(K\)-theory; algebraic cycles; Grothendieck topology; Chow group; motivic cohomology; localization property; Quillen-Lichtenbaum conjecture; étale cohomology; Bloch-Kato conjecture Marc Levine, Homology of algebraic varieties: an introduction to the works of Suslin and Voevodsky, Bull. Amer. Math. Soc. (N.S.) 34 (1997), no. 3, 293 -- 312. Research exposition (monographs, survey articles) pertaining to \(K\)-theory, Algebraic cycles, \(K\)-theory of schemes, Relations of \(K\)-theory with cohomology theories, Étale and other Grothendieck topologies and (co)homologies, Grothendieck topologies and Grothendieck topoi, Algebraic cycles and motivic cohomology (\(K\)-theoretic aspects) Homology of algebraic varieties: An introduction to the works of Suslin and Voevodsky | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \({\mathcal D}\) be a triangulated category which is essentially small, i.e., equivalent to a small one, such as the derived category \(D({\mathcal A})\) of a small abelian category \({\mathcal A}\). The author considers full triangulated subcategories \({\mathcal A}\) of \({\mathcal D}\) such that (i) \({\mathcal A}\) is closed under isomorphisms, and (ii) each object of \({\mathcal D}\) is a direct summand of an object in \({\mathcal A}\). The author first shows that there is a bijective correspondence between such subcategories \({\mathcal A}\) in \({\mathcal D}\) and subgroups \(H\) of the Grothendieck group \(K_0({\mathcal D})\). A subcategory \({\mathcal A}\) corresponds to the image of \(K_0 ({\mathcal A})\) in \(K_0({\mathcal D})\); a subgroup \(H\) corresponds to the full subcategory \({\mathcal A}_H\) whose objects are those \(a\) in \({\mathcal D}\) such that \(\overline a\in H\).
Now let \(X\) be a quasi-compact and quasi-separated scheme, such as an affine scheme or an algebraic variety. The derived category \(D(X)_{par f}\) of perfect complexes is essentially small. A thick triangulated subcategory \({\mathcal T}\) of \(D(X)_{par f}\) is called a tensor subcategory if for each object \(e\) in \(D(X)_{par f}\) and each object \(t\) in \({\mathcal T}\) the (derived) tensor product of \(e\) and \(t\) is also in \({\mathcal T}\). The second main result gives a bijective correspondence between the tensor subcategories \({\mathcal T}\) and the subspaces \(Y\) in \(X\) which are unions of closed subspaces with quasi-compact complement. Such a subspace \(Y\) corresponds to the tensor subcategory of perfect complexes which are acyclic at each point of \(X-Y\).
Combining these results, the author obtains a classification of tensor subcategories \({\mathcal A}\) of \(D(X)_{par f}\) satisfying (i) and (ii) above: they correspond bijectively to data \((Y,H)\) where \(Y\) is a subspace of \(X\) as above and \(H\) is a \(K_0(X)\)-submodule of the Grothendieck group of perfect complexes acyclic off \(Y\). The author's motivation for developing these results was an intuition about `higher-dimensional Cartier cycles' which, in his view, would have to correspond to perfect complexes in some triangulated subcategory of \(D(X)_{par f}\). triangulated category; derived cateogry; Grothendieck group; affine scheme; algebraic variety; perfect complexes; tensor subcategories; higher-dimensional Cartier cycles R. W. Thomason, The classification of triangulated subcategories. \textit{Compositio Math.} 105 (1997), no. 1, 1--27.MR 1436741 Zbl 0873.18003 Derived categories, triangulated categories, \(K\)-theory of schemes, Grothendieck groups (category-theoretic aspects), Applications of methods of algebraic \(K\)-theory in algebraic geometry, Grothendieck groups and \(K_0\) The classification of triangulated subcategories | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In algebraic geometry, adjunction theory is essentially the study of the interplay between the intrinsic geometry of a projective variety and the geometry induced by special projective embeddings of the variety. Thus the basic objects of study in adjunction theory are pairs, \((X,L)\), consisting of a projective variety \(X\) and an ample or very ample line bundle \(L\) over \(X\). Given a very ample line bundle \(L\) on a variety \(X\), there is a very rich geometry connected with the corresponding projective embedding. Namely, the various intersections of the (embedded) variety \(X\) with linear subspaces of the ambient projective space \(\mathbb{P}^N\), as well as other associated objects (e.g., the discriminant variety defined by the complete linear system \(|L |)\), provide lots of geometrical and numerical invariants characterizing the intrinsic geometry of \(X\), at least up to a certain extent. In this context, a special role is played by the so-called adjoint bundles of a given ample line bundle \(L\) on \(X\), and their significance has already become apparent in the study of algebraic surfaces by the old Italian school (Castelnuovo, Enriques, Fano, and others) of algebraic geometry. The classical period of adjunction theory (and of the method of adjoint linear systems) had reached its point of culmination and, simultaneously, its frontiers, when \textit{L. Roth} published his important work ``On the projective classification of surfaces'' [cf. Proc. Lond. Math. Soc., II. Ser. 42, 142-170 (1937; Zbl 0015.26904)]. The generalization of classical adjunction theory to higher-dimensional varieties, first initiated by (again) \textit{L. Roth} for threefolds [cf. ``Algebraic threefolds with special regard to problems of rationality'' (1955; Zbl 0066.14704)], underwent a very intensive and rapid development after the methods of modern algebraic geometry had become available.
Especially during the past twenty years, a tremendous progress has been achieved in studying ample divisors on varieties, and in the allied theory of adjoint systems and adjunction methods. Among the numerous contributors to this development, the authors of the present monograph under review, have acquired a particularly outstanding authority as leading experts in the field. The huge number of their papers devoted to this topic, and the just as huge number of important results obtained by them, did practically predestinate them for writing an encyclopedic treatise on this vastly developed subject.
In fact, the book under review provides a systematic, comprehensive and utmost detailed account on classical and modern adjunction theory of complex projective varieties. The authors present a monograph, which incorporates all characteristic features of a self-contained textbook, of a research report that leads to the very recent achievements in the field, and of an encyclopedia which encompasses both history and present-day state of the matter. The authors have worked in the results from nearly 700 research papers (which appeared between 1897 and 1994), including more than 50 articles published by themselves (sometimes with co-authors), and they have managed to keep the text essentially self-contained and consistent.
This is, mathematically and methodically, a great example of maximum efficiency in the literature on algebraic geometry. The text consists of 14 chapters. After a beautiful preface, in which the authors explain the history of the subject, concern and utility of adjunction theory within algebraic geometry, and the arrangement of the material presented in what follows, chapter 1 provides some background results from modern projective algebraic geometry found in most textbooks. Chapter 2 introduces to the various generalizations of the concept of ampleness for line bundles and the according positivity conditions. Chapter 3 discusses the properties of the basic varieties which occur in the adjunction-theoretic classification theory of projective varieties. The theory of Hilbert schemes and Mori's theory of extremal rays, as needed in the sequel, are made available in chapter 4. The following chapter 5 deals with special ample divisors on projective varieties and the restrictions which their existence imposes. Chapter 6 is devoted to rational curves on projective varieties and covers Mori's results on the condition of ``unbreaking'' for families of rational curves. Chapter 7 gives then an account on general adjunction theory via \(\mathbb{Q}\)-Cartier divisors and reduction methods (à la Beltrametti, Sommese, Fania, and others). The main technique here is Mori theory and the precise version of Kawamata's rationality theorem. The aspects and results of classical adjunction theory, which was contributed to by many algebraic geometers, is extensively presented, in the coherent framework of modern algebraic geometry, in chapters 8 to 12. This includes, to a large extent, the investigation of adjoint bundles, their spannedness, the adjunction mapping, adjunction theory for surfaces and threefolds, and the method of reduction for threefolds. Chapter 13 discusses some numerical results for polarized projective varieties, e.g., the double point formula for threefolds and a few Chern inequalities for ample divisors. The concluding chapter 14 studies some of the properties of special varieties occuring in adjunction theory. This chapter contains the basic structure results for scrolls, quadric fibrations, varieties with small invariants, projective manifolds with small defect, and for hyperplane sections of curves.
A seemingly complete bibliography, which is referred to constantly in the text, is particularly a welcome service to the reader.
Again: The material of the book is presented in encyclopedic thoroughness, indisputable rigour, and exemplary completeness. Quite undoubtedly, it will immediately become the standard text and reference book on adjunction theory in projective algebraic geometry. Bibliography; adjunction theory; threefolds; ampleness; Hilbert schemes; Mori's theory; Kawamata's rationality theorem; adjoint bundles; double point formula for threefolds; Chern inequalities M. C. Beltrametti, A. J. Sommese, \textit{The adjunction theory of complex projective varieties}, volume 16 of \textit{de Gruyter Expositions in Mathematics}. De Gruyter 1995. MR1318687 Zbl 0845.14003 Divisors, linear systems, invertible sheaves, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Research exposition (monographs, survey articles) pertaining to algebraic geometry The adjunction theory of complex projective varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We study the expectation of the number of components \(b_0(X)\) of a random algebraic hypersurface \(X\) defined by the zero set in projective space \(\mathbb{R} \operatorname{P}^n\) of a random homogeneous polynomial \(f\) of degree \(d\). Specifically, we consider invariant ensembles, that is Gaussian ensembles of polynomials that are invariant under an orthogonal change of variables.{
}Fixing \(n\), under some rescaling assumptions on the family of ensembles (as \(d \to \infty\)), we prove that \(\mathbb{E} b_0(X)\) has the same order of growth as \([\mathbb{E} b_0(X \cap \mathbb{R} \operatorname{P}^1)]^n\). This relates the average number of components of \(X\) to the classical problem of \textit{M. Kac} [Bull. Am. Math. Soc. 49, 314--320 (1943; Zbl 0060.28602)] on the number of zeros of the random univariate polynomial \(f |_{\mathbb{R} \operatorname{P}^1}\).{
}The proof requires an upper bound for \(\mathbb{E} b_0(X)\), which we obtain by counting extrema using random matrix theory methods from \textit{Y. V. Fyodorov} [``High-dimensional random fields and random matrix theory'', Markov Process. Relat. Fields 21, No. 3, 483--518 (2015), \url{https://arxiv.org/abs/1307.2379v2}], and it also requires a lower bound, which we obtain by a modification of the barrier method from \textit{A. Lerario} and \textit{E. Lundberg} [Int. Math. Res. Not. 2015, No. 12, 4293--4321 (2015; Zbl 1396.14049)] and \textit{F. Nazarov} and \textit{M. Sodin} [Am. J. Math. 131, No. 5, 1337--1357 (2009; Zbl 1186.60022)].{
}We also provide quantitative upper bounds on implied constants; for the real Fubini-Study model these estimates provide super-exponential decay (as \(n \to \infty\)) of the leading coefficient (in \(d\)) of \(\mathbb{E} b_0(X)\). real algebraic geometry; Gaussian field; harmonic polynomials; critical point theory; Hilbert's sixteenth problem Fyodorov, Yan V.; Lerario, Antonio; Lundberg, Erik, On the number of connected components of random algebraic hypersurfaces, J. Geom. Phys., 95, 1-20, (2015) Random matrices (probabilistic aspects), Hypersurfaces and algebraic geometry, Real algebraic sets, Random fields On the number of connected components of random algebraic hypersurfaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Correlation functions in quantum field theory are calculated using Feynman amplitudes, which are finite dimensional integrals associated to graphs. The integrand is the exponential of the ratio of the first and second Symanzik polynomials associated to the Feynman graph, which are described in terms of the spanning trees and spanning 2-forests of the graph, respectively.
In [\textit{O. Amini} et al., Izv. Math. 80, No. 5, 813--848 (2016; Zbl 1354.32012); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 80, No. 5, 5--40 (2016)], we related this ratio to the asymptotic of the Archimedean height pairing between degree zero divisors on degenerating families of Riemann surfaces. Motivated by this, we consider in this paper the variation of the ratio of the two Symanzik polynomials under bounded perturbations of the geometry of the graph. This is a natural problem in connection with the theory of nilpotent and SL2 orbits in Hodge theory.
Our main result is the boundedness of variation of the ratio. For this we define the exchange graph of a given graph which encodes the exchange properties between spanning trees and spanning 2-forests in the graph. We provide a complete description of the connected components of this graph, and use this to prove our result on boundedness of the variations. Symanzik polynomials; spanning trees and forests; exchange graph; Feynman amplitudes; height pairing on Riemann surfaces Graph polynomials, Feynman integrals and graphs; applications of algebraic topology and algebraic geometry, Trees, Combinatorial aspects of matroids and geometric lattices, Arithmetic varieties and schemes; Arakelov theory; heights, Height functions; Green functions; invariant measures in arithmetic and non-Archimedean dynamical systems, Period matrices, variation of Hodge structure; degenerations The exchange graph and variations of the ratio of the two Symanzik polynomials | 0 |
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