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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 The aim of this article is to give explicit formulae for various generating functions, including the generating function of torus-invariant primitive ideals in the big cell of the quantum minuscule Grassmannian of type \(B_n\).
Let \(\mathfrak g\) be a simple Lie algebra of rank \(n\) over the field of complex numbers, and let \(\pi:=\{\alpha_1,\ldots,\alpha_n\}\) be the set of simple roots associated to a triangular decomposition \(\mathfrak g=\mathfrak n^-\oplus\mathfrak h\oplus\mathfrak n^+\). Let \(W\) be the Weyl group associated to \(\mathfrak g\).
The aim of this article is to study the prime spectrum of so-called quantum Schubert cells from the point of view of algebraic combinatorics. Quantum Schubert cells have been introduced by \textit{C. De Concini, V. G. Kac} and \textit{C. Procesi} [Stud. Math., Tata Inst. Fundam. Res. 13, 41--65 (1995; Zbl 0878.17014)] as quantisations of enveloping algebras of nilpotent Lie algebras \(\mathfrak n_w:=\mathfrak n^+\cap\mathrm{Ad}_w(\mathfrak n^-)\), where \(\mathrm{Ad}\) stands for the adjoint action and \(w\in W\). These noncommutative algebras are defined thanks to the braid group action of \(W\) on the quantised enveloping algebra \(U_q(\mathfrak g)\) induced by Lusztig automorphisms. The resulting (quantum) algebra associated to a chosen \(w\in W\) is denoted by \(U_q[w]\). Here \(q\) denotes a nonzero element of the base field \(\mathbb K\), and we assume that \(q\) is not a root of unity. It was recently shown by \textit{M. Yakimov} [Proc. Am. Math. Soc. 138, No. 4, 1249--1261 (2010; Zbl 1245.16030)] that these algebras can be seen as the Schubert cells of the quantum flag varieties. Our aim is to study combinatorially the prime spectrum of the algebras \(U_q[w]\). quantum algebras; quantized enveloping algebras; primitive ideals; quantum Schubert cells; quantum flag varieties; algebraic combinatorics Ring-theoretic aspects of quantum groups, Quantum groups (quantized enveloping algebras) and related deformations, Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Ideals in associative algebras Enumeration of torus-invariant strata with respect to dimension in the big cell of the quantum minuscule Grassmannian of type \(B_n\). | 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 Let \(G\) be a simple adjoint group over the field \(\mathbb C\) of complex numbers. Let \(T\) be a maximal torus of \(G\). Let \(P\) be a parabolic subgroup of \(G\). In this article, we give a survey on the Geometric Invariant Theory related problems for the left action of \(T\) on \(G/P\). Schubert varieties; line bundles; semi-stable points S. S. Kannan, ``GIT related problems of the flag variety for the action of a maximal torus'' in Groups of Exceptional Type, Coxeter Groups and Related Geometries , Springer Proc. Math. Stat. 82 , Springer, New Delhi, 2014, 189-203. Grassmannians, Schubert varieties, flag manifolds, Geometric invariant theory GIT related problems of the flag variety for the action of a maximal torus | 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 under review starts with the space \({\mathrm{M}}_{\mathrm{n}}\) of all monic polynomials of degree \(n+1\) over \(K\) that vanish at \(0\), where \(K\) is an algebraically closed field. A lot of work has been made on subvarieties of \({\mathrm{M}}_{\mathrm{n}}\), and the author considers here the subvarieties \(Simp_n^m\), consisting of the morphisms with total ramification \(< m\). This subvariety is a Zariski open dense subset of \({\mathrm{M}}_{\mathrm{n}}\). The goal of the article is to compute the cohomology of \(Simp_n^m\), what is achieved in Theorem A. In order to state it, call \(\mathbf{p}(N)\) the number of partitions of a positive integer \(N\), and define \(\mathbf{c} : \mathbb{Z}^+ \rightarrow \mathbb{Z}^+ \) by \(\mathbf{c}(m) = \sum \mathbf{p}(n_1+1) \cdots \mathbf{p}(n_k+1)\), where the sum runs over \(k \geq 1\), \(n_1 + \dots + n_k=m\), \(n_1 \leq \dots \leq n_k\).
Then, if \(m \geq 1, n \geq 3m\), Theorem A gives the singular cohomology of \(Simp_n^m\), \(H^i\), and its étale cohomology, \(H^i_{ \acute{e}t}\), in the following terms:
(1) \(H^i(Simp_n^m(\mathbb{C});\mathbb{Q})\) is \(\mathbb{Q}\) for \(i=0\), \(\mathbb{Q}^{\oplus \mathbf{c}(m)}\) for \(i=2m-1\), and \(0\) otherwise. Besides, in the case \(i=2m-1\), \(H^{2m-1}(Simp_n^m(\mathbb{C});\mathbb{Q})\) is pure of weight \(-2m\) and Hedge type \((-m,-m)\).
(2) If \(\kappa\) is a field of characteristic \(0\) or greater than \(n+1\), \(H^i_{ \acute{e}t}(Simp_{n\; / \overline{\kappa}} ^m(\mathbb{C});\mathbb{Q}_\ell)\) is \(\mathbb{Q}_\ell(0)\) for \(i = 0\), \(\mathbb{Q}_\ell(-m)^{\oplus \mathbf{c}(m)}\) for \(i = 2m-1\), and \(0\) otherwise, with \(\ell\) coprime with the characteristic of \(\kappa\).
The proof of Theorem A is rather lengthy, and it runs throughout Sections 2 to 6 of the paper. However, the plan of the proof, and its development, are clearly explained. Its main tools are the topological properties of the poset that encodes the behavior of the ramification, and provides a stratification of \({\mathrm{M}}_{\mathrm{n}}\). As a noteworthy point of Theorem A we should note that the value of \(H^i\) is independent of \(n\). moduli of morphisms with fixed ramification behavior; Hurwitz spaces; polynomials; cohomology; point counts over finite fields Algebraic moduli problems, moduli of vector bundles, Étale and other Grothendieck topologies and (co)homologies, Positive characteristic ground fields in algebraic geometry, Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem), Étale cohomology, higher regulators, zeta and \(L\)-functions (\(K\)-theoretic aspects), Coverings of curves, fundamental group Cohomology of the space of polynomial maps on \(\mathbb{A}^1\) with prescribed ramification | 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 lecture is based on a former paper of the authors [J. Geom. Phys. 19, No. 3, 287-313 (1996; Zbl 0965.81096)]. \(b-c\) systems with integer spin on algebraic curves given by Weierstrass polynomials are described by an operator formalism. To that end Riemannian surfaces are considered and represented as an \(n\)-fold branching covering of the Riemannian sphere where \(n\) is the degree of the Weierstrass polynomial. From this an \(n\)-fold splitting of the Hilbert space into Fock spaces is deduced, such that in each Fock space only modes with the same monodromy characteristics propagate. The correlation functions are calculated. integer spin; Weierstrass polynomials; Riemannian surfaces; correlation functions Two-dimensional field theories, conformal field theories, etc. in quantum mechanics, Coverings of curves, fundamental group, Compact Riemann surfaces and uniformization Creation and annihilation operators for \(b-c\) systems on general algebraic 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 indicate how Hecke algebras, the Yang-Baxter equation, Hall-Littlewood polynomials, and Macdonald polynomials are related to the parameter \(y\) that Hirzebruch introduced in his study of the Riemann-Roch theorem. \(\chi_y\)-characteristic; flag variety; Riemann-Roch; Hall-Littlewood polynomial; Yang-Baxter equation; Hecke algebra; Macdonald polynomial; characteristic of Hirzebruch; spaces of cohomology; generating function; manifold; chern classes; cohomology ring; Grothendieck ring Combinatorial aspects of representation theory, Riemann-Roch theorems, Grassmannians, Schubert varieties, flag manifolds, Representations of finite symmetric groups About the ``\(y\)'' in the \(\chi_y\)-characteristic of Hirzebruch | 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 a novel implementation of the elliptic curve factoring method specifically designed for medium-size integers such as those arising by billions in the cofactorization step of the Number Field Sieve. In this context, our algorithm requires fewer modular multiplications than any other publicly available implementation. The main ingredients are: the use of batches of primes, fast point tripling, optimal double-base decompositions and Lucas chains, and a good mix of Edwards and Montgomery representations. elliptic curve method; cofactorization; double-base representation; twisted Edwards curve; Montgomery curve; CADO-NFS Factorization, Algebraic coding theory; cryptography (number-theoretic aspects), Cryptography, Applications to coding theory and cryptography of arithmetic geometry Faster cofactorization with ECM using mixed representations | 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 \(\mathrm{SU}(p, q)\)-flag domain is an open orbit of the real Lie group \(\mathrm{SU}(p, q)\) acting on the complex flag manifold associated to its complexification \(\mathrm{SL}(p + q, \mathbb{C})\). Any such flag domain contains certain compact complex submanifolds, called cycles, which encode much of the topological, complex geometric and representation theoretical properties of the flag domain. This article is concerned with the description of these cycles in homology using a specific type of Schubert varieties. They are defined by the condition that the fixed point of the Borel group in question is in the closed \(\mathrm{SU}(p, q)\)-orbit in the ambient manifold. Equivalently, the Borel group contains the \(AN\)-factor of some Iwasawa decomposition. We consider the Schubert varieties of this type which are of complementary dimension to the cycles. It is known that if such a variety has non-empty intersection with a certain base cycle, then it does so transversally (in finitely many points). With the goal of understanding this duality, we describe these points of intersection in terms of flags as well as in terms of fixed points of a given maximal torus. The relevant Schubert varieties are described in terms of Weyl group elements. Much of our work is of an algorithmic nature, but, for example in the case of maximal parabolics, i.e. Grassmannians, formulas are derived. flag domains; cycles; Schubert varieties; intersection pairing Grassmannians, Schubert varieties, flag manifolds On the intersection pairing between cycles in \(\mathrm{SU}(p,q)\)-flag domains and maximally real 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 \(k\) be an algebraically closed field of characteristic zero. The Grothendieck ring \(K_0({\mathcal{V}}/k)\) of algebraic varieties over \(k\) is generated (as an abelian group) by the isomorphism classes of schemes of finite type over \(k\) subject to the relations \([X]=[X\backslash Z] + [Z],\) where \(Z\subset X\) is a closed subscheme with the reduced structure. The product is defined as \([X]\cdot [Y]=[X\times Y].\) The main result of the paper asserts that for a pair of closed subschemes cut out (in certain way depending on a non-zero global section \(s\) of the appropriate homogenous variety) from the pair of Grassmanians of type \(G_2\) one has \(([X]-[Y])\cdot {\mathbb L} =0.\) Moreover, for the general choice of \(s\) one has \([X]\neq [Y] \) and both \(X\) and \(Y\) are smooth Calabi-Yau \(3\)-folds. Grothendieck ring; Grassmanian; Dynkin diagram; global section Applications of methods of algebraic \(K\)-theory in algebraic geometry, Grothendieck groups and \(K_0\), Varieties and morphisms, Grassmannians, Schubert varieties, flag manifolds, Calabi-Yau manifolds (algebro-geometric aspects) The class of the affine line is a zero divisor in the Grothendieck ring: via \(G_2\)-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 It has become quite common to study derived categories of coherent sheaves on varieties, with the hope that the derived category encodes interesting geometric information about the varieties. A fundamental question in this area is: Under which circumstances are the derived categories of two varieties equivalent? For some classes of varieties, this can only happen if both varieties are already isomorphic. A statement of this type is usually called a Derived Torelli Theorem.
For Enriques surfaces over fields of characteristic different from 2, a derived Torelli theorem holds: If \(X_1\) and \(X_2\) are two Enriques surfaces, we have
\[
D^b(X_1)\cong D^b(X_2)\quad\iff\quad X_1\cong X_2\,;
\]
as proven by \textit{T. Bridgeland} and \textit{A. Maciocia} [Math. Z. 236, No. 4, 677--697 (2001; Zbl 1081.14023)] in characteristic zero and by \textit{K. Honigs} et al. [Math. Res. Lett. 28, No. 1, 65--91 (2021; Zbl 1471.14040)] in positive characteristic. Line bundles \(L\) on Enriques surfaces are always exceptional objects, which means that \(\operatorname*{Ext}^*(L,L)\) is concentrated in degree zero and one-dimensional. Exceptional objects are of interest because they induce decompositions of the derived category with small pieces. For \(X\) a general Enriques surface, there is a collection \(\mathcal L=\{L_1,\dots, L_{10}\}\) of ten pairwise orthogonal line bundles; see [\textit{S.\ Zube}, Math.\ Notes 61, No.\ 6, 693--699 (1997; Zbl 0933.14023)]. Denoting the right-orthogonal complement by
\[
\operatorname{Ku}(X,\mathcal L):=\bigl\{E\in D^b(X)\mid \operatorname{Hom}_{D^b(X)}(L_i,E[m])=0\mid \text{ for all \(i=1,\dots, 10\), \(m\in \mathbb Z\)}\bigr\}\,,
\]
we get a semi-orthogonal decomposition
\[
D^b(X)=\bigl\langle \operatorname{Ku}(X,\mathcal L), L_1,\dots , L_{10}\bigr\rangle\,.\tag{\(\star\)}
\]
The authors of the article prove a refined Torelli theorem stating that, for two general Enriques surfaces \(X_1\) and \(X_2\) together with orthogonal collections of line bundles \(\mathcal L_1\) and \(\mathcal L_2\) of length 10, we have
\[
\operatorname{Ku}(X_1,\mathcal L_1)\cong \operatorname{Ku}(X_2,\mathcal L_2)\quad \implies \quad X_1\cong X_2\,.
\]
Let us give a brief sketch of the proof: In Section 2, the authors prove a useful general criterion for extending equivalences from admissible subcategories to bigger subcategories. Omitting some technical details, the key Proposition 2.5 says the following: Let \(D^b(X)=\langle \mathcal A, \mathcal A'\rangle\) and \(D^b(Y)=\langle \mathcal B, \mathcal B'\rangle\) be two semi-orthogonal decompositions of derived categories of smooth projective varieties, and let \(\Phi\colon \mathcal A\to \mathcal B\) be an equivalence. Then, given exceptional objects \(E\in \mathcal A'\) and \(F\in \mathcal B'\) satisfying
\[
\Phi(\alpha^! E)\cong \beta^! F\,,
\]
where \(\alpha\colon \mathcal A\hookrightarrow D^b(X)\) and \(\beta\colon \mathcal B\hookrightarrow D^b(Y)\) are the embeddings, one can extend \(\Phi\) to an equivalence \(\widehat \Phi\colon \langle \mathcal A,E\rangle \to \langle \mathcal B,F\rangle\).
The plan is to apply this Proposition to the decomposition \((\star)\). The main technical work which allows to do this is done in Section 4 (working in a slightly more general setup of Enriques categories). There, it is shown that, given a semi-orthogonal decomposition of the form \((\star)\), and denoting the embedding by \(\alpha\colon \operatorname{Ku}(X,\mathcal L)\hookrightarrow D^b(X)\), all the objects \(\alpha^!L_i\in \operatorname{Ku}(X,\mathcal L)\), for \(i=1,\dots, 10\), are \(3\)-spherical. Conversely, all \(3\)-spherical objects in \(\operatorname{Ku}(X,\mathcal L)\) are of this form, up to degree shifts.
This allows the authors to proceed as follows with the proof of their main theorem: Let \(\Phi\colon \operatorname{Ku}(X_1,\mathcal L_1)\to \operatorname{Ku}(X_2,\mathcal L_2)\) be an equivalence, and write \(\mathcal L_j=\{L^j_1,\dots,L^j_{10}\}\). As equivalences map \(3\)-spherical objects to \(3\)-spherical objects, we have \(\Phi(\alpha_1^! L^1_1)\cong \alpha_2^!L^2_i[m]\) for some \(i\in \{1,\dots ,10\}\) and some \(m\in \mathbb Z\). Hence, Proposition 2.5 applies and gives an extended equivalence
\[
\widehat \Phi\colon\bigl\langle \operatorname{Ku}(X_1,\mathcal L_1), L^1_1\bigr\rangle\to \bigl\langle\operatorname{Ku}(X_2,\mathcal L_2), L^2_i\bigr\rangle\,.
\]
Repeating this step ten times (with a small adjustment) gives the desired extended equivalence \(D^b(X_1)\cong D^b(X_2)\). Then, the (non-refined) derived Torelli theorem for Enriques surfaces can be applied to conclude \(X_1\cong X_2\).
The authors also use the general results of Sections 2 and 4 to give a simplified proof of a result of \textit{S. Hosono} and \textit{H. Takagi} [Kyoto J. Math. 60, No. 1, 107--177 (2020; Zbl 1475.14033)] relating the derived categories of Enriques surfaces and blow-ups of Artin--Mumford double solids. derived categories; Kuznetsov component; Enriques surfaces; Artin-Mumford double solids Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, \(K3\) surfaces and Enriques surfaces A refined derived Torelli theorem for Enriques 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 present book grew out of a series of lectures delivered by the two authors at the Summer School 1995 of the Graduiertenkolleg ``Geometry and nonlinear analysis'' at Humboldt University in Berlin. While the original lectures were designed to discuss some of the recent results on the geometry of moduli spaces of (semi-)stable coherent sheaves on an algebraic surface, the text at hand is a considerably elaborated and extended version of the initial notes. The outcome of the authors' rewarding and admirable effort at completing their lecture notes is now a book that serves several purposes at the same time. On the one hand, and in regard of its first part, it provides a textbook-like introduction to the theory of (semi-)stable coherent sheaves over arbitrary algebraic varieties and to their moduli spaces. On the other hand, mainly in view of its second part, the text has the character of both a research monograph and a comprehensive survey on some very recent results on those moduli spaces of (semi-)stable sheaves over (special) algebraic surfaces.
In both aspects, this book is rather unique in the existing literature on the classification theory of sheaves and vector bundles. Namely, for the first time in this central current area of research in algebraic geometry, a successful attempt has been undertaken to develop both the general, conceptual and methodical framework and the present state of knowledge in one of the most important special cases in a systematic, detailed, nearly complete and didactically processed presentation.
The text is divided into two major parts. After a careful introduction, which provides several motivations for studying sheaves on algebraic surfaces, in particular with a view to their significance in the differential geometry of four-dimensional manifolds and in gauge field theory (e.g., via Donaldson polynomials and Seiberg-Witten invariants), part I is devoted to the general theory of semi-stable sheaves and their moduli spaces. Chapter 1 introduces the basic concept of semi-stability for coherent sheaves over algebraic varieties, in the sense of D. Gieseker as well as in the (original) version of Mumford-Takemoto, and the fundamental material on Harder-Narasimhan filtrations, Jordan-Hölder filtrations, \(S\)-equivalence for semi-stable sheaves, and boundedness conditions. Flat families of sheaves, Grothendieck's Quot-scheme, the deformation theory of flags of sheaves, and Maruyama's openness-of-stability theorem are discussed in chapter 2, while chapter 3 deals with the most general form of the so-called Grauert-Mülich theorem and its application in establishing the boundedness of the set of semi-stable sheaves.
Moduli spaces for semi-stable sheaves, in their local and global aspects, is the subject of chapter 4. The authors discuss in detail C. Simpson's more recent approach to the construction of these moduli spaces, together with the related general facts from geometric invariant theory, and sketch the original construction by D. Gieseker and M. Maruyama likewise in an appendix. Furthermore, deformation theory is used to analyze the local structure of these moduli spaces, including dimension bounds and estimates for the expected dimension in the case of an algebraic surface. In another appendix the authors give an outlook to their own research contributions, in that they briefly describe moduli for ``decorated sheaves'' [cf. \textit{D. Huybrechts} and \textit{M. Lehn}, Int. J. Math. No. 2, 297-324 (1995; Zbl 0865.14004)]. This topic, though not systematically treated in the text, has recently found spectacular applications in conformal quantum field theory (e.g., in M. Thaddeus's proof of the famous Verlinde formula) and in non-abelian Seiberg-Witten theory.
The second part of the book, starting with chapter 5, mainly focuses on moduli spaces of semi-stable sheaves on algebraic surfaces. At first, the authors present various construction methods for stable vector bundle on surfaces, including Serre's correspondence between rank-2 vector bundles and codimension-2 subschemes, Maruyama's method of elementary transformations, and some illustrating examples. The geometry of moduli spaces of semi-stable sheaves on K3 surfaces is thoroughly explained in chapter 6, where in particular some very recent results by S. Mukai, A. Beauville, L. Göttsche-D. Huybrechts, K. O'Grady, J. Li, G. Ellingsrud-M. Lehn, and others are systematically compiled. Chapter 7 deals with the restriction of sheaves on surfaces to curves, focusing on the related work of H. Flenner, F. Bogomolov, and V. Mehta-A. Ramanathan in the 1980's. In chapter 8, the authors turn the attention to line bundles on moduli spaces and their Picard groups. The construction of determinantal line bundles and ampleness results for special line bundles on moduli spaces are presented by essentially following the approaches of J. Le Potier (1989) and J. Li (1993). As an application, the authors provide a profound comparison between the (algebraic) Gieseker-Maruyama moduli spaces of semi-stable vector bundles and the (analytic) Donaldson-Uhlenbeck compactification of the moduli spaces of Mumford-stable bundles. Chapter 9 is almost entirely devoted to K. O'Grady's recent work on the irreducibility and generic smoothness of moduli spaces for vector bundles on projective surfaces [cf. \textit{K. O'Grady}, Invent. Math. 123, No, 1, 141-207 (1996; Zbl 0869.14005)] and the related results by \textit{D. Gieseker} and \textit{J. Lie} [J. Am. Math. Soc. 9, 107-151 (1996; Zbl 0864.14005)].
Chapter 10, entitled ``Symplectic structures'', turns to differential forms on moduli spaces of stable sheaves on surfaces. After a lucid survey of the technical background material such as Atiyah classes, trace maps, cup products, the Kodaira-Spencer map, etc., the authors describe the tangent bundle of the smooth part of a moduli space by means of the universal family of vector bundles. Then, via the explicit construction of closed differential forms on moduli spaces, Mukai's theorem on the existence of a non-degenerate symplectic structure on the moduli space of stable sheaves on a K3 surface is derived.
The concluding chapter 11 deals with the birational properties of moduli spaces of semi-stable sheaves on surfaces. The main result presented here is a simplified proof of \textit{J. Li}'s recent theorem [Invent. Math. 115, No. 1, 1-40 (1994; Zbl 0799.14015)] stating that moduli spaces of semi-stable sheaves on surfaces of general type are also of general type. Other results on the birational type of such moduli spaces are surveyed in a brief sub-section, and the treatise concludes with two instructive examples showing how the Serre correspondence can be used to obtain information about the birational structure of moduli spaces of sheaves on a K3 surface. Actually, both examples are variations on two recent theorems due to T. Nakashima (1993) and K. O'Grady (1995), respectively, and their discussion is based upon an elegant combination of the results from chapter 8 and 10 in the book.
Altogether, the present text fascinates by comprehensiveness, rigor, profundity, up-to-dateness and methodical mastery. The bibliography comprises 263 references, most of which are really referred to in the course of the text. Each chapter comes with its own specific introduction and, always at the end, with a list of extra comments, hints to the original literature, and remarks on related topics, further developments and current research problems. The authors have successfully tried to keep the presentation of this highly advanced material as self-contained as possible, so that the text should be accessible for readers with a solid background in algebraic geometry. Active researchers in the field will appreciate this book as a valuable source and reference for their work. vector bundles on projective surfaces; stable coherent sheaves; moduli spaces; gauge field theory; Donaldson polynomials; Seiberg-Witten invariants; Grauert-Mülich theorem; semi-stable sheaves; geometric invariant theory; conformal quantum field theory; Verlinde formula; Seiberg-Witten theory; Picard groups; determinantal line bundles; Gieseker-Maruyama moduli spaces; Donaldson-Uhlenbeck compactification; differential forms on moduli spaces of stable sheaves; birational properties Hu D.~Huybrechts and M.~Lehn. \newblock \em Geometry of moduli spaces of sheaves, Vol. E31 of \em Aspects in Mathematics. \newblock Vieweg, 1997. Algebraic moduli problems, moduli of vector bundles, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Research exposition (monographs, survey articles) pertaining to algebraic geometry, Vector bundles on surfaces and higher-dimensional varieties, and their moduli The geometry of moduli spaces of sheaves | 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 blender is a closed convex cone of real homogeneous polynomials that is also closed under linear changes of variable. Non-trivial blenders only occur in even degree. Examples include the cones of ``psd forms'', ``sos forms'', convex forms and sums of \(2u\)-th powers of forms of degree \(v\). The paper is devoted to a general presentation of blenders and their properties and to the analyze the extremal elements of some particular blenders. positive polynomials; sums of squares; sums of higher powers; convex polynomials; convex cones B. Reznick, \textit{Blenders}, Notions of Positivity and the Geometry of Polynomials (P. Brändén, M. Passare, M. Putinar, eds.), Trends in Mathematics, 345-373, Springer, Basel (2011) Sums of squares and representations by other particular quadratic forms, Forms of degree higher than two, Waring's problem and variants, Real algebraic and real-analytic geometry, Convexity of real functions of several variables, generalizations, Convex functions and convex programs in convex geometry Blenders | 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 Grassmann manifolds have the well-known Schubert cell decomposition. Here the volumes of some Schubert cells standardly embedded into a Grassmann manifold with a natural Riemannian structure are calculated. Grassmann manifolds; volume of Schubert cells Projective differential geometry, Grassmannians, Schubert varieties, flag manifolds Volumes of Schubert cells of Grassmann 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 Let \(\Sigma\) be a set of prime numbers which is either of cardinality one or equal to the set of all prime numbers. In this paper, we prove that various objects that arise from the \textit{geometry} of the configuration space of a hyperbolic curve over an algebraically closed field of characteristic zero may be \textit{reconstructed group-theoretically} from the pro-\(\Sigma\) \textit{fundamental group} of the configuration space. Let \(X\) be a hyperbolic curve of type \((g,r)\) over a field \(k\) of characteristic zero. Thus, \(X\) is obtained by removing from a proper smooth curve of genus \(g\) over \(k\) a closed subscheme [i.e., the ``divisor of cusps''] of \(X\) whose structure morphism to \(\operatorname{Spec} (k)\) is finite étale of degree \(r\); \(2g-2+r>0\). Write \(X_n\) for the \(n\)-th configuration space associated to \(X\), i.e., the complement of the various diagonal divisors in the fiber product over \(k\) of \(n\) copies of \(X\). Then, when \(k\) is \textit{algebraically closed}, we show that the \textit{triple} \((n,g,r)\) and the \textit{generalized fiber subgroups} -- i.e., the subgroups that arise from the various \textit{natural morphisms} \(X_n \to X_m [m < n]\), which we refer to as \textit{generalized projection morphisms} -- of the pro-\(\Sigma\) \textit{fundamental group} \(\Pi_n\) of \(X_n\) may be \textit{reconstructed group-theoretically} from \(\Pi_n\) whenever \(n \geq 2\). This result \textit{generalizes} results obtained previously by the first and third authors and A. Tamagawa to the case of \textit{arbitrary hyperbolic curves} [i.e., without restrictions on \((g,r)]\). As an application, in the case where \((g,r)= (0,3)\) and \(n \geq 2\), we conclude that there exists a \textit{direct product decomposition}
\[
\mathrm{Out}(\Pi_n) = \mathrm{GT}^{\Sigma} \times \mathfrak{S}_{n + 3}
\]
-- where we write ``\(\mathrm{Out}(-)\)'' for the group of outer automorphisms [i.e., \textit{without any auxiliary restrictions}!] of the profinite group in parentheses and \(\mathrm{GT}^{\Sigma}\) (respectively, \(\mathfrak{S}_{n + 3})\) for the pro-\(\Sigma\) \textit{Grothendieck-Teichmüller group} (respectively, symmetric group on \(n + 3\) letters). This direct product decomposition may be applied to obtain a \textit{simplified purely group-theoretic equivalent definition} -- i.e., as the \textit{centralizer} in \(\mathrm{Out}(\Pi_n)\) of the \textit{union of the centers of the open subgroups} of \(\mathrm{Out}(\Pi_n)\) -- of \(\mathrm{GT}^{\Sigma}\). One of the key notions underlying the theory of the present paper is the notion of a pro-\(\Sigma\) \textit{log-full subgroup} -- which may be regarded as a sort of \textit{higher-dimensional analogue} of the notion of a pro-\( \Sigma\) \textit{cuspidal inertia subgroup of a surface group} -- of \(\Pi_n\). In the final section of the present paper, we show that, when \(X\) and \(k\) satisfy certain conditions concerning ``\textit{weights}'', the pro-\(l\) log-full subgroups may be \textit{reconstructed group-theoretically} from the natural outer action of the absolute Galois group of \(k\) on the geometric pro-\(l\) fundamental group of \(X_n\). anabelian geometry; configuration space; generalized fiber subgroup; Grothendieck-Teichmüller group; hyperbolic curve; log-full subgroup Families, moduli of curves (algebraic), Coverings of curves, fundamental group Group-theoreticity of numerical invariants and distinguished subgroups of configuration space groups | 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 finite stratification of the Grassmanninan \(\mathrm{Gr}(k,n)\) to so-called positroid varieties \(\Pi_f\) is a refinement of the usual Schubert stratification. Knutson-Lam-Speyer computed that the cohomology class of such a variety \(\Pi_f\) is represented by the affine Stanley symmetric function \(\tilde{F}_f\). The paper under review extends this result to the image of a positroid variety under the (rational) map \(Z_{\mathrm{Gr}}:\mathrm{Gr}(k,n)\to \mathrm{Gr}(k,k+m)\) induced by a linear map \(Z\) from \(n\) dimensions to \(k+m\) dimensions. The closure of the image of \(\Pi_f\) is denoted by \(Y_f\), and is called an amplituhedron variety (when the dimension does not drop under the map). The result of the present paper is that the cohomology class of \(Y_f\) is a truncated version of \(\tilde{F}_f\). Truncation is defined through the shape of the partitions occuring in a Schur expansion. positroid varieties; Schubert calculus; Stanley symmetric functions; amplituhedron variety Lam, T., Amplituhedron cells and Stanley symmetric functions, Commun. Math. Phys., 343, 1025, (2016) Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations Amplituhedron cells and Stanley symmetric 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 the interplay between graph theory and the topology of isolated complex surface singularities is studied. To begin with it is well known that the dual graph \((\Gamma, w,g)\) of a good resolution \(\pi: \widetilde {V}\to V\) of a normal surface singularity \((V,p)\) determines \(\widetilde {V}\) up to diffeomorphism. Here the vertices of the graph \(\Gamma\) correspond to the irreducible components \(E_i\) of the exceptional fibre of \(\pi\), the edges to the intersections of these components, the coordinates \(w_i\) of the vector \(w\) of weights to the self-intersection numbers and the coordinates \(g_i\) of the vector \(g\) to the genera of the Riemann surfaces \(E_i\). Moreover the intersection matrix \(E= (E_i, E_j)\) is always negative definite.
On the other hand the intersection matrix \(E= \text{adj} (\Gamma)+ \text{diag} (w)\) of an (abstract) weighted graph \((\Gamma, w)\) is negative definite for almost all negative weights \(w_i\). Especially an (abstract) double-weighted graph \((\Gamma, w,g)\), \(w\in \mathbb{Z}^n\), \(g\in \mathbb{N}^n\), without loops is the dual graph of a normal surface singularity if and only if its intersection matrix is negative definite. Finally, the authors exploit this interplay in the case of Gorenstein singularities. A normal surface singularity is said to be numerically Gorenstein if the tangent bundle of \(V-p\) is topologically trivial over \(\mathbb{C}\) which amounts to say that the canonical class \(K\in H^2 (V, \mathbb{Q})\) is an integral linear combination of the \(E_i\) satisfying the adjunction formula \(2g-2= w+EK\). For an (abstract) weighted graph \((\Gamma,w)\) with non-singular intersection matrix \(E\) then it is shown that the set of vectors \(g\) of genera such that the adjunction formula \(2g-2= 2+EK\) has an integral solution \(K\) is infinite and parametrized by the integral solutions \(X\) of the congruence \(EX\equiv w(2)\). Finally the graph manifold \(X(\Gamma)\) obtained by plumbing admits a spin-structure if and only if \(K\) is even. For even weight \(w\) this is realized for infinitely many genera \(g\). Moreover if \(K=0\), then all weights equal \(-2\), all genera are 0 and \(\Gamma\) is one of the classical Dynkin diagrams \(A_n\), \(D_n\), \(E_6\), \(E_7\), \(E_8\).
Thus the authors obtain a new characterization of rational double points since \(K=0\) is equivalent to a spin(3)-structure on \(X(\Gamma)\). plumbing; complex surface singularities; weighted graph; Gorenstein singularities; normal surface singularity; spin-structure; rational double points Larrión F., Seade J.: Complex surface singularities from the combinatorial point of view. Topol. Appl. 66(3), 251--265 (1995) Singularities of surfaces or higher-dimensional varieties, Specialized structures on manifolds (spin manifolds, framed manifolds, etc.), Local analytic geometry, Planar graphs; geometric and topological aspects of graph theory Complex surface singularities from the combinatorial point of view | 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 According to the approach of Rosenberg, Van den Bergh and Smith, a non-commutative space or quasi-scheme is a Grothendieck category \(\text{Mod-}X\), to be viewed as the category of sheaves over some (nonexistent) space \(X\). The original motivation of the author was the fact that there does not exist a suitable notion of locally free sheaf for general non-commutative spaces. Bimodules connecting non-commutative spaces were introduced by Van den Bergh; the author introduces the notion of Frobenius bimodule between non-commutative spaces. If we apply this definition in the situation where the spaces are the categories of modules over two rings \(R\) and \(S\), then we recover the classical notion of \((R,S)\)-bimodule.
Frobenius bimodules with certain additional properties are studied, namely dimension preserving Frobenius bimodules (Section 4) and right localizing Frobenius bimodules (Section 5). In Section 6, Frobenius bimodules connecting Noetherian schemes are investigated, and related to sheaf bimodules studied by Nyman and Van den Bergh. Rank functions are studied in Section 7; a glueing theorem is proved in Section 8. In Section 9, Frobenius bimodules connecting a space to itself are studied. In Section 10, it is shown that there is a duality between the categories \(\text{Frob}(X,Y)\) and \(\text{Frob}(Y,X)\), consisting of respectively Frobenius \((X,Y)\)-bimodules and Frobenius \((Y,X)\)-bimodules. noncommutative spaces; Frobenius bimodules; sheaves; Noetherian schemes; noncommutative vector bundles; categories of modules; Grothendieck categories Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30--16S40]; rings, modules, etc. on which these act, Rings arising from noncommutative algebraic geometry, Noncommutative algebraic geometry, Bimodules in associative algebras, Module categories in associative algebras, Associative rings of functions, subdirect products, sheaves of rings, Grothendieck categories Frobenius bimodules between noncommutative 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 By a classical result of \textit{A. Beauville} [Invent. Math. 55, 121--140 (1979; Zbl 0403.14006)] if the canonical image of a smooth surface of general type \(X\) is a surface \(\Sigma\), denoting by \(S\) a minimal desingularization of \(\Sigma\), either \(p_g(S)=0\) or the resolution map \(S \to \Sigma\) is the canonical map of \(S\).
It is very easy to construct examples with \(p_g(S)=0\), but it is much more difficult to give examples of the latter case with degree \(d \geq 2\), being \(d\) the degree of the canonical map of \(X\): in this case the dominant rational map \(X \rightarrow S\) is called a {good canonical cover} of degree \(d\). Beauville in the above mentioned paper could give only one example, and its paper motivated many authors in constructing other examples of good canonical covers.
In this paper the authors construct three sequences of good canonical covers of degree \(2\) with \(X\) regular and unbounded invariants \(p_g\) and \(K^2\). Only sporadic examples of surfaces with these properties were previously known.
Sequences of examples were already known, but only of irregular surfaces. For example, the same authors [Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 29, No. 4, 905--938 (2000; Zbl 1016.14020)] gave a method for constructing infinite sequences of good canonical covers with \(d=2\), but all examples constructed by this method have \(q(X)\geq 2\).
Roughly speaking, the authors' idea is to find actions of a group \(G\) on some of those good canonical covers \(X \rightarrow S\), {killing the irregularity} of \(X\) and letting the quotient \(X/G \rightarrow S/G\) remain a good canonical cover.
They could find three sequences of examples, always using the group \(G=\mathbb{Z}/3\mathbb{Z}\). In the last section they could show that in one of these three cases the surface \(X\) is simply connected. canonical maps; double covers Surfaces of general type, Coverings in algebraic geometry Regular canonical 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 We discuss recent developments on geometric theory of ramification of schemes and sheaves. For invariants of \(\ell \)-adic cohomology, we present formulas of Riemann-Roch type expressing them in terms of ramification theoretic invariants of sheaves. The latter invariants allow geometric computations involving some new blow-up constructions. conductor; \(\ell \)-adic sheaf; wild ramification; Grothendieck-Ogg-Shafarevich formula; Swan class; characteristic class Takeshi Saito, Wild ramification of schemes and sheaves, Proceedings of the International Congress of Mathematicians. Volume II, Hindustan Book Agency, New Delhi, 2010, pp. 335 -- 356. Étale and other Grothendieck topologies and (co)homologies, Varieties over finite and local fields, Ramification and extension theory Wild ramification of schemes and sheaves | 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 geometric properties of varieties associated with invariant subspaces of nilpotent operators. There are algebraic groups acting on these varieties, and we give dimensions of orbits of these actions. Moreover, a combinatorial characterization of the partial order given by degenerations is described. degenerations; partial orders; Hall polynomials; nilpotent operators; invariant subspaces; Littlewood-Richardson tableaux Group actions on varieties or schemes (quotients), Combinatorial aspects of algebraic geometry, Representations of quivers and partially ordered sets, Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers, Graph algorithms (graph-theoretic aspects), Invariant subspaces of linear operators Operations on arc diagrams and degenerations for invariant subspaces of linear operators | 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 short biography of Alexander Grothendieck and survey of his work in algebraic geometry features a personal tone by recollections of the second author and a letter sent to him by Grothendieck, dated 3 February 2010 and reproduced in facsimile. The survey gives a chronological account, focussing on homological algebra [\textit{A. Grothendieck}, Tohoku Math. J. (2) 9, 119--221 (1957; Zbl 0118.26104)], schemes, the algebraic fundamental group, descent theory [\textit{A. Grothendieck}, in: Séminaire de géométrie algébrique du Bois Marie 1960/61: revêtements étales et groupe fondamental (SGA~1). Berlin-Heidelberg-New York: Springer-Verlag (1971; Zbl 0234.14002); Fondements de la géométrie algébrique: extraits du séminaire Bourbaki 1957--1962. Paris: Secrétariat mathématique (1962; Zbl 0239.14002)], Grothendieck topologies [\textit{M. Demazure}, in: Séminaire de géométrie algébrique 1962/64: schémas en groupes (SGA~3), I: Propriétés générales des schémas en groupes. Berlin-Heidelberg-New York: Springer-Verlag (1970; Zbl 0207.51401)], étale cohomology [\textit{L. Illusie}, in: Alexandre Grothendieck: a mathematical portrait. Somerville, MA: International Press. 175--192 (2014; Zbl 1303.14007)], and the monodromy theorem [\textit{J.-P. Serre} and \textit{J. Tate}, Ann. Math. (2) 88, 492--517 (1968; Zbl 0172.46101)]. In particular, schemes with nilpotents in the structure sheaf are presented through the example of the Picard functor. homological algebra; scheme; algebraic fundamental group; descent theory; Grothendieck topology; étale cohomology; monodromy theorem; nilpotent in the structure sheaf; Picard functor Biographies, obituaries, personalia, bibliographies, Research exposition (monographs, survey articles) pertaining to algebraic geometry, History of algebraic geometry, Étale and other Grothendieck topologies and (co)homologies Life and work of Alexander Grothendieck | 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 C be an irreducible, smooth, projective curve defined over the field of complex numbers. We call C elliptic-hyperelliptic (e.h. for short) if it admits a degree two morphism \(\pi: C\to E\) onto an elliptic curve. We denote by \(M_ g^{eh}\) the moduli space of e.h. curves of genus g. The aim of this note is to present a proof of the following theorem: \(M_ 4^{eh}\) is rational.
We proceed as follows: In section 1 the canonical model of a generic e.h. curve C (of genus \(4\)) is shown to be complete intersection of a unique cubic cone R and a unique quadric. By looking at the tangent space to the canonical space at the vertex of R, in section \(2,\) we associate to C a pair \((Z,\gamma)\), where Z and \(\gamma\) are smooth coplanar curves of degree 3 and 2 respectively, and we are able to show that \(M_ 4^{eh}\) is birational to \(((Z,\gamma))/PGL(3).\) - After fixing a quadratic form defining \(\gamma\) we can prove that \(\{(Z,\gamma)\}/PGL(3)\) is birational to \(H^ 0(P^ 1,{\mathcal O}_{{\mathbb{P}}^ 1}(6))/G_{\ell_ 0}\) where \(G_{\ell_ 0}\) is a \({\mathbb{C}}^*\)-extension of \({\mathbb{Z}}_ 2.\)
In section 3 we compute the representation of \(G_{\ell_ 0}\) on \(H^ 0(P^ 1,{\mathcal O}_{{\mathbb{P}}^ 1}(6))\) and we show that its \(G_{\ell_ 0}\)-invariant field is purely transcendental over \({\mathbb{C}}\) completing the proof of the theorem. genus four double covers of elliptic curves; rationality of moduli space of elliptic-hyperelliptic curves Bardelli, Fabio; Del Centina, Andrea: The moduli space of genus four double covers of elliptic curves is rational. Pac. J. Math. 144, No. 2, 219-227 (1990) Families, moduli of curves (algebraic), Complete intersections, Algebraic moduli problems, moduli of vector bundles The moduli space of genus four double covers of elliptic curves is rational | 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 purpose of this paper is to prove the following
Theorem: Let \(M\) be a complex projective manifold, \(R\in H_2(M,\mathbb Z)\), \(g,k\geq 0\). Let \({\mathcal C}_{R,g,k}(M)\) be the moduli space of stable maps of genus \(g\) with \(k\) marked points and representing class \(R\). Then the homology class associated to the algebraic virtual fundamental class (a Chow class on \({\mathcal C}_{R,g,k}(M))\) as in the paper by \textit{K. Behrend} [Invent. Math. 127, 601-617 (1997; Zbl 0909.14007)] and the symplectic virtual fundamental class as in the paper by \textit{B. Siebert} [``Gromov-Witten invariants for general symplectic manifolds'', preprint \url{http://arXiv.org/abs/dg-ga/?9608005}]\ coincide. Gromov-Witten invariants; virtual fundamental class; Grothendieck duality; derived category; moduli space; homology class Siebert, B., Algebraic and symplectic Gromov--Witten invariants coincide, Ann. Inst. Fourier (Grenoble) 49 (1999), 1743--1795. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Homotopy theory and fundamental groups in algebraic geometry, Fine and coarse moduli spaces, Enumerative problems (combinatorial problems) in algebraic geometry Algebraic and symplectic Gromov-Witten invariants coincide | 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 projective variety defined over \(\mathbb{C}\) with only ordinary double points and such that \(\omega_X \cong \mathcal{O}_X\). The paper under review studies the problem of when such a variety is smoothable, i.e., when there is a flat projective morphism \( f : \mathcal{X} \rightarrow \Delta\), where \(\Delta\) is the unit disk, such that \(f^{-1}(0) = X\), \(f^{-1}(t)\) is smooth for \(t \not= 0\) and \(\mathcal{X}\) is smooth. By using the \(T^1\)-lifting property, \textit{Y. Kawamata} [J. Algebraic Geom. 1, 183--190 (1992; Zbl 0818.14004)] has shown that \(X\) has unobstructed deformations and consequently its versal deformation space \(\mathrm{Def}(X)\) is smooth. Therefore the existence of a smoothing of \(X\) is equivalent to the existence of a first order smoothing. In dimension three, \textit{R. Friedman} [Math. Ann. 274, 671--689 (1986; Zbl 0576.14013)] has given a topological condition for the existence of a first order smoothing in terms of a small resolution \( Y @>{f}>> X\) of \(X\). Let \(C_i\), \(i=1,\dots , k\), be the exceptional curves of such a resolution. Then Friedman showed that a first order smoothing of \(X\) exists if and only if there is a relation \(\sum_{i=1}^k \delta_i [C_i]=0\) in \(H_2(Y, \mathbb{R})\), where \([C_i]\) is the class of \(C_i\) and \(\delta_i \not= 0\), for all \(i\). \textit{G. Tian} [Essays on mirror manifolds. Cambridge, MA: International Press. 458--479 (1992; Zbl 0829.32012)] has shown that if such a relation exists, then the first order smoothing lifts to an actual smoothing.
In this paper, the authors obtain similar criteria for the existence of first order smoothings in the case when \(\dim X=n=2m+1\) is odd. Since in higher dimensions ordinary double points do not have small resolutions, in place of a small resolution the authors consider the blow up \(Y @>{f}>> X\) of \(X\) along its singular points. Let \(P_i\), \(i=1, \dots , k\), be the singular points of \(X\) and \(Q_i=f^{-1}(P_i)\) the \(f\)-exceptional set. These are \((n-1)\)-dimensional quadrics with standard \((n-1)/2=m\)-dimensional planes \(E_i\), \(F_i\) (if \(n=3\), these are just the rulings of \(Q_i\)). Let \(A_i\), \(B_i\) be their homology classes. If \(n=3\), the authors show that a first order smoothing of \(X\) exists if and only if there is a relation \(\sum_{i=1}^k \delta_i (A_i-B_i)=0\) in \(H_2(Y, \mathbb{R})\) with all \(\delta_i \not= 0\). More generally, let \(e \in \mathrm{Ext}_X^1(\Omega_X, \mathcal{O}_X)\) be a first order deformation of \(X\). The authors define a map \(\phi : \mathrm{Ext}_X^1(\Omega_X, \mathcal{O}_X) \rightarrow H^{m,m}(Q)=\bigoplus_{i=1}^k H^{m,m}(Q_i)\) which they call the Yukawa map and they show that \(\phi(e)=(\delta_1 (A_1-B_1), \dots , \delta_k(A_k-B_k))\). They show that there is a relation \(\sum_i \delta_i (A_i-B_i)=0\) in \(H_{n-1}(Y)\) and that \(e\) is a first order smoothing if and only if \(\delta_i \not=0\) for all \(i\). Calabi-Yau; smoothing; ordinary double points Calabi-Yau manifolds (algebro-geometric aspects), Deformations of complex structures, Formal methods and deformations in algebraic geometry, Compact complex \(n\)-folds Smoothing nodal Calabi-Yau \(n\)-folds | 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 a supersingular abelian variety \(A\) of dimension~\(g\) over the finite field \({\mathbb F}_q={\mathbb F}_{p^n}\). Bearing in mind the case of a supersingular elliptic curve over a prime field, when the trace of Frobenius (on the Tate module \(T_\ell(E)\)) is zero as long as \(p>3\), we can ask what conditions the supersingularity imposes on the characteristic polynomial \(P_A(X)\) of Frobenius on \(T_\ell(A)\).
A look at the examples where this polynomial has been computed directly immediately suggests that the answer should be that all the odd coefficients vanish, i.e.\ \(P_A(X)\) is an even function of \(X\), under not too stringent conditions. It is easy to prove that this is indeed true provided \(p\) is large enough, but simply copying the proof of the vanishing of the trace only gives a result for rather large~\(p\), roughly \(p>{{2g}\choose{g}}^2\).
In this paper the authors use a more refined, but still largely elementary, argument to show that \(P_A(X)\) is even as long as \(n\) is odd and \(p>2g+1\). They note that according to examples given in [\textit{G. McGuire} et al., Funct. Approximatio, Comment. Math. 51, No. 2, 415--436 (2014; Zbl 1304.14025)], the extra condition that \(n\) is odd cannot be avoided.
The proof, like the simpler version, relies on the result of Manin and Oort that says that the roots of \(P_A(X)\) are all of the form \(\sqrt{q}\times \zeta\), where \(\zeta\) is some root of unity. They combine this with the elementary result that the \(n\)th cyclotomic polynomial \(\Phi_n(X)\) is even if and only if \(4|n\), and a result from [loc. cit.] that describes the minimum polynomial of numbers of the form \(\sqrt{\pm q}\times \xi\), where \(\xi\) is a primitive \(4t\)-th root of unity.
Finally, they briefly describe some of the immediate and potential consequences of their result for Iwasawa theory, as developed by \textit{S.-i. Kobayashi} [Invent. Math. 152, No. 1, 1--36 (2003; Zbl 1047.11105)] and \textit{F. E. I. Sprung} [J. Number Theory 132, No. 7, 1483--1506 (2012; Zbl 1284.11147)]. abelian varieties; supersingular; cyclotomic polynomials Finite ground fields in algebraic geometry, Polynomials in number theory, Abelian varieties of dimension \(> 1\) On the parity of supersingular Weil 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 paper, bringing together the connection between the Zolotarev polynomials and the Painlevé VI equations from one hand and the potential theory from other hand, the authors introduce a new type of deformation (the so called iso-harmonic deformation) of annua ardomains with a marked point in the extended complex plane. The introduced iso-harmonic deformation allows the authors to derive a solution of the Painlevé VI equation with parameters \(\alpha=\gamma=1/8, \beta=-1/8, \delta=3/8\) in a new way. In particular, starting with the Zolotarev polynomials and varying the elliptic curve that supports the Zolotarev polynomials they obtain a family of elliptic curves with the same property. It turns out that the function \(f(x)\) which determines the position of the zero of the differential of the third kind \(\Omega\) on the pointed family of elliptic curves is a solution of the Painlevé VI equation with parameters \(\alpha=\gamma=1/8, \beta=-1/8, \delta=3/8\). Next, the authors involve the obtained result in the potential theory. In particular, they deform an annular domain \(V_h\) (which is the conformal image of the complement of a union of two intervals) and the pole \(c(h)\) of the Green function of \(V_h\) keeping the harmonic measures of \(V_h\) invariant. It turns out that under such a deformation the critical point of the Green function of \(V_h\) with a pole at \(c(h)\) solves the same Painlevé equation VI with parameters \(\alpha=\gamma=1/8, \beta=-1/8, \delta=3/8\). The authors call such a deformation an iso-harmonic deformation. Painlevé VI equations; Okamoto transformations; elliptic curves; abelian differentials; Zolotarev polynomials; Green functions; annular domains; harmonic measures Painlevé and other special ordinary differential equations in the complex domain; classification, hierarchies, Isomonodromic deformations for ordinary differential equations in the complex domain, Elliptic functions and integrals, Relationships between algebraic curves and integrable systems Deformations of the Zolotarev polynomials and Painlevé VI equations | 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 sequence of rational functions in a variable \(q\) is \(q\)-holonomic, see \textit{D. Zeilberger} [J. Comput. Appl. Math. 32, No. 3, 321--368 (1990; Zbl 0738.33001)] and \textit{H. S. Wilf} and \textit{D. Zeilberger} [Invent. Math. 103, No. 3, 575--634 (1991; Zbl 0739.05007)] if it satisfies a linear recursion with coefficient polynomials in \(q\) and \(q^n\).
In virtue of a fundamental result by Wilf-Zeilberger, Quantum Topology turns out to provide us with a plethora of \(q\)-holonomic sequences of natural origin. In particular, the present paper takes into account the \(q\)-holonomic sequence of Jones polynomials of a knot and its parallels, \textit{S. Garoufalidis} and \textit{Thang T.Q. Lê} [Geom. Topol. 9, 1253--1293 (2005; Zbl 1078.57012)]).
The author associates a tropical curve, see [\textit{J. Richter-Gebert, B. Sturmfels} and \textit{T. Theobald}, in: G. L. Litvinov (ed.) et al., Idempotent mathematics and mathematical physics. Proceedings of the international workshop, Vienna, Austria, February 3--10, 2003. Providence, RI: American Mathematical Society (AMS) Contemporary Mathematics 377, 289--317 (2005; Zbl 1093.14080)] and [\textit{D. Speyer} and \textit{B. Sturmfels}, Math. Mag. 82, No. 3, 163--173 (2009; Zbl 1227.14051)] to each \(q\)-holonomic sequence; in particular, to every knot \(K\) a tropical curve is associated, via the Jones polynomial of \(K\) and its parallels. As a consequence, a relation is established between the AJ Conjecture [see the author, in: C. Gordon (ed.) et al., Proceedings of the Casson Fest. Coventry: Geometry and Topology Monographs 7, 291--309 (2004; Zbl 1080.57014)] and the Slope Conjecture [the author, Quantum Topol. 2, No. 1, 43--69 (2011; Zbl 1228.57004)], which relate the Jones polynomial of \(K\) and its parallels respectively to the \(SL(2; \mathbb C)\) character variety and to slopes of incompressible surfaces.
The paper gives also an explicit computation of the tropical curve for the \(4_1\), \(5_2\) and \(6_1\) knots, verifying in these cases the duality between the tropical curve and a Newton subdivision of the A-polynomial of the knot. Knots; Jones polynomials; AJ Conjecture; Slope Conjecture; A-polynomial; non-commutative A-polynomial; Jones slope; tropicalization; tropical curve; tropical geometry; Newton polygon; quantization; BSP states; twist knots 6. S. Garoufalidis, Knots and tropical curves, in Interactions Between Hyperbolic Geometry, Quantum Topology and Number Theory, Contemporay Mathematics, Vol. 541 (American Mathematical Society Providence, 2011), pp. 83-101. genRefLink(16, 'S0218216516500577BIB006', '10.1090%252Fconm%252F541%252F10680'); Knots and links in the 3-sphere, Invariants of knots and \(3\)-manifolds, Topology of general 3-manifolds, , Generalized hypergeometric series, \({}_pF_q\) Knots and tropical 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 Given two rational points on the graph of a cubic polynomial with rational coefficients, it is well known that one can construct another rational point by finding the third intersection point of the curve with the chord through the two points. Marked on the graph of \(y^2=x^4-12\) is the rational point \((2,2)\). However, the chord-tangent method is not directly applicable since \(y^2=x^4-12\) is a quartic and not a cubic. So how are we to generate other rational points? polynomials with rational coefficients; rational points Rational points New points from old | 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 construct an example of a manifold homeomorphic to the connected sum \({\mathbb C}{\mathbb P}^2\# 13\overline {\mathbb C}{\mathbb P}^2\) but not diffeomorphic to it. Note that, for all \(k>9\), \textit{R. Friedman} and \textit{J. W. Morgan} [J. Differ. Geom. 27, No. 2, 297--369 (1988; Zbl 0669.57016)] constructed algebraic surfaces that are homeomorphic to \({\mathbb C}{\mathbb P}^2\# k\overline {\mathbb C}{\mathbb P}^2\) but not homeomorphic to it. However, the author's example is not in the Friedman-Morgan list since this example has no symplectic structure. Seiberg-Witten invariants; symplectic 4-manifolds; double branched cover; Spin-structures Differentiable structures in differential topology, Symplectic manifolds (general theory), Elliptic surfaces, elliptic or Calabi-Yau fibrations, \(K3\) surfaces and Enriques surfaces, Low-dimensional topology of special (e.g., branched) coverings, Group actions on manifolds and cell complexes in low dimensions, Applications of global analysis to structures on manifolds Exotic smooth structure on \(\mathbb C\mathbb P^2\sharp 13\overline{\mathbb C\mathbb P}^2\) | 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 A Klein surface is the quotient of a Riemann surface \(M\) by an antiholomorphic involution \(\sigma\). This pair \((M, \sigma)\) can also be viewed as a real algebraic curve. Its topological classification is given by a triple \((g,n,a)\), where \(g\) is the genus of \(M\), \(n\) is the number of connected components of the fixed-point set of \(\sigma\) in \(M\), and \(a\) gives the orientability of \(M/\sigma\).
This very long and detailed paper is devoted to the study of the moduli spaces of semi-stable real and quaternionic vector bundles for a given topological type \((g,n,a)\). They can be embedded into the symplectic quotient corresponding to the moduli variety of semi-stable holomorphic vector bundles of fixed rank \(r\) and degree \(d\), on a smooth complex projective curve.
So the authors generalize the method given by Atiyah and Bott, in order to get the goal of the paper, which is to compute the mod 2 Poincaré polynomials of these moduli spaces when \(r\) and \(d\) are coprime. After a very technical preparation throughout the paper this is obtained in the final Section 6.
An Appendix is added in which the polynomials are explicitly given for low values of the rank \(r \leq 4\). Klein surfaces; Yang-Mills equations; vector bundle; Poincaré polynomials Liu, Chiu-Chu Melissa; Schaffhauser, Florent, The Yang-Mills equations over Klein surfaces, J. topol., 6, 3, 569-643, (2013) Vector bundles on curves and their moduli, Topology of real algebraic varieties, Klein surfaces The Yang-Mills equations over Klein 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 This paper is a continuation of the author's work on various aspects of elimination and resultant. For the earlier relevant articles see Adv. Math. 37, 212-238 (1980; Zbl 0527.13005) and 90, No. 2, 117-263 (1991; Zbl 0747.13007). There is a subsequent article dealing with similar issues [Adv. Math. 126, No. 2, 119-250 (1997; see the following review)]. These papers are rather long and thus difficult to go over completely in this short review. I will just describe the basic problem that is being tackled. Let \(\kappa\) be a scheme and \(E\) a rank \(n\) locally free sheaf over \(\kappa\). Let \(r\geq 1\) be an integer and let \(d_1,\ldots,d_r\) be positive integers. Let
\[
A=\text{Sym}_\kappa\Bigl(\bigoplus_{i=1}^r \text{Sym}^{d_i}(E)^*\Bigr)
\]
and \(S=\text{Spec} A\). Let \(C=A\otimes_\kappa\text{Sym}_\kappa(E)\). One has the canonical section
\[
{\mathcal O}_\kappa\to\text{Sym}^{d_i}(E)^*\otimes\text{Sym}^{d_i}(E)
\]
and thus one has a natural map \(f_i:C(-d_i)\to C\), a \(C\)-module homomorphism. Essentially \(f_i\) corresponds to the `universal homogeneous polynomial' of degree \(d_i\). Let \(B=C/(f_1,\ldots, f_r)\) and \(X=\text{Proj} B\). We have the natural projective morphism from \(X\to S\) and let \(T\) be the image of \(X\). The basic issue of elimination theory from the view point of invariant theory is to describe \(T\), in terms of the representations of \(\Aut_\kappa(E)\). The author does this in great detail and in full generality. In particular no Noetherian hypothesis is made and in author's words \textit{sans contorsions perpétuelles}. The stress here is to keep track of the \(\Aut(E)\) structure throughout. In the case when \(r=n\) above, we have the classical case of resultant and the author describes methods to see which points are singular etc. Needless to say, these get connected to Jacobian ideals, Grothendieck's residue and utilises some of the classical isomorphisms like Kronecker isomorphism (if \(E,F\) are vector bundles of rank \(n,m\), then \(\det(E\otimes F)=(\det E)^m\otimes(\det F)^n\), the equality being natural). In conclusion, this paper is a very exhaustive treatment of the subject, where references in this generality are hard to come by. elimination theory; invariant theory; Jacobian ideals; Grothendieck's residue; Kronecker isomorphism Jouanolou, J.P., Aspects invariants de l'élimination, Adv. math., 114, 1, 1-174, (1995) Actions of groups on commutative rings; invariant theory, Determinantal varieties, Polynomials over commutative rings, Complete intersections, Vector and tensor algebra, theory of invariants Invariant aspects of elimination | 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 Exponent polynomials are expressions of the form \(P\bigl(e^{h(X_ 1,\dots,X_ n)},X_ 1,\dots,X_ n\bigr)\), where \(P(U,X_ 1,\dots,X_ n)\) and \(h(X_ 1,\ldots,X_ n)\) are polynomials with integer coefficients. Given a system of polynomials in exponent inequalities, the paper presents an algorithm for testing if the system has one solution over the reals \(\mathbb{R}^ n\). Assuming that the degree of all the involving \(P\) and \(h\) polynomials is less than \(d\), that the bit lengths of the integer coefficients are less than \(M\), and that the number of inequalities is less than \(k\), it is reasonable to estimate a bound for the size of the system by \(L=Mkd^ n\) (dense representation). With this notation, the algorithm presented here runs in time polynomial in \(M(nkd)^{n^ 4}\) (i.e. is subexponential in \(L\), as it is bounded by \(L\) to some power which is polynomial in \(\log (L))\). The work extends and uses previous work on the purely algebraic case (systems of polynomial inequalities), in particular requires computation with infinitesimals and some results from non-standard analysis. algebraic complexity; decidability; exponent polynomials; polynomials in exponent inequalities; non-standard analysis N. Vorobjov. The complexity of deciding consistency of systems of polynomial in exponent inequalities.J. Symbolic Comput.,13, 139--173, 1992. Analysis of algorithms and problem complexity, Symbolic computation and algebraic computation, Computational aspects in algebraic geometry, Real algebraic sets, Nonstandard models of arithmetic, Computational aspects and applications of commutative rings The complexity of deciding consistency of systems of polynomials in exponent inequalities | 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 present book offers one more introduction to algebraic geometry, although during the past twenty years a rather great amount of textbooks on this subject has appeared. Thus one might ask the question of what is the specific character of this new textbook.
Generally speaking, algebraic geometric is amongst the oldest disciplines in mathematics. However, in the recent decades, it has undergone a radical change concerning its foundations, rigor, methods, interrelations with other branches in mathematics, and applications. In particular, algebraic geometry is intimately related to commutative algebra, algebraic number theory, complex analysis, differential geometry, topology, and --- nowadays --- also to special topics in mathematical physics. With regard to this widespread spectrum of interrelations and roots, foundational concepts and methods, purposes for application, and due to the rapid development that algebraic geometry is still undergoing, any introduction to this subject must be highly selective. And, in fact, it is precisely the selection of material, which makes the recent textbooks and standard references differ, apart from the level of the used methods and the completeness of the presented results.
The present textbook aims to introduce various aspects of algebraic geometry from the transcendental (i.e., complex-analytic and differential-geometric) point of view. This means, it does not intend to provide an introduction to ``abstract'' algebraic geometry from Grothendieck's scheme-theoretic viewpoint. Instead this textbook is developed around the algebraic theory of complex manifolds and varieties, with the main emphasis on the low-dimensional cases of curves and surfaces. In this regard, it is conceptually closely related to the much more advanced and comprehensive standard text ``Principles of algebraic geometry'' by \textit{Ph. Griffiths} and \textit{J. Harris} [New York 1978; Zbl 0408.14001)]. However, the present exposition is much more selective, and written at the second year graduate level. As the author himself points out, the book is basically designed as a guide to complex algebraic geometry for the nonexpert. --- With respect to this aim of his, the author has done an excellent job.
In chapter 1 he introduces preliminary materials from commutative algebra, projective algebraic geometry, and algebraic topology. This includes: affine and projective varieties, some ideal theory, analytic varieties, dimension theory, the degree of a variety in projective space, simplicial homology, intersection numbers, De Rham cohomology, Poincaré duality, and the Hodge decomposition theorem for complex manifolds. Here some proofs are carried out (e.g. Hilbert's basis theorem and Hilbert's Nullstellensatz); as for some other fundamental results, the author refers to the standard literature.
Chapter 2 provides the reader with various complex-analytical methods and techniques such as sheaves and their cohomology, complex vector bundles and Chern classes, line bundles, divisors, linear systems, Kähler manifolds and their Hodge theory, Hermitean vector bundles, and specific bundles over projective spaces. Special emphasis is put on computing Hodge numbers and Chern classes in several particular cases. --- The material here, although not given with complete proofs, is very well arranged and instructively presented. The standard reference for further reading is, again, the book of Griffiths and Harris.
Chapter 3 contains an exposition of algebraic curves and compact Riemann surfaces. Basically, this includes plane projective curves, the Plücker formulae, meromorphic functions and differential forms on Riemann surfaces (the Riemann-Hurwitz formula, the residue theorem, etc.), divisors, linear systems, projective embeddings of Riemann surfaces, elliptic curves, a discussion of Jacobians, Abel's theorem, Torelli's theorem, Weierstrass points, hyperelliptic curves, the Riemann-Roch theorem for curves, and some outlook to the Brill-Noether theory and special divisors.
Chapter 4 is devoted to the theory of compact algebraic surfaces and their Enriques classification. After a beautiful discussion of the intersection pairing on compact topological 4-manifolds, the author discusses the basic invariants of surfaces, (bi-)rational maps, the blow- up process, the notion of Kodaira dimension, ample divisors, ruled surfaces, rational surfaces, unirational surfaces, the Albanese variety of a surface, and the Enriques classification of surfaces of non-general type. Subsequently, a rather detailed treatment of \(K\)-3 surfaces and the Torelli theorem satisfied by them is given, and the Chern number geography for surfaces of general type is reviewed afterwards. The chapter ends with an outlook to complex spaces and singular surfaces. --- As for chapter 3 and chapter 4, which together form one main part of the book, proofs of the basic standard theorems are generally given. In addition, many more advanced topics are touched upon, outlined, or discussed with respect to their recent developments (e.g., moduli of \(K\)- 3 surfaces, the Miyaoka-Yau-Bogomolov inequality, etc.). It is, in fact, this combination of providing basic material with rigor and discussing more advanced (and recent) topics along this way, which gives the book its particular flavour and value. The interested reader is referred, as for further reading, to the existing more advanced references (or research articles), and he gets provided with quite a lot of motivation and profound basic knowledge for that.
The concluding chapter 5 deals with some fundamental techniques from Hermitean differential geometry and their applications in algebraic geometry. This reflects the author's special research interests and, due to this fact, offers some never-before-published material on the moving frame theoretic treatment of submanifolds in projective space. This chapter may be of particular interest for both algebraic geometers and differential geometers. The exposition starts with Grassmannians and their topological properties. This includes an account on metrics, Schubert calculus, unitary frames, local Hermitean geometry and curvature forms of Grassmannians, Chern numbers, and the universal bundle construction. The next topic concerns embedded curves in \(\mathbb{P}^ n\) and the Plücker formulae, whereas the following section is devoted to the theory of (higher) osculating maps of projective complex submanifolds. The author presents a proof of Weyl's formulae (and a generalization of them) relating Chern forms of osculating bundles to osculating Kähler forms. This may be thought of as a higher-dimensional analogue of the Plücker formulae given before, and represents really new material. The rest of the chapter discusses formulae of Gauss-Bonnet type for projective hypersurfaces (i.e., formulae which relate the total curvatures of hypersurfaces to their Chern numbers) in great detail and, as a nice application, surfaces in \(\mathbb{P}^ 3\) and \(\mathbb{P}^ 5\). --- Chapter 5 forms the other main part of the book and offers a lot of interesting material which can barely be found elsewhere.
The author has added, just for the convenience of the reader, two appendices. Appendix I (written by \textit{Robert Fisher}) gives some background material on complexification and complex differential forms, and appendix II discusses (with proofs) elliptic functions and complex elliptic curves.
Each chapter in the book starts with its own introduction, in which the aim, the organization of the material, and hints to further references are explained.
All in all, the present textbook is a highly welcome addition to the already existing ones. It is certainly particularly valuable for the analytically oriented beginner, who wants a profound introduction, without getting the whole material in full generality and completeness at the beginning. The text only assumes some familiarity with algebraic topology, complex function theory, and differential geometry. In this regard, the book is also useful for any working mathematician or physicist, who is a non-expert in algebraic geometry, but wants to get acquainted with algebro-geometric methods for further applications in his field of research. Finally, the book also provides an excellent text for teaching purposes at the graduate level in algebraic or complex-analytic geometry. transcendental algebraic geometry; Kaehler manifolds; Hodge theory; Hermitean vector bundles; algebraic curves; compact Riemann surfaces; compact algebraic surfaces; Torelli theorem; Chern number; moving frame; Schubert calculus; Grassmannians; complex-analytic geometry Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Curves in algebraic geometry, Transcendental methods, Hodge theory (algebro-geometric aspects), Transcendental methods of algebraic geometry (complex-analytic aspects), Riemann surfaces; Weierstrass points; gap sequences, Compact complex surfaces, Surfaces and higher-dimensional varieties Complex algebraic geometry. An introduction to 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 No review copy delivered. Schubert codes; linear codes associated to Schubert varieties; minimum weight codewords; dual Schubert codes Linear codes (general theory), Applications to coding theory and cryptography of arithmetic geometry, Geometric methods (including applications of algebraic geometry) applied to coding theory Majority logic decoding for certain Schubert codes using lines in 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 An elliptic curve in twisted Jacobi intersections curve is an elliptic curve over a field \(K\) defined by \(au^2+v^2 = 1\), \(bu^2+w^2 = 1\) for \(a,b\in K\) with \(ab(a-b)\neq 0\). This paper presents the division polynomials for this type of elliptic curves and gives theirs basic properties. Furthermore, the specific formulae for the division polynomials is given. division polynomials; twisted Jacobi Intersections curves Applications to coding theory and cryptography of arithmetic geometry, Algebraic coding theory; cryptography (number-theoretic aspects) Division polynomials for twisted Jacobi intersections 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 \(GL_ n\) be the group of \(n \times n\) invertible complex matrices, and \(P\) a parabolic subgroup of \(GL_ n\). In this paper we give a geometric description of the cohomology ring of a Schubert subvariety \(Y\) of \(Gl_ n/P\). Our main result (theorem 3.1) states that the coordinate ring \(A(Y \cap Z)\) of the scheme-theoretic intersection of \(Y\) and the zero scheme \(Z\) of the vector field \(V\) associated to a principal regular nilpotent element \({\mathfrak n}\) of \({\mathfrak g}{\mathfrak l}_ n\) is isomorphic to the cohomology algebra \(H^*(Y;\mathbb{C})\) of \(Y\). This theorem was conjectured for any reductive algebraic group \(G\) by \textit{E. Akyildiz}, \textit{J. B. Carell} and \textit{D. I. Liebermann} in Compos. Math. 57, 237- 248 (1986; Zbl 0613.14035), and it was proved for the Grassmannian manifolds by \textit{E. Akyildiz} and \textit{Y. Akyildiz} in J. Differ. Geom. 29, No. 1, 135-142 (1989; Zbl 0692.14031). We were recently informed that \textit{D. H. Peterson} has just proved that \(GL_ n\) is exactly the algebraic group \(G\) where the cohomology ring of any Schubert subvariety \(Y\) of the space \(G/B\) is isomorphic to \(A(Y \cap Z)\). Here \(B\) stands for a Borel subgroup of \(G\). It is also interesting to note that the cohomology ring of the union of two Schubert subvarieties in \(GL_ n/P\) may not admit such a description. This result is due to \textit{J. B. Carrell}. cohomology ring of a Schubert variety E. Akyıldız, A. Lascoux, and P. Pragacz, Cohomology of Schubert subvarieties of \?\?_{\?}/\?, J. Differential Geom. 35 (1992), no. 3, 511 -- 519. Grassmannians, Schubert varieties, flag manifolds, (Co)homology theory in algebraic geometry Cohomology of Schubert subvarieties of \(GL_ n/P\) | 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 classical problem in algebraic complexity is to find lower bounds for polynomial evaluation, i.e. for the problem of evaluating a complex polynomial \(f\in C[x_ 1,\ldots,x_ m]\) at a point \(\xi\in C^ m\).
Given \(f\in C[x_ 1,\ldots,x_ m]\), one says that an algebraic computation tree \(T\) computes \(f\) when for every \(\xi\in C^ m\) the output of \(T\) with input \(\xi\) is \(f(\xi)\). The complexity of polynomial evaluation for a given \(f\) is the minimal depth of all algebraic decision trees that compute \(f\). Additive (or multiplicative) complexity is defined in the same way but counting only additions and substractions (resp. nonscalar multiplications or divisions) when considering the depth of the computation tree.
Many results have been provided in the last decades that show lower bounds for this problem that depend on \(f\). In particular, the case of polynomials having coefficients that are algebraically independent over \(\mathbb{Q} \) has been largely studied and for this case the following bounds hold
\begin{align*}
L_+(f)&\geq \binom{d+m}{m}-1,\\ L_*(f)&\geq \biggl[ \binom{d+m}{m}-1\biggr]/2,
\end{align*}
where \(L_ +(f)\) is the additive complexity of \(f\) and \(L_ *(f)\) its multiplicative complexity.
More recently, attention has been paid to an \textit{a priori} simpler problem, to decide just whether \(f(\xi)\) equals zero. The main result of the reviewed paper is that for the case of algebraically independent coefficients testing for zero is not easier than evaluating at a point.
In fact, the authors consider a hypersurface \(X\subset C^ m\) having irreducible components \(X_ 1,\ldots,X_ t\) satisfying
\begin{align*}
&X_i=\text{Zeroset}(f_i), \quad f_i\in C[x_1,\ldots,x_m]\text{ irreducible}\\ &\deg(f_i)=d_i\quad (i=1,\ldots,t)
\end{align*}
such that the polynomials \(f_ 1,\ldots,f_ t\) can be chosen such that all their coefficients are algebraically independent over \(\mathbb{Q} \), and they derive the bounds
\begin{align*}
C_{+,=}(X)&= \sum_{i=1}^t\biggl[\binom{d_i+m}{m}- 1\biggr],\\ C_{*,=}(X)&\geq \frac12 \sum_{i=1}^t \biggl[ \binom{d_i+m}{m}-1\biggr],
\end{align*}
where \(C_{+,=}\) denotes the branching additive complexity (we now count additions, substractions and equality tests) and \(C_{*,=}\) the multiplicative branching complexity (i.e. the minimal number of nonscalar multiplications, divisions and equality tests). Moreover, they show that the second bound is sharp for min\(_{1\leq i\leq t} d_ i\to\infty\) keeping \(m\) and \(t\) fixed.
A related problem is posed in a natural way when considering real varieties instead of complex ones. For this case the authors show that the same bounds hold for irreducible varieties. In fact, for
\begin{align*}
&X=\text{Zeroset}(f), \quad f\in\mathbb{R}[x_1,\ldots,x_m]\text{ irreducible}\quad \deg(f)=d\\ \end{align*}
with the coefficients of \(f\) algebraically independent over \(\mathbb{Q}\) the following bounds hold
\begin{align*}
C_{+,\leq}(X)&=\binom{d+m}{m}-1,\\ C_{*,\leq}(X)&\geq \frac12 \biggl[ \binom{d+m}{m}-1\biggr],
\end{align*}
where we note that we allow inequality tests of the form \(y\geq 0\) in the decision trees and these tests are now considered for computing the additive and multiplicative branching complexities.
A final result shows that the irreducibility hypothesis is a necessary assumption since for the reducible zero-dimensional case, both additive and multiplicative complexities turn out to be logarithmic in \(t\) (the number of points). zerosets of polynomials; algebraic complexity; lower bounds for polynomial evaluation; additive and multiplicative branching complexities Bürgisser, P.; Lickteig, T.; Shub, M.: Test complexity of generic polynomials. J. complexity 8, 203-215 (1992) Analysis of algorithms and problem complexity, Computational aspects of algebraic surfaces, Real and complex fields Test complexity of 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 A brief survey of work of \textit{V. Voevodsky} (and the author) on the construction of motivic cohomology, and of recent work on Bloch's higher Chow groups is presented.
Fix a field \(F\) and write \(Sch/F\) and \(Sm/F\) for the categories of schemes (resp. smooth schemes) of finite type over \(F\). Besides the usual properties of a cohomology theory \({\mathcal H}\) on \(Sch/F\) one should have transfer homomorphisms \(Tr_{X/S} : {\mathcal H} (X) \to {\mathcal H} (S) \) for finite surjective morphisms \(X \to S\) with integral \(X\) and smooth and irreducible \(S\). The \(Tr_{X/S}\) should satisfy some obvious compatibility properties. Formalizing one may define presheaves with transfers on \(Sch/F\). For smooth schemes \(X\) and \(Y\) one has the notion of (finite) correspondences \(Cor (X,Y)\), and one defines the additive category \(SmCor/F\) with objects from \(Sm/F\) and morphisms \(\Hom_{SmCor/F} (X,Y) = Cor (X,Y)\). Via the graph of a morphism \(X \to Y\) in \(Sm/F\) one has a canonical functor \(Sm/F \to SmCor/F\). A presheaf with transfers (PWT) on \(Sm/F\) is a contravariant additive functor \({\mathcal H} : SmCor/F\to Ab\). For a Grothendieck topology \({\mathcal T}\), \({\mathcal H}\) is called a \({\mathcal T}\)-sheaf with transfers on \(Sm/F\) if the composed functor \(Sm/F \to SmCor/F @>{\mathcal H}>> Ab\) is a sheaf in the corresponding topology. Of particular importance is the Nisnevich topology. In particular, the category of Nisnevich sheaves with transfers \(Shv_{\text{Nis}} (SmCor/F)\) is abelian. The presheaf \(L(X)\) with \(L(X) (S) =\) the free abelian group generated by closed integral subschemes \(Z \subset X \times S\) that are finite and surjective over a component of \(S\), is a Nisnevich sheaf. For \(X\in Sm/F\) and \(K\) a complex (bounded from above) of Nisnevich sheaves with transfers one has, for any \(i \in \mathbb{Z}\), \(\Hom_{D^-(Shv_{\text{Nis}} (SmCor/F))} (L(X), K[i]) = H^i_{\text{Nis}} (X,K)\). For a PWT \({\mathcal F}\) on \(X\) one defines a new one \({\mathcal H}om (X, {\mathcal F})\) by \({\mathcal H}om (X, {\mathcal F}) (S) = {\mathcal F} (X \times S)\). If \({\mathcal F}\) is a \({\mathcal T}\)-sheaf, then so is \({\mathcal H}om (X, {\mathcal F})\). \({\mathcal F}\) is called homotopy invariant if \({\mathcal F} \simeq {\mathcal H}om (\mathbb{A}^1, {\mathcal F})\). \({\mathcal F}\) is called contractible if there are points \(0,1 \in \mathbb{A}^1\) with corresponding homomorphisms \(i_0, i_1 : {\mathcal H}om (\mathbb{A}^1, {\mathcal F}) \to {\mathcal F}\) such that there exists a presheaf homomorphism \(\varphi : {\mathcal F} \to {\mathcal H}om (X, {\mathcal F})\) with \(i_0 \varphi = 0\) and \(i_1 \varphi = id\). For such \({\mathcal F}\) and perfect field \(F\) one has comparison isomorphisms for the Zariski and Nisnevich cohomologies.
The triangulated category of motives over \(F\), \(DM (F)\), is defined as the full subcategory of \(D^- (Shv_{\text{Nis}} (SmCor/F))\) consisting of complexes with homotopy invariant cohomology sheaves. It can be shown that \(DM (F)\) is equivalent to the localization of \(D^- (Shv_{\text{Nis}} (SmCor/F))\) by the thick subcategory of complexes quasi-isomorphic to complexes of contractible sheaves. For \(X\) in \(Sch/F\) one defines its motive \(M (X)\) as the image of \(L(X)\) in \(DM (F)\). One gets a splitting \(M (\mathbb{P}^1) = M (\text{Spec} F) \oplus \widetilde M (\mathbb{P}^1)\), and one defines the Tate motive \(\mathbb{Z} (1)\) by \(\mathbb{Z} (1) = \widetilde M (\mathbb{P}^1) [-2]\). One defines, for any scheme \(X\), its motivic cohomology \(H^i_{\mathcal M} (X, \mathbb{Z} (n)) = \Hom_{DM (F)} (M(X), \mathbb{Z} (n) [i])\). For smooth \(X\) one gets \(H_{\mathcal M}^i (X, \mathbb{Z} (n)) = H^i_{\text{Nis}} (X, \mathbb{Z} (n)) = H^i_{\text{Zar}} (X, \mathbb{Z} (n))\), fitting in Beilinson's picture of motivic cohomology. For \(F\) a field of characteristic zero and \(X \in Sm/F\) equidimensional, Voevodsky showed that Bloch's higher Chow group \(CH^q (X,n) = H^{2q - n}_{\mathcal M} (X, \mathbb{Z} (q))\). Actually, Voevodsky proved more: For equidimensional \(X \in Sch/F\), the higher Chow groups of \(X\) coincide with motivic Borel-Moore homology. One may also look at the higher Chow groups with finite coefficients, \(CH^q (X,n; \mathbb{Z}/m)\). Then, for \(d\)-equidimensional quasi-projective \(X \in Sch/F\), \(F\) algebraically closed of zero characteristic, one has \(CH^q (X,n; \mathbb{Z}/m) = H_c^{2(d - q) + n} (X, \mathbb{Z}/m (d - q))^\#\), where \(^\#\) denotes the dual \(\Hom (-, \mathbb{Q}/ \mathbb{Z})\) and where \(\mathbb{Z}/m (i) = \mathbb{Z} (i) \otimes^L \mathbb{Z}/m\). For smooth \(X\) this simplifies to \(CH^q (X,n; \mathbb{Z}/m) = H^{2q - n}_{\text{ét}} (X, \mathbb{Z}/m (q))\). Using the Bloch-Lichtenbaum spectral sequence for \(X \in Sm/ \mathbb{C}\), i.e., \(E_2^{p,q} = CH^{-q} (\text{Spec} \mathbb{C} (X), - p - q\); \(\mathbb{Z}/m) \Rightarrow K_{- p - q} (\mathbb{C} (X), \mathbb{Z}/m)\), and a result of Thomason, one obtains a proof of the Quillen-Lichtenbaum conjecture for curves and surfaces, i.e., \(\dim X \leq 2\): The canonical homomorphism \(K_i (X, \mathbb{Z}/m) \to K_i^{\text{top}} (X(\mathbb{C}), \mathbb{Z}/m)\) is an isomorphism for \(i \geq \dim X\).
Other subjects discussed are Voevodsky's \(h\)-topologies on the category of schemes and eventual comparison isomorphisms, and sheaves of equidimensional cycles. proof of Quillen-Lichtenbaum conjecture for curves and surfaces; higher Chow groups; transfer homomorphisms; presheaf with transfers; Grothendieck topology; Nisnevich topology; triangulated category of motives; Tate motive; motivic cohomology; motivic Borel-Moore homology; sheaves of equidimensional cycles A. Suslin, Algebraic \(K\)-theory and Motivic Cohomology, Proc. International Congress of Mathematicians, Zürich 1994, vol. 1, Birkhäuser, 1995, pp. 342-351. \(K\)-theory in geometry, Generalizations (algebraic spaces, stacks), Étale and other Grothendieck topologies and (co)homologies, \(K\)-theory and homology; cyclic homology and cohomology Algebraic \(K\)-theory and motivic cohomology | 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 mainly deals with minimal algebraic surfaces of general types with \(K^2=2p_g-1\). We prove that for \(p_g\geq 7\) all these surfaces are birational to a double cover of some rational surfaces, and all but a finite classes of them have a unique fibration of genus 2; then we study their structures by determining their branch loci and singular fibres. We study similarly surfaces with \(p_g=5,6\). Lastly, we show that when \(p_g\geq 13\) all these surfaces are simply-connected. branch locus; minimal algebraic surfaces of general types; double cover; fibration; singular fibres Xianfang L.: Algebraic surfaces of general type with K 2 = 2p g 1 and p g 5. Acta Math. Sinica New Ser. 12(3), 234--243 (1996) Surfaces of general type, Families, moduli, classification: algebraic theory Algebraic surfaces of general type with \(K^ 2= 2p_ g- 1\), \(p_ g\geq 5\) | 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 \(P_1,\dots,P_n\) be \(n\) Laurent-polynomials on the \(n\)-dimensional algebraic torus \(({\mathbb C}^*)^n\), and assume that their associated Newton polyhedra \(\Delta_1,\dots,\Delta_n\) are in general position with respect to each other. Consider a rational \(n\)-form \(\omega\) on the torus that is regular on the complement of the hypersurface defined by the product \(P_1\cdots P_n\). In a previous paper [Dokl. Math. 54, No.~2, 700--702 (1996); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 350, No.~3, 298--300 (1996; Zbl 0898.32004)], \textit{O. A. Gelfond} and \textit{A. G. Khovanskii} announced an explicit formula for the sum of the Grothendieck residues of the form \(\omega\) at all the common zeros of the Laurent-polynomials \(P_1,\dots, P_n\) in terms of a sum over the vertices of the polyhedron arising as the Minkowski-sum of \(\Delta_1,\dots,\Delta_n\). The formula involves combinatorial coefficients of the vertices and certain residues of \(\omega\) at the vertices. In the paper under review this formula is proved using toric compactification and topological arguments. Grothendieck residues; Newton polyhedra; toric varieties Gelfond O.A., Khovanskii A.G.: Toric geometry and Grothendieck residues. Mosc. Math. J 2(1), 99--112 (2002) Toric varieties, Newton polyhedra, Okounkov bodies, Residues for several complex variables Toric geometry and Grothendieck residues. | 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 interesting paper, the Ringel-Hall algebra (and some variants of it) is studied for the category \(\text{coh}(X)\) of coherent sheaves over \(X\), where \(X\) is a smooth projective curve over a finite field \(\mathbb{F}_q\). More precisely, the complex valued functions on isomorphism classes of objects of \(\text{coh}(X)\) carry an algebra structure, which is analogous to the classical Hall algebra of a discrete valuation ring [see \textit{I. G. Macdonald}, Symmetric functions and Hall polynomials. 2nd ed. (Oxford, Clarendon Press) (1998; Zbl 0899.05068)] and to Ringel's Hall algebras for module categories over hereditary algebras. Among those functions are `automorphic forms', i.e. functions on vector bundles, and Eisenstein series. The main results of the paper give precise relations for certain generating functions which are quite similar to relations in quantum affine algebras.
An important (and motivating) example is that of the projective line \(\mathbb{P}^1\) over \(\mathbb{F}_q\). A special case of \textit{A. A. Beilinson}'s theorem [Funct. Anal. Appl. 12, 214-216 (1979); translation from Funkts. Anal. Prilozh. 12, No. 3, 68-69 (1978; Zbl 0424.14003)] asserts that \(\text{coh} (\mathbb{P}^1)\) is derived equivalent to the module category of a finite dimensional hereditary algebra (of Dynkin type \(A_1^{(1)})\), the Kronecker algebra. \textit{J. A. Green}'s theorem [Invent. Math. 120, 361-377 (1995; Zbl 0836.16021)] which extends Ringel's theorem [\textit{C. M. Ringel}, Invent. Math. (1990; Zbl 0735.16009)] gives an isomorphism between the composition algebra (a subalgebra of the Hall algebra) and the positive part \(U_q(\widehat {\mathfrak n}_+)\) of the quantum affine algebra associated with the same Dynkin diagram \(A_1^{(1)}\). Kapranov's `Ringel-Hall algebra' construction for \(\text{coh} (\mathbb{P}^1)\) produces a different `nilpotent' subalgebra of the same quantum affine algebra.
For general \(X\) as above, \(\text{coh} (X)\) still is a hereditary category and much of the technology developed by Ringel and Green for module categories of hereditary algebras still works.
The new algebras produced in this way relate quantum affine algebras and automorphic forms on \(X\) by many striking analogies. These analogies are collected in a table in section 5.3 (which also poses the open problem to fill in the remaining entries of the table by new analogies). For example, positive roots correspond to cusp eigenforms, and root space decomposition of \(U_q(\widehat{\mathfrak n}_+)\) corresponds to spectral decomposition of the algebra of automorphic forms.
To pass from the `positive part' \(U_q(\widehat{\mathfrak n}_+)\) to the full quantum affine algebra \(U_q (\widehat {\mathfrak g})\), one may apply the technique of forming the Drinfeld double. This has been done by \textit{Jie Xiao} [J. Algebra 190, 100-144 (1997; Zbl 0874.16026)]. The present paper complements this approach by adding the Heisenberg double as an intermediate step. Finally, the main result (theorem 6.7) states several identities between certain generating functions which hold true in the Drinfeld double of the Hall algebra of automorphic forms. These identities are very similar to relations valid in quantum affine algebras, in particular if one defines these algebras by Drinfeld's loop realization. curves over finite fields; Ringel-Hall algebra; coherent sheaves; quantum affine algebras; automorphic forms; Drinfeld double; Heisenberg double M. Kapranov, ''Eisenstein series and quantum affine algebras,'' J. Math. Sci. (New York), vol. 84, iss. 5, pp. 1311-1360, 1997. Relationship to Lie algebras and finite simple groups, Representations of quivers and partially ordered sets, , Quantum groups (quantized enveloping algebras) and related deformations, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20] Eisenstein series and quantum affine 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 In this paper, and in characteristic \(0\), the authors describe the smooth locus of the moduli space of linear series with prescribed vanishing sequences in at most two marked points. In the particular case of no marked points, this result specializes to the Gieseker-Petri theorem, proved previously in [\textit{D. Gieseker}, Invent. Math. 66, 251--275 (1982; Zbl 0522.14015)] and [\textit{ D. Eisenbud} and \textit{J. Harris}, Invent. Math. 74, 269--280 (1983; Zbl 0533.14012)], and recently in [\textit{S. Payne } and \textit{D. Jensen}, Algebra Number Theory 8, No. 9, 2043-2066 (2014; Zbl 1317.14139)].
The main idea of the proof is based on degeneration to a chain of elliptic curves and studying the corresponding the moduli space of limit linear series, introduced by Eisenbud and Harris. In parallel, the smoothness conditions of a point impose that the point must lie on the smooth locus of Schubert cycles and the smooth locus of Schubert varieties is characterized. degeneration; Grassmannian; Schubert varieties; almost-transverse flags; linear series; ramification Special divisors on curves (gonality, Brill-Noether theory), Grassmannians, Schubert varieties, flag manifolds The Gieseker-Petri theorem and imposed ramification | 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 present recent results about double reflection and incircular nets. The building blocks are pencils of quadrics, related billiards and quad graphs. double reflection; incircular nets; integrability; geometry Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry, Relations of finite-dimensional Hamiltonian and Lagrangian systems with topology, geometry and differential geometry (symplectic geometry, Poisson geometry, etc.), Relationships between algebraic curves and integrable systems Pencils of quadrics, billiard double-reflection and confocal incircular 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 What the authors call \textsl{The Main Theorem} of this enlightening paper can be conventionally split into two parts. The first is concerned with a certain symmetric structure of the \(n\)th exterior power of a polynomial ring; the second with a general determinantal formula evoking Giambelli's formula in classical Schubert Calculus on Grassmann Schemes or Jacobi-Trudy formula in the theory of symmetric polynomials. In spite of being strongly related with classical and widely investigated subjects, the result is new and shed further light on the beautiful algebraic properties of exterior powers of a module.
To describe the main theorem of the paper in a more precise way, here is a piece of notation. Let \(\bigotimes^n_AA[X]\) and \(\bigwedge^n_AA[X]\) be, respectively, the tensor and exterior \(n\)th-power of a polynomial algebra in one indeterminate with coefficients in \(A\), a commutative ring with unit. Also, let \(R:=A[X_1,\dots, X_n]\) be the ring of polynomials in \(n\) indeterminates. Because of the natural identification between \(\bigotimes^n_AA[X]\) and \(R\), the former is naturally an \(S:=R^{\text{sym}}\)-module, where \(R^{\text{sym}}\) is the \(A\)-algebra of the symmetric polynomials in \(R\). The first part of the main theorem then says that there exists a unique \(S\)-module structure on \(\bigwedge^n_AA[X]\) such that the canonical projection \(\bigotimes^nA[X]\rightarrow \bigwedge^nA[X]\) is \(S\)-linear. The \textsl{symmetric structure} of \(\bigwedge^n_AA[X]\) is described in a very explicit way: it turns out that \(\bigwedge^nA[X]\) is a free \(S\)-module of rank \(1\) generated by \(\phi:=X^{n-1}\wedge X^{n-2}\wedge\ldots\wedge X^0\) (\(X^0=1_A\)). As a consequence, if \(f_1,\dots, f_{n}\) are \(n\) arbitrary elements of \(A[X]\), the \(n\)-vector \(f_1(X)\wedge\ldots\wedge f_n(X)\) must be an \(S\) multiple of \(\phi\) and at this point the second part of the main theorem comes into play.
It shows that such a multiple can be computed by means of a very general and beautiful determinantal formula (that alluded to in the title) involving the coefficients of the polynomials \(f_i\)s only. We omit to write down the general determinantal formula, as it appear in the paper, which would require some additional explanations, but we mention a remarkable particular case: when \(f_i=X^{h_i+n-i}\) (\(1\leq i\leq n\)), one gets \( X^{h_1+n-1}\wedge X^{h_2+n-2}\wedge\ldots\wedge X^{h_{n}}=s_{h_{1},\ldots,h_n}\cdot\phi \) where, if \(s_h\) is the complete symmetric polynomial of degree \(h\), then \(s_{h_1,h_2,\ldots, h_n}\) is the usual Schur-polynomial \(\det(s_{h_i+j-1})\), which can be interpreted as the classical Giambelli's formula of Schubert calculus on Grassmann schemes. Hence Laksov and Thorup's determinantal formula can be seen as the ultimate and most natural generalization of it.
As one may expect from the nature itself of the results, the topic of this paper is related with many different subjects in mathematics, such as combinatorics, representation theory, geometry\dots To emphasize such a wide interplay, the authors care to prove the main theorem using different techniques within different frameworks. The most combinatorial in character is certainly that proposed in Section~2, based on a Pieri type formula enjoyed by the action of complete symmetric polynomials on the natural basis of \(\bigwedge^n_AA[X]\). That of Section~3, instead, relies on the isomorphism between \(\bigwedge^n_AA[X]\) and the ring of alternating polynomials. Section 4 proposes another proof based on symmetrization: let \(\xi\) be the residue class of \(T\) modulo \(P=\prod(T-X_i)\) in the ring \(S[\xi]=S[T]/(P)\). Then \(\bigwedge^nS[\xi]\) is naturally an \(S\)-module and remarkably such a module structure coincides with the symmetric structure defined in Section~1 of the paper, described in the first part of this review. Within the same framework, Section~5 proposes a very short proof of the main theorem which has the nice feature of implying Jacobi-Trudy formula. Finally, last two sections are devoted to look at the main theorem using the divided difference operators as well as the theory of universal splitting algebras, related with the work of Grothendieck on the homology of flag schemes. determinantal formula; Schubert calculus; exterior algebras; Giambelli's formula; Grassmann schemes; symmetric structures; symmetric functions; symmetrizing operators; divided difference operators; intersection theory; universal splitting algebras Laksov, D. and Thorup, A., A determinantal formula for the exterior powers of the polynomial ring, Indiana Univ. Math. J. 56 (2007), 825--845. Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations A determinantal formula for the exterior powers of the polynomial ring | 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 \(f_t : {\mathbb C}^n \to {\mathbb C}\), \(t \in [0,1]\), be a family of polynomial maps, \(f_t \in {\mathbb C}[t,x_1, \dots, x_n]\), \(n \geq 2\), having only isolated singularities and \(\mu(t)\) the (affine) Milnor number (the sum of the local Milnor numbers at critical points of \(f_t\)) and \(\lambda(t)\) the Milnor number at infinity (the sum of the local Milnor numbers at critical points of \(f_t\) at infinity).
Suppose that \(\mu(t)\), \(\lambda (t)\), the number of critical points and the number of critical points at infinity for fixed \(t\) do not depend on \(t\). Moreover, suppose that the critical values at infinity depend continuously on \(t\). Then the fibrations \(f_0^{-1}({\mathbb C} \smallsetminus B\bigl(0)\bigr) \to {\mathbb C} \smallsetminus B(0)\) and \(f_1^{-1}\bigl({\mathbb C} \smallsetminus B(1)\bigr) \to {\mathbb C} \smallsetminus B(1)\) are fibre homotopy equivalent and for \(n \not= 3\) differentiably isomorphic. Here \(B(0)\), resp.\ \(B(1)\), is the set of critical points of \(f_0\), resp.\ \(f_1\).
Under the additional assumption that \(\lambda(t) = 0\) and \(n \not= 3\), it is proved that \(f_0\) and \(f_1\) are topologically equivalent. In the case \(n = 2\) the same holds without requiring \(\lambda(t) = 0\) but \(\deg(f_t)\) not depending on \(t\). \(\mu\)-constant theorem; family of polynomials; singularities at infinity; Milnor number; Milnor number at infinity [3]A. Bodin, Invariance of Milnor numbers and topology of complex polynomials, Comment. Math. Helv. 78 (2003), 134--152. Equisingularity (topological and analytic), Singularities of curves, local rings Invariance of Milnor numbers and topology of complex 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 Let \(p\) be a prime number. Using the connection between bounding binomial exponential sums and bounding the number of solutions of certain equations over finite fields, in this work the authors obtain new bounds for \[S_{k,n}(a, b)=\sum_{x=0}^{p-1}e^{2\pi ix/p}(ax^k+bx^n),\] where \(k\) and \(n\) are positive integers, and \(a,b\) are arbitrary integer coefficients. This is achieved through a new method which provides sharper bounds for the number of solutions of the associated equations, and also for the degrees of their irreducible components. binomial exponential sums; rational points on curves; factors of polynomials Polynomials over finite fields, Exponential sums, Finite ground fields in algebraic geometry Binomial exponential sums | 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 a multilinear polynomial in several noncommuting variables with coefficients in an arbitrary field \(K\). Kaplansky conjectured that for any \(n\), the image of \(p\) evaluated on the set \(M_n(K)\) of \(n\) by \(n\) matrices is a vector space. In this paper, we settle the analogous conjecture for a quaternion algebra. noncommutative polynomials; Kaplansky conjecture; quaternion algebra Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.), Noncommutative algebraic geometry, Ordinary and skew polynomial rings and semigroup rings The images of noncommutative polynomials evaluated on the quaternion algebra | 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 extends Viro's method of glueing polynomials in order to keep singular or critical points in the process. The input for the glueing method is a subdivision \(\{\Delta_i\}\) of a nondegenerate Newton polyhedron and a compatible system of polynomials \(F_i\) with support on the \(\Delta_i\). A sufficient condition for the existence of a polynomial with the same singularities as the \(F_i\) is roughly that for each \(i\) the equisingular locus in the space of all polynomials is smooth and transversal to the space of polynomials with the given Newton diagram and coinciding with \(F_i\) for all monomials in \(\Delta_i\).
As example plane curves with the maximal number of cusps are constructed for degree eight (\(\kappa=15\)) and nine (\(\kappa=20\)). Another application is the asymptotically complete solution to the problem of possible collections of critical points of real polynomials in two variables without critical points at infinity. glueing polynomials; Newton polyhedron; equisingular locus; plane curves; maximal number of cusps; critical points E. Shustin, Gluing of singular and critical points. \textit{Topology}\textbf{37} (1998), 195-217. MR1480886 Zbl 0905.14008 Global theory and resolution of singularities (algebro-geometric aspects), Equisingularity (topological and analytic), Topological properties in algebraic geometry, Singularities in algebraic geometry, Singularities of curves, local rings Gluing of singular and critical 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 The authors study double points of plane curves either using their implicit equation or their parametrization.
Recall that a singularity oftype \(A_s\) for a plane curve is a double point that can be resolved via \(r\) blown-ups if \(s=2r-\varepsilon\), \(\varepsilon =0,1\) and the desingularization yields two points if \(\varepsilon=1\) and only one point if \(\varepsilon=0\).
The authors generalize result from their previous paper [\textit{A. Bernardi} et al., J. Symb. Comput. 86, 189--214 (2018; Zbl 1390.14183)] about type of singularities of points to points on any plane curve. Then there is presented an algorithm which classifies double points of any plane curve. This algorithm is based on studying the osculating curves to a curve at double point.
The paper also shows an example which illustrates how to build a plane rational curve with double points of chosen type using projection techniques.
This example gives also a counterexample to Lemma 4.2 in [loc. cit.] and this way the authors show that there was a mistake in that paper. plane curves; double points; curvilinear schemes Singularities of curves, local rings, Divisors, linear systems, invertible sheaves Remarks on double points 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 An important invariant of a finitely generated group \(\Gamma\) is its moduli space of Lie group-valued representations. From the case when the Lie group is \(\mathrm{GL}_n(\mathbb{C})\), these moduli spaces are called \textit{character varieties}. The study of character varieties relates to the theory of bundles of various types, locally homogeneous geometry, knot theory, and mathematical physics.
Here is how to define a character variety (over \(\mathbb{C}\)). Start with a complex reductive affine algebraic group \(G\) and a finitely generated (discrete) group \(\Gamma\). The group \(G\) acts by conjugation on the affine variety of homomorphisms \(\mathrm{Hom}(\Gamma, G)\). The \textit{\(G\)-character variety of \(\Gamma\)} is the affine (geometric invariant theoretic) quotient \(\mathfrak{X}_\Gamma(G):=\mathrm{Hom}(\Gamma, G)/\!\!/G\) by this action.
As \(\mathfrak{X}_\Gamma(G)\) is a complex affine variety, its cohomology admits a mixed Hodge structure (MHS). The MHS is encoded by the mixed Hodge polynomial \(\mu_{\Gamma, G}\), which generalizes the Poincaré polynomial.
The paper under review addresses the case when \(\Gamma\cong \mathbb{Z}^r\). Although for many cases of \(r\) and \(G\), the variety \(\mathfrak{X}_{\mathbb{Z}^r}(G)\) is irreducible and even normal, it is known to not always be irreducible, and is not known to always be normal. So the authors consider a normalization of a special component, denoted by \(\mathfrak{X}^*_{\mathbb{Z}^r}(G)\), that contains the identity representation. In general, \(\mathfrak{X}^*_{\mathbb{Z}^r}(G)\cong T^r/W\) where \(T\) is a maximal torus in \(G\) and \(W\) is the Weyl group of \(G\). The first main theorem in this well-written and interesting paper is an explicit computation of the mixed Hodge polynomial of \(\mathfrak{X}^*_{\mathbb{Z}^r}(G)\).
Specializing the mixed Hodge polynomial in a certain way one obtains a polynomial, called the \(E\)-polynomial. The \(E\)-polynomial, with respect to compactly supported cohomology, is related to arithmetic geometry. Let \(\mathbb{F}_q\) be a field of order \(q\). If the variety in question admits a model over \(\mathbb{Z}\) and the function that counts the number of \(\mathbb{F}_q\)-points is a polynomial in \(q\), then the \(E\)-polynomial and the counting function coincide. The second main theorem in this paper computes the (compactly supported) \(E\)-polynomial of \(\mathfrak{X}_{\mathbb{Z}^r}(\mathrm{SL}(n,\mathbb{C}))\) explicitly, in terms of partitions.
Some of the main results in this paper have recently been generalized by the reviewer and the authors of this paper [``Mixed Hodge structures on character varieties of nilpotent groups'', Preprint, \url{arXiv:2110.07060}]). In particular, we compute the MHS of the identity component of \(G\)-character varieties when \(\Gamma\) is nilpotent (which includes the abelian case). We also show there is only one irreducible component containing the identity representation (in the abelian case) and moreover that the normalization step used in the paper under review is unnecessary.
Finally, the reviewer notes that the paper under review, in addition to being mathematically interesting, can serve as a very good introduction to MHSs on character varieties. Overall, the paper is a fine piece of scholarship! free abelian group; character variety; mixed Hodge structures; Hodge-Deligne polynomials; equivariant E-polynomials; finite quotients Character varieties, Mixed Hodge theory of singular varieties (complex-analytic aspects), Representations of finite symmetric groups, Group actions on varieties or schemes (quotients) Hodge-Deligne polynomials of character varieties of free abelian groups | 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 \(p \geq 5\) be a prime number. We generalize the results of \textit{E. de Shalit} [Contemp. Math. 165, 135--148 (1994; Zbl 0863.14015)] about supersingular \(j\)-invariants in characteristic \(p\). We consider supersingular elliptic curves with a basis of 2-torsion over \(\overline{\mathbb{F}}_p\), or equivalently supersingular Legendre \(\lambda\)-invariants. Let \(F_p(X, Y) \in \mathbb{Z} [X, Y]\)be the \(p\)-th modular polynomial for \(\lambda\)-invariants. A simple generalization of Kronecker's classical congruence shows that \(R(X) : = \frac{F_p(X, X^p)}{p}\)is in \(\mathbb{Z} [X]\). We give a formula for \(R(\lambda)\) if \(\lambda\) is supersingular. This formula is related to the Manin-Drinfeld pairing used in the \(p\)-adic uniformization of the modular curve \(X(\Gamma_0(p) \cap \Gamma(2))\). This pairing was computed explicitly modulo principal units in a previous work of both authors. Furthermore, if \(\lambda\) is supersingular and is in \(\mathbb{F}_p\), then we also express \(R(\lambda)\)in terms of a CM lift (which is shown to exist) of the Legendre elliptic curve associated to \(\lambda\). rigid analytic geometry; Mumford uniformization; semi-stable curves; modular curves; modular polynomials; supersingular elliptic curves Modular and Shimura varieties, Rigid analytic geometry, Other analytic theory (analogues of beta and gamma functions, \(p\)-adic integration, etc.), Elliptic curves, Complex multiplication and abelian varieties, Algebraic moduli of abelian varieties, classification Congruence formulas for Legendre modular 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 [For the entire collection see Zbl 0723.00046.]
We examine the extended Riccati flow on a Grassmannian near the topological closure of the stable manifold of any invariant locus of dimension at least two. Such a closure is an example of a singular (in the sense of algebraic geometry) Schubert variety. The flow exhibits sensitive dependence on initial conditions near the singularities of this variety. The spectrum of Lyapunov exponents is derived from the spectrum of the infinitesimal generator of the flow. matrix Riccati differential equation; Riccati flow; Schubert variety Dynamics induced by flows and semiflows, Manifolds of solutions, Ergodic theorems, spectral theory, Markov operators, Grassmannians, Schubert varieties, flag manifolds, Structural stability and analogous concepts of solutions to ordinary differential equations The Riccati flow near the `edge' | 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 \(D\) be a central division algebra over a field \(F\). The author studies the rigidity of the motivic decompositions of the Severi-Brauer varieties of \(D\), with respect to the ring of coefficients and to the base field \(F\), where \(D\) is a central division algebra over \(F\). He first shows that if the ring of coefficients is a field, these decompositions only depend on its characteristic. In the second part, the author shows that if \(D\) remains division over a field extension \(E/F\), the motivic decompositions of several Severi-Brauer varieties of \(D\) remain the same when extending the scalars to \(E\). upper motives; Grothendieck motives; central simple algebras; Severi-Brauer varieties De Clercq, C., \textit{motivic rigidity of Severi-Brauer varieties}, J. Algebra, 373, 30-38, (2013) (Equivariant) Chow groups and rings; motives, Skew fields, division rings Motivic rigidity of Severi-Brauer 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 These lectures give a detailed and almost self-contained introduction to algebraic stacks. A great part of the paper is devoted to preliminary technical topics, both from category theory (like Grothendieck topologies, fibred categories and stacks) and algebraic geometry (like faithfully flat descent). All this machinery is finally used to present the definition and some basic properties first of algebraic spaces and then of algebraic stacks. Grothendieck topologies; fibred categories; stacks; faithfully flat descent; algebraic spaces; algebraic stacks Generalizations (algebraic spaces, stacks), Local structure of morphisms in algebraic geometry: étale, flat, etc., Stacks and moduli problems, Fibered categories, Grothendieck topologies and Grothendieck topoi, Research exposition (monographs, survey articles) pertaining to algebraic geometry Lectures on algebraic stacks | 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 Author's abstract. We describe a number of geometric contexts where categorification appears naturally:coherent shaves, constructible shaves and shaves of modules over quatizations. In each case, we discuss how ``index formulas'' allow us to easily perform categorical calculations, and readily relate classical constructions of geometric representations theory to categorical ones.
From the introduction: This paper is structurated around 3 different geometric contexts which naturally lead to categorification:
In Section 2 (\(K\)-theory), we consider categories of coherent sheaves and algebraic \(K\)-theory. This is arguably the firs place in the literature where the modern philosophy of categorification appears, and the most likely to be somewhat familiar to the general reader.
In Section 3 (The function -sheaf correspondence), we consider categories of constructible sheaves and the function-sheaf correspondence. While perhaps a more specialized taste, this is actually an incredibly powerful theory, with connections to deep number theory. In this author's opinion, it is one any aspiring categorifier should know a bit of.
In Section 4 (Symplectic resolutions), we consider categories of sheaves of modules over quantizations. This is the least familiar context, and one still under development. Unlike to other two examples, we have not had the benefit of having Grothendieck around to help us with it. However, progress on it has been made, which we will briefly discuss here. categorification; \(K\)-theory; Grothendieck group; coherent shves; Functoriality; Chern charcater and index formula; Weyl character formula; Euler characteristic; étale cohomology; Grothendieck trace formula; Grassmannians and \(\mathfrak{sl}_2\); flag varieties; Hall algebras Webster, Ben, Knot invariants and higher representation theory, Memoirs of the Amer. Math. Soc., 250, v+141 pp., (2017) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], \(K\)-theory of schemes, Étale and other Grothendieck topologies and (co)homologies, Characteristic classes and numbers in differential topology, Definitions and generalizations in theory of categories Geometry and categorification | 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 cite from the text: This paper completes part of \textit{P. Dèbes} and \textit{M. Fried}, Arithmetic variation of fibers in families: Hurwitz monodromy criteria for rational points, J. Reine Angew. Math. 409, 106-137 (1990; Zbl 0699.14033). It is a tools paper, illustrating with one main (and several support) examples how to analyze a certain style of question. When a diophantine or geometric problem about curves defines possible exceptional cases, how do you find if the exceptions do exist? In practice the production of illustrative curves occurs with extra data. That data usually includes a reference variable defining the curve as a cover. That is why, in practice, Hurwitz-style families of covers are natural. Our illustrations especially discuss tools for treating the value sets of genus 0 covers. This will show elementary use of Hurwitz families and braid group actions as does [\textit{P. Dèbes}, Györy, Kálman (ed.) et al. Number theory in progress. Proc. Int. Conf. 1977, Volume I: Diophantine problems and polynomials. Berlin, de Gruyter, 75-102 (1999; Zbl 0941.14007), \textit{M. D. Fried}, Sémin. Théor. Nombres, Paris/Fr. 1987-88, Prog. Math. 81, 77-117 (1990; Zbl 0729.12009); Finite Fields Appl. 1, 326-359 (1995; Zbl 0869.11093); Contemp. Math. 186, 15-32 (1995; Zbl 0833.12002); \textit{G. Malle} and \textit{B. H. Matzat}, Inverse Galois theory, Springer, Berlin (1999; Zbl 0940.12001) and \textit{H. Völklein}, Groups as Galois groups, Cambridge Univ. Press (1996; Zbl 0868.12003)]. [\textit{J.-P. Serre}, Topics in Galois theory, Boston, Bartlett and Jones Publishers (1992; Zbl 0746.12001)] has simpler examples not requiring Hurwitz families and braid groups.
This paper culminates an era in which the monodromy method has proved itself innumerable times on many problems of renown. It is time to say these (as delineated here) are the elementary tools. The program modular towers of \textit{M. D. Fried} [Contemp. Math. 186, 111-171 (1995; Zbl 0957.11047)] has outlined the next stage of problems. It requires a set of tools (like modular representation theory and higher monodromy) that divides it from the tools here. Still, there are many problems around which require neat analysis for which the tools here suffice.
(Authors' summary) Suppose \(C\) is an algebraic curve, \(f\) is a rational function on \(C\) defined over \(\mathbb{Q}\), and \({\mathcal A}\) is a fractional ideal of \(\mathbb{Q}\). If \(f\) is not equivalent to a polynomial, then Siegel's theorem gives a necessary condition for the set \(C(\mathbb{Q}) \cap f^{-1} ({\mathcal A})\) to be infinite: \(C\) is of genus 0 and the fiber \(f^{-1} (\infty)\) consists of two conjugate quadratic real points. We consider a converse. Let \({\mathcal P}\) be a parameter space for a smooth family \(\Phi:{\mathcal T} \to{\mathcal P} \times \mathbb{P}^1\) of (degree \(n)\) genus 0 curves over \(\mathbb{Q}\). That is, the fiber \({\mathcal T}_p\) of points of \({\mathcal T}\) over \({\mathbf p}\times \mathbb{P}^1\) has genus 0 for \({\mathbf p}\in {\mathcal P}\). Assume a Zariski dense set of \({\mathbf p} \in{\mathcal P} (\mathbb{Q})\) have fiber \(\Phi_{\mathbf p}^{-1} ({\mathbf p}\times \infty)\) over \(\infty\) consisting of two conjugate quadratic real points. The family \(\Phi\) is then a Siegel family. We ask when the conclusion of Siegel's theorem -- \(\Phi_{\mathbf p} (\mathbb{Q})\cap {\mathcal A}\) is infinite -- holds for a Zariski dense subset of \({\mathbf p}\in {\mathcal P}(\mathbb{Q})\).
We show how braid action on covers and Hurwitz spaces can tackle this. It refines a unirationality criterion for Hurwitz spaces. A particular family, \(_{10}\Phi'\), of degree 10 rational functions, illustrates this. It arises as the exceptional case for a general result on Hilbert's irreducibility theorem. \textit{M. D. Fried} [Applications of the classification of simple groups to monodromy. II: Davenport and Hilbert-Siegel problems, preprint, 1-55 (1986)] says the only indecomposable polynomials \(f(y)\in \mathbb{Q}[y]\) with \(f(y)-t\) reducible in \(\mathbb{Q}[y]\) for infinitely many \(t\in {\mathcal A}\setminus f(\mathbb{Q})\) have degree 5. We show the family \(_{10}\Phi'\) satisfies the converse to Siegel's theorem. Thus, exceptional polynomials of degree 5 in (loc. cit.) do exist.
We suspect this result generalizes, thus codifying arithmetic accidents occurring in \(_{10}{\mathcal P}'\). To illustrate, we've cast this paper as a collection of elementary group theory tools for extracting from a family of covers special cases with specific arithmetic properties. Examples of Siegel and Néron families show the efficieney of the tools, though each case leaves a diophantine mystery. Siegel's theorem; Siegel family; braid action on covers; Hurwitz spaces; unirationality criterion; rational functions; Hilbert's irreducibility theorem; exceptional polynomials P. Dèbes and M. Fried,Integral specialization of families of rational functions, Pacific Journal of Mathematics190 (1999), 45--85. Hilbertian fields; Hilbert's irreducibility theorem, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory) Integral specialization of families of rational 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 complex quasi-projective variety and let \(X^{(n)}\) denote its symmetric product. Let \(T_{y*}:K_0(var/X)\rightarrow H_{ev}^{BM}(X)\oplus\mathbb{Q}[y]\) be the un-normalized motivic Hirzebruch class transformation, where \(H_{ev}^{BM}(X)\) denotes the Borel-Moore homology in even degrees and \(K_0(var/X)\) the relative Grothendieck group of complex algebraic varieties over \(X\) (see [\textit{J.-P. Brasselet} et al., J. Topol. Anal. 2, No. 1, 1--55 (2010; Zbl 1190.14009)]). For any positive integer \(r\), let \(d^r:X\rightarrow X^{(r)}\) denote the composition of the diagonal embedding \(X\hookrightarrow X^r\) with the projection \(X^r\rightarrow X^{(r)}\), and let \(\Psi_r\) denote the \(r\)-th homological Adams operation, which on \(H_{2k}^{BM}(X^{(r)},\mathbb{Q})\) is defined by multiplication by \(1/r^k\), together with \(y\mapsto y^r\). The main result of this paper is the folloing generating series formula for the Hirzebruch classes of the symmetric powers \(\mathcal{M}^{(n)}\in D^b\text{MHM}(X^{(n)})\) of a fixed complex of mixed Hodge modules on the variety \(X\): \(\sum_{n\geq 0}T_{(-y)*}(\mathcal{M}^{(n)})\cdot t^n=\text{exp}\left(\sum_{r\geq 1}\Psi_r(d_*^rT_{(-y)*}(\mathcal{M}))\cdot\frac{t^r}{r}\right)\). If one lets \(\mathcal{M}\) be the constant Hodge sheaf \(\mathbb{Q}_X^H\), the above formula becomes the expression \(\sum_{n\geq 0}T_{(-y)*}(X^{(n)})\cdot t^n=\text{exp}\left(\sum_{r\geq 1}\Psi_r(d_*^rT_{(-y)*}(X))\cdot\frac{t^r}{r}\right)\) for the motivic Hirzebruch classes. symmetric product; characteristic classes; Grothendieck group Cappell , S. E. Maxim , L. Schüurmann , J. Shaneson , J. L. Yokura , S. Characteristic classes of symmetric products of complex quasi-projective varieties Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Algebraic cycles Characteristic classes of symmetric products of complex quasi-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 A classical result of Borel states that the integer cohomology ring of a full flag variety \(Fl(n)=GL_n/B\) is isomorphic to \({\mathbb Z}[x_1,\dots,x_n]/I\), where \(I\) is the ideal generated by all symmetric polynomials without constant term. The generators \(x_1,\dots,x_n\) correspond to the first Chern classes of the line bundles \({\mathcal V}_i/{\mathcal V}_{i-1}\), where \({\mathcal V}_\bullet\) is the tautological vector bundle over \(Fl(n)\). On the other hand, this ring also has a natural additive basis consisting of the \textit{Schubert classes}, formed by the flags satisfying certain incidence conditions on their intersection with a given flag. A classical problem of Schubert calculus is to express these classes via the generators \(x_1,\dots,x_n\). This was done by Bernstein-Gelfand-Gelfand and Demazure. There are natural generalizations of this setting to full flag varieties for other classical groups, \(SO(n)\) and \(Sp(n)\), due to Harris-Tu, Fulton and the others.
In this paper the author considers a similar problem for one of the five exceptional groups, namely, for \(G_2\). Geometrically, the full flag variety \(X\) of \(G_2\) is formed by the pairs of nested subspaces of dimensions 1 and 2 in a seven-dimensional vector space \(V\), that are isotropic with respect to a generic alternating trilinear form on \(V\). This variety also admits a Schubert decomposition. As in the classical case, one can consider two quotient line bundles associated with the tautological vector bundle on \(X\); the first Chern classes of these bundles generate \(H^*X\). The author expresses the classes of the Schubert varieties, corresponding to the 12 elements of the Weyl group of type \(G_2\), in terms of these generators. Some (apparently new) explicit geometric descriptions of the full flag variety of type \(G_2\) are also provided. degeneracy locus; equivariant cohomology; flag variety; Schubert variety; Schubert polynomial; exceptional Lie group; octonions Anderson, D., Chern class formulas for \(G_2\) Schubert loci, Trans. Amer. Math. Soc., 363, 12, 6615-6646, (2011) Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Exceptional groups, Symmetric functions and generalizations Chern class formulas for \(G_{2}\) Schubert loci | 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 show that a generic real projective \(n\)-dimensional hypersurface of odd degree \(d\), such that \(4(n-2)=\binom{d+3}{3}\), contains ``many'' real 3-planes, namely, in the logarithmic scale their number has the same rate of growth, \(d^3\log d\), as the number of complex 3-planes. This estimate is based on the interpretation of a suitable signed count of the 3-planes as the Euler number of an appropriate bundle. real algebraic geometry; enumerative geometry; real Schubert calculus Classical problems, Schubert calculus, Topology of real algebraic varieties, Hypersurfaces and algebraic geometry Abundance of 3-planes on real projective 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 Let \(C\) be a smooth integral algebraic curve of genus \(g \geq 2\), \(C^{(2)}\) the second symmetric product of \(C\), and \(\chi\) and \(\delta\) the fibre and the diagonal numerical classes, respectively, in the Néron-Severi space \(N^1(C^{(2)})_{\mathbb{R}}\). In this paper the author investigates the restriction of the nef cone \(\text{Nef}(C^{(2)})\) on the plane spanned by \(\chi\) and \(\delta\). Let \(\tau(C) := \inf \{ \mu \geq 0 \mid (\mu + 1)\chi - (\delta/2) \text{ is nef on } C^{(2)} \}\). Among many other results, the author proves the following.
{Theorem.} Let \(k\) be a nonnegative integer. If \(C\) is a smooth integral curve of genus \(g > \max \{2 k + 1, 4 k - 3\}\) then \(\tau(C) \geq g - k\) if and only if there exists a smooth integral curve \(H\) of genus \(q \leq k/2\) such that \(C\) is a double covering of \(H\). Furthermore, in this case the curve \(H\) is unique up to isomophisms adn \(\tau(C) = g - 2q\). double coverings; nef cone; symmetric product of curves Chan, K.: A characterization of double covers of curves in terms of the ample cone of second symmetric product. J. Pure Appl. Algebra \textbf{212}(12):2623-2632 (2008) Special divisors on curves (gonality, Brill-Noether theory), Coverings of curves, fundamental group A characterization of double covers of curves in terms of the ample cone of second symmetric product | 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 \(\mathbb Sch_k\) be the category of separated schemes of finite type over a field \(k\), and \(\mathbb Fat_k\) be its full subcategory of schemes connected and finite over \(k\), that is, of fat points over \(k\). As usual, \(\mathbb L=[\mathbb A^1_k]\) is the Lefschetz motif in a Grothendieck ring.
Generalizing of the classical Grothendieck ring \(\mathrm{Gr}(\mathbb Var)_k\) over an algebraically closed field \(k\), \textit{H. Schoutens} proposed the theory of schemic Grothendieck rings [Schematic Grothendieck rings. Volume I and II. \url{websup-port1.citytechcuny.edu/faculty/hschoutens/PDF/SchemicGrothendieckRingPartI.pdf}]. A sieve \(\mathcal X:\mathbb Sch_k\to\mathbb Sets\) is called subschemic if there exists morphism of schemes \(\varphi:X\to Y\), such that \(\mathcal X=Im\phi^o\), where \(\phi^o\) is the induced morphism of sieves. \(\mathcal X\) is called formal sieve, if for any \(m\in\mathbb Fat_k\) there is a subschemic sieve \(\mathcal Y_m\subset\mathcal X\), so that \(\mathcal Y_m(m)=\mathcal X(m)\). All formal sieves form a subcategory \(\mathbb Form_k\), and as shown in Schoutens [loc. cit.], it is a Grothendieck pre-topology. Moreover, the Grothendieck ring of the formal site \(\mathrm{Gr}(\mathbb Form)_k\) could be defined, and for any \(m\in\mathbb Fat_k\) there is a surjective homomorphism \(\sigma_m:\mathrm{Gr}(\mathbb Form)_k\to\mathrm{Gr}(\mathbb Var)_k\). For any fat point, let \(S_m:=\{s\mathbb L^i:\sigma_m(s)=1,i\mathbb N\}\), it is stable under multiplication, and take the localization of \(\mathrm{Gr}(\mathbb Form)_k\) by it. As \(\sigma_m\) induces \(\sigma'_m:S_m^{-1}\mathrm{Gr}(\mathbb Form)_k\to\mathrm{Gr}(\mathbb Var)_k[\mathbb L^{-1}]\), and there exists a filtration of \(S_m^{-1}\mathrm{Gr}(\mathbb Form)_k\) by subgroups \(\mathcal F_i:=\{T:\dim(\sigma'_m(T))<i\}\), we can define \(S^{-1}_m\widehat{\mathrm{Gr}(\mathbb F}orm)_k\) to be the completion with respect to it.
As shown by Schoutens ([loc. cit.]), the truncated arc space could be generalized when \(\mathrm{Spec }k[t]/(t^n)\) is replaced by a fat point \(m\), defining the generalized arc space of a sieve \(\mathcal X\) along m to be the functor \(\nabla_m\mathcal X(-)=\mathcal X(-\times_km):\mathbb Fat_k\to\mathbb Sets\). If \(X\in\mathbb Sch_k\) and \(n\in\mathbb N\), let the \(n\)-jet of \(X\) at \(p\in X\) be \(J_p^nX:=\mathrm{Spec}(\mathcal O_{X,p}/m^n_p)\in\mathbb Sch_{k(p)}\), where \(p\) has residue field \(k(p)\), and let the auto-arc space of \(X\) at \(p\) of order \(n\) be \(\mathcal A_n(X,p):=\nabla_{\mathcal J_p^n(X)}\mathcal J_p^n(X)\). As proved then, if \(X\) is smooth at \(p\), there is a canon- ical isomorphism between \(\mathcal A_n(X,p)\) and the auto-arc space of \(\mathcal A_n(\mathbb A^d_{k(p)},q)\) at some \(k(p)\)-rational point \(q\), for \(d=\dim_pX\). About the converse, the author proposes the following conjecture about how much the auto-arc spaces could measure the smoothness of \(X\) at a point. It claims that if \(X\in\mathbb Var_k\), \(p\in X\), and if for all \(n\gg 0\) it holds that \(\mathcal A_n(X, p)^{\mathrm{red}}\simeq\mathbb A^r_{k(p)}\) for some \(r\in\mathbb N\), then \(X\) is smooth at \(p\).
Next is introduced the auto Igusa-Zeta series associated to \(X\) at \(p\), \(\zeta_{X,p}^{\mathrm{auto}}(t):=\sum_{n=0}^\infty[\mathcal A_{n+1}(X,p)]\mathbb L^{d.l(J_p^{n+1}X)}t^n\) as the generating series for the sequence of \(\mathcal A_n(X,p)\), where for \(m\in\mathbb Fat_k\), \(l(m):=\dim\mathcal O_m(m)\), and its reduced version \( \bar{\zeta}_{X,p}(t)\in\mathrm{Gr}(\mathbb Var)_k[\mathbb L^{-1}]\). From the the isomorphism above, supposed that \(X\) is smooth at \(p\) and \(d=\dim_pX\), one has \(\bar{\zeta}_{X,p}(t)=\mathbb L^{-d}\frac{1}{1-t}= \bar{\zeta}_{\mathbb A^d_k,p}(t)\). Thus, for \(f:X\to Y\) étale at \(p\), one has \(\bar{\zeta}_{X,p}(t) = \bar{\zeta}_{Y,f (p)}(t)\), and, as before a conjecture is proposed about the converse, that is, if \(\bar{\zeta}_{X,p}=\bar{\zeta}_{Y,q}\) at some point \(q\), then \((X,p)\) is analytically isomorphic to \((Y,q)\).
Then this conjecture is investigated, and the spaces \(\mathcal A_n(C,p)^{\mathrm{red}}\) are calculated in the case of cuspidal cubic or node \(C\). Another conjecture is proposed giving a formula for \(\mathcal A_n(C,p)^{\mathrm{red}}\) in the case of a reduced curve \(C\) with one singularity. The calculations are using the Sages script, proposed by the author [Sage mathematics software. Sage development team. \url{http://www.sagemath.org}] for computing the arc space \(\nabla_mX\) for any affine scheme \(X\) and fat point \(m\), in \(\mathrm{char }k=0\). The script is provided in the last section of the article.
After showing, for \(X\in\mathbb Sch_k\) and any \(n\in N\), that there are natural morphisms \(\mathcal A_n(X, p)^{\mathrm{red}}\to\mathcal A_{n-1}(X,p)^{\mathrm{red}}\), and giving characterizations when \(Y_n\) is smooth over \(J_p^nX\), in the final sections are discussed two ways to introduce the motivic volume of an infinite auto-arc space. For a closed \(Y\subset X\) defined by ideal \(I_Y\), let \(Y_n\subset X\) be defined by \(I_Y^n\), \(n\in\mathbb N\), and put \(Y_0:=(Y_n)^{\mathrm{red}}\). Assuming that \(Y_n\in\mathbb Sch_{k(p)}\) is affine of pure dimension \(d\), and there is a smooth morphism \(Y_n\to J_p^nX\) for some \(p\), the infinite auto-arc space of \(Y_n\) along \((X,p)\) is defined as \(\mathcal A=\mathcal A_{X,p}(Y_0):=\varprojlim(\nabla_{J(n)}Y_n)^{\mathrm{red}}\).
On this space is defined a natural motivic volume at level \(n\), \(\nu_{X,p}^{\mathrm{auto}}(\mathcal A,n):=[\nabla_{J(n)}Y_n^{\mathrm{red}}]\mathbb L^{-d_n}\), and the motivic integral along the length function, \(\int_{\mathcal A}\mathbb L^{-l}d\nu^{\mathrm{auto}}\). Then is proved that for \(Y_n\) as above the integral equals \([Y_0]\mathbb L^{-\dim(Y_0)}\bar{\zeta}_{X,p}(\mathbb L^{-1})\). In a second approach resembling more the classical motivic integration are defined the adjusted motivic volume of \(\mathcal A\), and just as before, the corresponding motivic integral of the length function. Then the auto Poincaré series is introduced, with its rationality discussed using some results from [\textit{J. Denef} and \textit{F. Loeser}, Invent. Math. 135, No. 1, 201--232 (1999; Zbl 0928.14004)].
In the last section is provided the author's Sage code [loc. cit.], which computes the arc space of an affine scheme with respect to a fat point, in the case when \(\mathrm{char }k=0\). sieve; fat point; Grothendieck pre-topology; auto-arc space; auto Igusa-zeta function; motivic integral Arcs and motivic integration, Computational aspects of algebraic curves, Symbolic computation and algebraic computation, Grothendieck topologies and Grothendieck topoi On the auto Igusa-zeta function of an algebraic curve | 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 investigates flat families over discrete valuation rings describing the degeneration of projective toric varieties into reducible schemes with toric components. The combinatorial counterpart is given by a certain kind of non-compact polyhedra; the irreducible components of the special fibers correspond to the compact faces of the corresponding polytope. The whole construction is similar to perturbing the equation of a reducible toric divisor.
As an application, the author obtains a formula for the number of solutions with prescribed order of a sufficiently general system of Laurent polynomials. This result generalizes the well-known formula of Kouchnirenko. discrete valuation rings; degeneration of projective toric varieties; non-compact polyhedra; toric divisor; number of solutions of Laurent polynomials A. L. Smirnov, Torus schemes over a discrete valuation ring, Algebra i Analiz 8 (1996), no. 4, 161 -- 172 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 8 (1997), no. 4, 651 -- 659. Toric varieties, Newton polyhedra, Okounkov bodies, Formal power series rings, Families, fibrations in algebraic geometry, Valuation rings Torus schemes over a discrete valuation ring | 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 concerns the Bruhat-Renner decomposition of reductive monoids and is a survey on the recent developments on the associated Bruhat-Chevalley orders, Stanley-Reisner rings and Hecke algebras. The author also introduces the concept of a triangular Hecke algebra. Quite a few relevant examples are presented. The author is a main and original contributor to the development. In the paper, he also poses two conjectures about the associated Hecke algebras. The interested reader is also advised to read \textit{L. E. Renner}'s complementary paper in these Proceedings [ibid. 125-143 (2004; Zbl 1061.20059)]. Bruhat-Chevalley orders; Bruhat-Renner decompositions; Cohen-Macaulay rings; conjugacy classes; finite groups of Lie type; finite monoids of Lie type; Gorenstein rings; Hecke algebras; Kazhdan-Lusztig polynomials; linear algebraic monoids; reductive groups; reductive monoids; Putcha lattices of cross-sections; Renner monoids; representation theory; shellability; Stanley-Reisner rings; Weyl groups Semigroups of transformations, relations, partitions, etc., Hecke algebras and their representations, Linear algebraic groups over arbitrary fields, Combinatorics of partially ordered sets, Group actions on varieties or schemes (quotients), Representation of semigroups; actions of semigroups on sets, Representations of finite groups of Lie type Bruhat-Renner decomposition and Hecke algebras of reductive monoids. | 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 gives a long series of remarks about \(K\)-algebras, where \(K\) is an (usually) algebraically closed field of characteristic zero. These remarks relate to the weak nullstellensatz (\(V(I)\) is nonempty for every proper ideal \(I\) of \(K[X_1, \dots, X_n]\)) and use several problems from \textit{W. Fulton}'s 1969 book on algebraic curves [Algebraic curves. New York-Amsterdam: W.A. Benjamin, Inc. (1969; Zbl 0181.23901)] and elementary notions from functional analysis. zeros of polynomials; spectrum; Gelfand-Mazur Theorem; ideals; characters Polynomials over commutative rings, Extension theory of commutative rings, Relevant commutative algebra, General theory of commutative topological algebras, Ideals, maximal ideals, boundaries, Structure and classification of commutative topological algebras Some remarks on Hilbert's (weak) nullstellensatz | 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 expanded version of the article can be found in the booklet of \textit{L. Gatto} [``Schubert calculus: an algebraic introduction''. Publicações Matemáticas do IMPA (2005; Zbl 1082.14054)]. We cite from the review of that book:\dots he presents a new point of view on Schubert calculus on a Grassmann manifold. In this interpretation the Chow ring of the Grassmann manifold of \(k\)-dimensional subspaces on an \(n\)-space is the \(k\)th exterior product of an \(n\)-dimensional vector space \(M\) with a fixed base \(e_1,\dots,e_n\), and the Chern classes are presented as certain-differential operators on the exterior product. These operators the author calls Schubert derivations, and are obtained from the operator \(D_1 :M\to M\) given by \(D_1(e_i) = e_{i+1}\) for \(i = 1,\dots,n-1\) and \(D_1(e_n) = 0\).
The treatment gives an easy and natural approach to Schubert calculus, that is well adopted to computations. In particular it gives a satisfactory explanation of the determinants appearing in Giambelli's formula.
It is refreshing and surprising that it is possible to take a new perspective on this classical part of geometry, particularly taken into account the massive amount of work in the field, coming from many different parts of mathematics like geometry, algebra and combinatorics. Chow ring; Schubert derivation; Schubert varieties; Pieri's formula; Giambelli's formula Gatto, L.: Schubert calculus via Hasse-Schmidt derivations. Asian J. Math. 3, 315-322 (2005) Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds Schubert calculus via Hasse-Schmidt 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 subject of this paper are Calabi-Yau manifolds which are obtained by resolving the singularities of double coverings of \(\mathbb P^3\) branched along an octic surface. These surfaces are allowed to be reducible and to have isolated and non-isolated singularities of certain specified types, more general than those considered in the paper of \textit{S. Cynk} and \textit{T. Szemberg} [in: Singularities Symposium-Łojasiewicz 70, Cracow 1996, and Seminar Singularities and Geometry, Warsaw 1996, Banach Cent. Publ. 44, 93-101 (1998; Zbl 0915.14025)].
The main result of this paper states (similar to the paper of Cynk and Szemberg) the existence of a resolution of such a double solid which is Calabi--Yau. The topological Euler number is computed for such manifolds and a table is presented, which shows the possible values. Using the methods and results of the paper of Cynk and Szemberg mentioned above and by allowing isolated singularities on the branch octic, the author is able to fill most of the gaps in the table of possible Euler numbers which was presented in his joint paper with T. Szemberg (loc. cit.). octic arrangements; Calabi-Yau resolution; topological Euler number; small resolution of double solid; isolated singularities S. Cynk, Double coverings of octic arrangements with isolated singularities , Adv. Theor. Math. Phys. 3 (1999), 217-225. Calabi-Yau manifolds (algebro-geometric aspects), Coverings in algebraic geometry, Topological properties in algebraic geometry, Singularities of surfaces or higher-dimensional varieties Double coverings of octic arrangements with isolated singularities | 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 Bordiga surface S is a projective plane birationally embedded into \(P^ 4\) by quartics through given 10 points, so that its degree equals 6. The present paper studies when such surface is a set-theoretical complete intersection of a cubic and a quartic. This relates to the existence of a double structure on S, and the latter may relate to a vector bundle of rank 2 on \(P^ 4\) (which splits in the present case). The author proves that such a double structure exists if and only if S contains a certain curve C of degree 4 and is contained in the secant variety of C. Here C is either a normal rational curve or a union of two conics which intersect at one point. The 10 points on \(P^ 2\) to be blown up are also determined in these cases. Bordiga surface; set-theoretical complete intersection; double structure Rathmann, J.: Double structures on bordiga surfaces. Comm. algebra 17, 2363-2391 (1989) Rational and ruled surfaces, Complete intersections Double structures on Bordiga 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 From authors' abstract: If \(\Sigma\) is a smooth genus two curve, \(\Sigma\subset\text{Pic}^1(\Sigma)\) the Abel embedding in the degree one Picard variety, \(|2\Sigma|\) the projective space parametrizing divisors on \(\text{Pic}^1(\Sigma)\) linearly equivalent to \(2\Sigma\), and \(\text{Pic}^0(\Sigma)_2=G\cong(\mathbb Z/2\mathbb Z)^4\) the subgroup of points of order two in the Jacobian variety \(J(\Sigma)=\text{Pic}^0(\Sigma)\), then \(G\) acts on \(|2\Sigma|\) and the quotient variety \(|2\Sigma|/G\) parametrizes two fundamental moduli spaces associated with the curve \(\Sigma\). Namely, Narasimhan-Ramanan's work implies an isomorphism of \(|2\Sigma|/G\) with the space \(\mathcal M\) of (\(S\)-equivalence classes of semistable, even) \(\mathbb P^1\) bundles over \(\Sigma\), and Verra has defined a precise birational correspondence between \(|2\Sigma|/G\) and Beauville's compactification of \(\mathcal P^{-1}(J(\Sigma))\), the fiber of the classical Prym map over \(J(\Sigma)\).
In this paper, we give a new (birational) construction of the composed Narasimhan-Ramanan-Verra map \(\alpha:\mathcal M\dasharrow\mathcal P^{-1}(J(\Sigma))\), defined purely in terms of the geometry of a (generic stable) \(\mathbb P^1\) bundle \(X\to\Sigma\) in \(\mathcal M\), and also an explicit rational inverse map \(\beta:\mathcal P^{-1}(J(\Sigma))\dasharrow\mathcal M\). The map \(\alpha\) may be viewed as an analog for Prym varieties of Andreotti's reconstruction of a curve \(C\) of genus \(g\) from the branch locus of the canonical map on the symmetric product \(C^{(g-1)}\). Prym varieties; étale double cover; Prym theta divisor; Narasimhan-Ramanan-Verra map Smith, No article title, Pac. J. Math., 188, 353, (1999) Jacobians, Prym varieties On the geometry of two dimensional Prym 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 This paper was written and narrowly circulated back in 2008. It studies gauge theories with gauge groups being products of unitary groups \(U(r)\times U(r')\) on five dimensional spaces \(X\times S^1\), where \(X\) is a complex surface. The moduli space of \(U(r)\) instantons is partly compactified to the space \(\mathcal{M}(r)\) of torsion free rank \(r\) sheaves on a compactification of \(X\), and the matter in the bifundamental representation is described by a natural sheaf formed by \(\mathrm{Ext}\) groups on \(\mathcal{M}(r)\times\mathcal{M}(r')\). It defines a Fourier-Moukai operator between \(K\)--groups \(\Phi_{\mathrm{Ext}}\!\!: K_{G\times T}(\mathcal{M}(r))\to K_{G\times T}(\mathcal{M}(r'))\), where \(G\) is the group of constant gauge transformations and \(T\) is a maximal torus of \(\mathrm{GL}(2,\mathbb{C})\).
For \(r=r'=1\) the correct \(G\times T\) equivariant cohomology was computed by Carlsson and Okounkov, this paper generalizes the computation to \(K\)--theory. It is in the spirit of BPS/CFT correspondence, which reduces BPS protected observables to correlation functions in a \(q\)--deformed two dimensional CFT. The operator \(\Phi_{\mathrm{Ext}}\) is computed through the Macdonald-Mehta-Cherednik identity, which expresses the values of of the Hermitian form \(\langle f,\Theta g\rangle_{\Delta}\) in the basis of the Macdonald polynomials. Here \(\langle\cdot,\cdot\rangle_{\Delta}\) is the Macdonald's Hermitian inner product for the root system of \(\mathrm{GL}(N)\), and \(\Theta\) is the theta function of its weight lattice. As an application the authors compute the partition functions for \(A_r\)--type quiver \(U(1)^{r+1}\) theories by reducing instanton sums to the trace of a product of vertex operators.
As a side issue it is observed that the Wiener-Hopf factorization of \(\Theta\) produces an operator that takes the orthogonal Macdonald polynomials to the interpolation Macdonald polynomials. For general lattices the support of the factorization is a half-space of the lattice, but for \(\mathrm{GL}(N)\) it is much smaller, the positive orthant of \(\mathbb{Z}^N\). This explains why the theory of interpolation Macdonald polynomials is much richer for \(\mathrm{GL}(N)\) than say for \(\mathrm{SL}(N)\). instantons; BPS/CFT correspondence; Macdonald-Mehta-Cherednik identity; interpolation Macdonald polynomials; Fourier-Moukai operator E. Carlsson, N. Nekrasov and A. Okounkov, \textit{Five dimensional gauge theories and vertex operators}, arXiv:1308.2465 [INSPIRE]. Mirror symmetry (algebro-geometric aspects), Relationships between surfaces, higher-dimensional varieties, and physics, Quantum field theory on lattices, Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.), Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) Five dimensional gauge theories and vertex operators | 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 The main results of the paper is the following classification theorem:
{Theorem 1}. Suppose that \(X\) is a smooth affine 3-fold over an algebraically closed field \(k\) having characteristic unequal to \(2.\) The map assigning to a rank \(2\) vector bundle \(\mathcal E\) on \(X\) the pair \((c_{1}(\mathcal E), c_{2}(\mathcal E))\) of Chern classes gives a pointed bijection
\[
{\mathcal V}_{2} \rightarrow {\text{ Pic}}(X) \times {\text{CH}}^2(X).
\]
As a consequence the authors derive the cancellation law for vector bundles of rank \(2\) on \(X\) i.e. if \({\mathcal E}_{1}, {\mathcal E}_{2} \in {\mathcal V}_{2}(X)\) satisfy \({\mathcal E}_{1}\oplus {\mathcal E} \cong {\mathcal E}_{2}\oplus {\mathcal E},\) then \({\mathcal E}_{1}\cong {\mathcal E}_{2}.\) The techniques for the proof include the methods of obstruction theory for describing the set of \({\mathbb A}^{1}\)-homotopy classes of maps to the corresponding Grassmannian. For the result it is enough to describe the first ``nonstable'' \(\mathbb A^{1}\)-homotopy sheaf of the symplectic group. Then the authors use the F. Morel's \(\mathbb A^{1}\)-homotopy classification of vector bundles to reduce the result to cohomology vanishing statements. Finally they prove the required vanishing statements. vector bundles; \({\mathbb A}^{1}\)-homotopy; Grothendieck-Witt groups; obstruction theory A. Asok J. Fasel A cohomological classification of vector bundles on smooth affine threefolds http://arxiv.org/abs/1204.0770 Motivic cohomology; motivic homotopy theory, Obstruction theory in algebraic topology, Higher symbols, Milnor \(K\)-theory, Projective and free modules and ideals in commutative rings, Stability for projective modules A cohomological classification of vector bundles on smooth affine 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 Let \(G\) be a connected reductive group over an algebraically closed field \(k\) of characteristic not 2; let \(\theta\in\Aut (C)\) be an involution and \(K=G^\theta\subseteq G\) the fixed point group of \(\theta\) and let \(P\subseteq G\) be a parabolic subgroup. The set \(K \setminus G/P\) of \((K,P)\)-double cosets in \(G\) plays an important role in the study of Harish Chandra modules. \textit{M. Brion} and the author [Can. J. Math. 52, 265--292 (2000; Zbl 0972.14039)] gave a description of the orbits of symmetric subgroups in a flag variety \(G/P\) mainly using geometric arguments. For general \(P\), it is difficult to describe the combinatorics of the decomposition of the closure of a double coset in terms of \(K\times P\) double cosets. However, in some special cases one can describe the combinatorics of the closures of the double cosets in more detail. This paper discusses the special case that \(P\) contains a \(\theta\)-stable Levi factor \(L\) and the set of roots of the connected center \(S\) of \(L\) is a root system with Weyl group \(W(S)=N_G (S)/Z_G(S)\). Here \(N_G(S)\) (resp. \(Z_G(S))\) is the normalizer (resp. centralizer) of \(S\) in \(G\). In this case the combinatorics of the Weyl Group can be used to describe the closures of the double cosets of a part of the double coset space which includes the open and closed orbits and we get a number of results similar to the case that \(P=B\) a Borel subgroup. This root system condition on \(P\) is satisfied in many cases. For example in the case that \(P\) is a minimal parabolic \(k_0\)-subgroup of \(G\) or a minimal \(\theta\)-split parabolic subgroup of \(G\) or a minimal \((\theta,k_0)\)-split parabolic \(k_0\)-subgroup of \(G\). Here \(k_0\subseteq k\) is a subfield of \(k\) and \(G,\theta\) are defined over \(k_0\). double cosets; Weyl group; root system; parabolic subgroup Helminck, A.: Combinatorics related to orbit closures of symmetric subgroups in flag varieties. CRM proc. Lecture notes 35, 71-90 (2004) Homogeneous spaces and generalizations, Grassmannians, Schubert varieties, flag manifolds, Representation theory for linear algebraic groups Combinatorics related to orbit closures of symmetric subgroups in 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 Ternary real-valued quartics in \(\mathbb{R}^{3}\) that are invariant under octahedral symmetry are considered. The geometric classification of these surfaces is given. A new type of surface emerges from this classification. ternary quartic surfaces; octahedral group; chambers; homogeneous polynomials Special surfaces, Real algebraic sets, Computational aspects of algebraic surfaces Geometric classification of real ternary octahedral quartics | 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 Schubert cycle of a Grassmannian \(G(r,n)\). Here the author gives a combinatorial proof of Hodge's postulation formula giving the Hilbert function of \(X\) with respect to the Plücker embedding of \(G(r,n)\). The main point of the paper is to introduce algebraists to some combinatorial techniques which seem to be important in this area. Hilbert polynomial; Schubert cycle; Grassmannian; Hilbert function Ghorpade, S. R.: A note on Hodge's postulation formula for Schubert varieties. Geometric and combinatorial aspects of commutative algebra (Messina, 1999), 211-220 (2001) Grassmannians, Schubert varieties, flag manifolds, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series A note on Hodge's postulation formula for 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 A hypersurface is factorial if every Weil divisor on \(X\) is a Cartier divisor. Let \(X\) be a normal hypersurface in \({\mathbb{P}}^4\) of degree \(d\geq 3\) that has at most isolated singular points. C. Ciliberto and V. Di Gennaro conjectured that a hypersurface \(X\) is factorial if \(X\) has at most isolated ordinary double points,
\[
|{\text{Sing}}(X)|\leq 2(d-1)(d-2),
\]
and \(X\) contains neither planes nor quadric surfaces. In this paper, with an independent geometric method, the author proves the following result announced by Youngho Woo: \(X\) is factorial if
\[
|{\text{Sing}}(X)|< (d-1)^2,
\]
\(X\) has at most isolated ordinary double points and \(X\) contains no planes. hypersurfaces; ordinary double points; factorial property I. A. Cheltsov, On a conjecture of Ciliberto , Sb. Math. 201 (2010), no. 7-8, 1069-1090. \(3\)-folds, Hypersurfaces and algebraic geometry On a conjecture of Ciliberto | 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 we construct free resolutions of a certain class of closed subvarieties of affine space of symmetric matrices (of a given size). Our class covers the symmetric determinantal varieties (i.e., determinantal varieties in the space of symmetric matrices), whose resolutions were first constructed by \textit{T. Jozefiak} et al. [Astérisque 87--88, 109--189 (1981; Zbl 0488.14012)]. Our approach follows the techniques developed by \textit{M. Kummini} et al. [Pac. J. Math. 279, No. 1--2, 299--328 (2015; Zbl 1342.14103)], and uses the geometry of Schubert varieties. Schubert varieties; Lagrangian Grassmannian; free resolutions Grassmannians, Schubert varieties, flag manifolds, Syzygies, resolutions, complexes and commutative rings, Singularities of surfaces or higher-dimensional varieties, Linear algebraic groups over the reals, the complexes, the quaternions, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Global theory and resolution of singularities (algebro-geometric aspects) Free resolutions of some Schubert singularities in the Lagrangian 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 Toric degenerations are a particularly useful tool for describing algebraic properties of varieties in terms of combinatorics of polytopes and polyhedral fans. The goal of the paper under review is to construct a family of toric degenerations for Richardson varieties inside the Grassmannian. To do this, the authors consider a family of matching fields, which were originally introduced by Sturmfels and Zelevinsky for studying certain Newton polytopes. They associate a weight vector to each block diagonal matching field and characterise when the corresponding initial ideal is toric, thus providing a family of toric degenerations for Richardson varieties.
Given a Richardson variety \(X_{w}^v\) and a weight vector \(\mathbf{w}_\ell\) arising from a matching field, they consider two ideals: an ideal \(G_{k,n,l}|_{w}^{v}\) obtained by restricting the initial of the Plücker ideal to a smaller polynomial ring, and a toric ideal defined as the kernel of a monomial map \(\phi_{l}|_{w}^{v}\). First they characterise the monomial-free ideals of form \(G_{k,n,l}|_{w}^{v}\). Next they construct a family of tableaux in bijection with semi-standard Young tableaux which leads to a monomial basis for the corresponding quotient ring. Finally, they prove that when \(G_{k,n,l}|_{w}^{v}\) is monomial-free and the initial ideal \(\mathrm{in}_{\mathbf{w}_l}(I(X_{w}^v))\) is generated by degree two polynomials, then the ideals \(\mathrm{in}_{\mathbf{w}_l}(I(X_{w}^v))\), \(G_{k,n,l}|_{w}^{v}\) and \(\ker(\phi_{l}|_{w}^{v})\) are all equal, and provide a toric degeneration for the Richardson variety \(X_{w}^v\) . Gröbner and toric degenerations; Grassmannians; semi-standard Young tableaux; Schubert varieties; Richardson varieties; standard monomial theory Grassmannians, Schubert varieties, flag manifolds, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Fibrations, degenerations in algebraic geometry Standard monomial theory and toric degenerations of Richardson 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 recently proved that the arc-analytic type of a singular Brieskorn polynomial determines its exponents. This last result may be seen as a real analogue of a theorem by Yoshinaga and Suzuki concerning the topological type of complex Brieskorn polynomials. In the real setting it is natural to investigate further by asking how the signs of the coefficients of a Brieskorn polynomial change its arc-analytic type. The aim of the present paper is to answer this question by giving a complete classification of Brieskorn polynomials up to the arc-analytic equivalence. The proof relies on an invariant of this relation whose construction is similar to the one of Denef-Loeser motivic zeta functions. The classification obtained generalizes the one of Koike-Parusiński in the two variable case up to the blow-analytic equivalence and the one of Fichou in the three variable case up to the blow-Nash equivalence. singular Nash function germs; arc-analytic equivalence; Brieskorn polynomials; motivic zeta functions; virtual Poincaré polynomial Arcs and motivic integration, Singularities in algebraic geometry, Nash functions and manifolds, Topology of real algebraic varieties, Equisingularity (topological and analytic) Complete classification of Brieskorn polynomials up to the arc-analytic equivalence | 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 any algebraic scheme over a field, and let \(\alpha\in K^0(X)\). Then \(\alpha\) has a well-defined rank \(\text{rk} \alpha\), and Chern classes \(c_k(\alpha)\). We consider \(\alpha\otimes [{\mathcal L}]\), where \({\mathcal L}\) is an arbitrary line bundle on \(X\).
Theorem. \(c_{\text{rk} \alpha+1} (\alpha)= c_{\text{rk} \alpha+1} (\alpha\otimes [{\mathcal L}])\). Grothendieck group; tensor product of vector bundles Aluffi, Paolo; Faber, Carel, A remark on the Chern class of a tensor product, Manuscripta Math., 0025-2611, 88, 1, 85-86, (1995) Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Homology of classifying spaces and characteristic classes in algebraic topology, Applications of methods of algebraic \(K\)-theory in algebraic geometry A remark on the Chern class of a tensor product | 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 \(f: (\mathbb C^n,0) \rightarrow (\mathbb C,0)\) be a germ of a holomorphic function with an isolated critical point at the origin and let \(G\) be a finite abelian group acting faithfully on \((\mathbb C^n,0)\) and preserving \(f\). The authors define the quantum cohomology group \(\mathcal H_{f,G}\) of the pair \((f,G)\) similarly to the paper [\textit{H. Fan} et al., Ann. Math. (2) 178, No. 1, 1--106 (2013; Zbl 1310.32032)], where this notion was introduced in the framework of quantum singularity theory (FJRW-theory) for quasihomogeneous function germs. The paper under review aims to transfer or adapt the basic notions and results of the usual singularity theory to FJRW-theory. Using the theory of group rings, the authors define consistently orbifold versions of the monodromy operator on \(\mathcal H_{f,G}\), the Milnor fibre and vanishing cohomology group (the Milnor lattice in \(\mathcal H_{f,G}\)), the Seifert form and the intersection form, and so on. In conclusion, they discuss some examples with invertible polynomials related to the orbifold Landau-Ginzburg models and the Berglund-Hübsh-Henningson duality (see, e.g., [\textit{P. Berglund} and \textit{M. Henningson}, Nucl. Phys., B 433, No. 2, 311--332 (1995; Zbl 0899.58068)]), the behavior of the Milnor lattice under the corresponding mirror symmetry, etc. quantum singularity theory; FJRW-theory; quantum cohomology group; Milnor fibre; Seifert form; invertible polynomials; Coxeter-Dynkin diagrams; Landau-Ginzburg models; Berglund-Hübsh-Henningson duality; Milnor lattice; mirror symmetry Group actions on affine varieties, Topology and geometry of orbifolds, Local complex singularities, Topological invariants on manifolds, Symmetries, equivariance on manifolds Orbifold Milnor lattice and orbifold intersection form | 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 There are \(3264\) conics tangent to five given conics in the projective plane \(\mathbb P^2\) and there exist five explicit conics so that all \(3264\) complex solutions are real [\textit{W. Bruns} et al., Adv. Math. 244, 171--206 (2013; Zbl 1295.13010); \textit{F. Ronga} et al., Rev. Mat. Univ. Complutense Madr. 10, No. 2, 391--421 (1997; Zbl 0921.14036)]. In this paper the authors study such tangency questions in one dimension higher. They consider quadrics (i.e. quadratic surfaces) in \(\mathbb P^3\). \textit{H. Schubert} [Kalkül der abzählenden Geometrie. (Mit e. Vorw. von Steven L. Kleiman u. e. Bibliogr. d. Arbeiten von Hermann Schubert, d. von Werner Burau zsgest. wurde). Reprint. Heidelberg, New York: Springer-Verlag (1979; Zbl 0417.51008)] found that there are 666841088 quadrics tangent to nine given quadrics in \(\mathbb P^3\). Their goal is to decide whether there exist nine real quadrics so that all complex solutions are real. In this article they present first steps towards answering that question.
Schubert's calculus predicts the number of complex solutions to a system of polynomial equations that depend on geometric figures like lines and planes in \(\mathbb P^3\). In this article they study these polynomial equations and present practical tools for solving them. The main interest is in solutions over the real numbers \(\mathbb R\). They discuss the associated polynomial systems and they state two conjectures about their reality. quadratic surfaces; polynomial equations; numerical methods; Schubert's problems; Schubert calculus; cohomology ring of flag varieties; Schubert triangle Classical problems, Schubert calculus, Enumerative problems (combinatorial problems) in algebraic geometry, Iterative numerical methods for linear systems, Combinatorial aspects of commutative algebra Tangent quadrics in real 3-space | 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 automorphism group of a \(K3\) surface is always a discrete finitely generated group by \textit{H. Sterk}'s theorem [Math. Z. 189, 507--513 (1985; Zbl 0545.14032)] (and work of Lieblich-Maulik in the case of positive characteristic). Yet it is usually extremely hard to compute, compare, for instance, the prototype calculations by \textit{E. B. Vinberg} for the two ``most algebraic'' complex \(K3\) surfaces [Math. Ann. 265, 1--21 (1983; Zbl 0537.14025)].
The paper under review pioneers the computation of the automorphism group of the supersingular \(K3\) surface \(X\) of Artin invariant \(1\) in characteristic \(3\), i.e. it has maximal Picard number \(\rho(X)=22\) and discriminant \(\mathrm{disc NS}(X)=-9\). For instance, \(X\) can be given as the Fermat quartic surface (with 112 lines defined over \(\mathbb F_9\)). Note that the given model already has a huge, yet finite projective automorphism group \(\mathrm{PGU}_4(\mathbb F_9)\) of size 13,063,680.
The main result of the present paper is that the above finite group can be supplemented by two involutions to generate the full automorphism group of \(X\). The involutions arise as deck transformations for suitable double sextic models of \(X\), building on previous work by the second author. The techniques are based on Conway theory (in particular chamber decompositions) and Borcherd' method (as employed, for instance, by the first author for the generic Jacobian Kummer surface). Presently, however, the Néron-Severi lattice is not reflective which causes the calculations to become much more involved; in particular, they heavily depend on machine assistance. It is emphasized that the methods can be adapted to many other \(K3\) surfaces. supersingular \(K3\) surface; automorphism group; Fermat quartic; double sextic; Borcherds' method; Conway theory; chamber decomposition Kondō, S.; Shimada, I., The automorphism group of a supersingular K3 surface with Artin invariant 1 in characteristic 3, Int. Math. Res. Not. IMRN., 2014, 1885-1924, (2014) \(K3\) surfaces and Enriques surfaces, Automorphisms of surfaces and higher-dimensional varieties The automorphism group of a supersingular \(K3\) surface with Artin invariant 1 in characteristic 3 | 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 variety over a finite field \(\mathbb{F}\) and let \(\text{Sym}^n(X)\) be the \(n\)th symmetric power of \(X\). A famous result by \textit{B. Dwork} [Am. J. Math. 82, 631--648 (1960; Zbl 0173.48501)] states that the zeta-function
\[
Z_X(t)= \sum_i |\text{Sym}^n(X)| t^n
\]
is a rational function on \(t\). If \(k\) is any field one can consider the Grothendieck ring \(A(k)= K_0({\mathcal V}_k)\) for varieties over \(k\), i. e., the ring of \(\mathbb{Z}\)-combinations of isomorphism classes of \(k\)-varieties modulo the relations
\[
[X]=[Y]+ [X-Y]
\]
for closed varieties \(Y\subset X\). Then the motivic zeta function is defined, as a power series in the ring \(A(k)\), by
\[
\zeta_X(t)= \sum_i [\text{Sym}^n(X)] t^n.\tag{1}
\]
By a result of \textit{M. Kapranov} [``The elliptic curve in the S-duality theory and Eisenstein series for Kac-Moody groups'', \url{arXiv:math/0001005}], if \(X\) is a curve then the function in (1) is rational, while the same statement is false for higher-dimensional varieties.
The aim of this paper is to prove the rationality of motivic zeta functions for curves with a finite abelian group action.
Let \(G\) be a fixed algebraic group and let \({\mathcal V}^G_k\) be the category of varieties with \(G\)-actions. An object in \({\mathcal V}^G_k\) is a pair \((V,\sigma)\), with \(X\in{\mathcal V}_k\) and \(\sigma: G\times X\to X\) an algebraic group action of \(G\) on \(X\). The Grothendieck group \(K_0({\mathcal V}_k)\) is the free abelian group on isomorphism classes in \({\mathcal V}^G_k\)1 modulo the relations
\[
[X,\sigma]= [Y,\tau]+ [X- Y,\sigma],
\]
where \((Y,\tau)\) is a closed \(G\)-invariant subspace of \((X, \sigma)\). Multiplication in \(K_0({\mathcal V}^G_k)\) is defined by
\[
[X,\sigma] [Y,\tau]= [X\times Y,\,\sigma\times\tau],
\]
where \(\sigma\times\tau: G\times X\times Y\to X\times Y\) is defined by \((\sigma\times\tau)(g,x,y)= (\sigma(g,x), \tau(g,y))\). Let \(A(k,G)\) be the Grothendieck ring obtained in such a way. Then the author proves the following:
Theorem 1. Let \(G\) be finite abelian group of order \(m\) and let \(C\) be a non-singular projective curve over an algebraically closed field \(k\) of characteristic \(p\), with \(p\) not dividing \(m\). Let \(\sigma: G\times C\times C\) be a group action on \(C\). Then the motivic zeta function
\[
\zeta_{(C,\sigma)}= \sum_i [\text{Sym}^n(C,\sigma)] t^n
\]
is rational, as a power series in the ring \(A(k,G)\). Grothendieck ring for varieties with group actions; motivic zeta-functions; \(K\)-theory; Picard bundle; equivariant bundles; Weil conjectures; invariant theory Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Motivic cohomology; motivic homotopy theory, \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture Rationality of motivic zeta functions for curves with finite abelian group actions | 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 proves existence theorems for higher order singularities of a finite morphism to \({\mathbb{P}}^ m\) and deduces a result on simple connectivity of varieties admitting a finite morphism of bounded singularity. - The singularities are obtained by successive degeneration of double points. Our main tool is R. Schwarzenberger's notion of generalized secant sheaves and the connectedness theorem by W. Fulton and the author. morphism to projective space; higher order singularities of a finite morphism; simple connectivity of varieties; successive degeneration of double points Singularities in algebraic geometry, Rational and birational maps, Ramification problems in algebraic geometry, Topological properties in algebraic geometry Higher order singularities of morphisms to projective space | 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 semisimple algebraic group over an algebraically closed field \(k\) of characteristic zero. Let \({\mathcal B}\) be the variety of Borel subgroups. Consider the projective embedding \({\mathcal B}\to \mathbb{P}(H^0 ({\mathcal B}, L_\rho)^*)\), where \(L_\rho\) is the line bundle associated to the Steinberg weight \(\rho\), which is the half sum of positive roots. In this paper we show that any positive dimensional projective space contained in the image of \({\mathcal B}\) is necessarily of dimension one.
Furthermore we determine exactly these lines. Indeed we show that if \(\ell\subset {\mathcal B}\) is such a line, then there exists a minimal parabolic subgroup \(P\subset G\) such that \(\ell\) is the set of Borel subgroups which are contained in \(P\). In particular this implies that the possible homology classes of such lines correspond under the usual identification of the root lattice with \(H_2({\mathcal B}, \mathbb{Z})\) to the set of simple roots.
The result is obtained as an application of some properties of the intersection of Schubert cycles in the cohomology ring of \({\mathcal B}\). flag varieties; Borel subgroups; Schubert cycles Grassmannians, Schubert varieties, flag manifolds, Projective techniques in algebraic geometry Projective spaces in 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 This fascinating and important paper develops model-theoretically a theory of motivic integration over valued fields of residue characteristic 0. The authors work with the geometry of algebraically closed valued fields, and to some extent with `V-minimal' expansions of such fields, such as the rigid analytic expansions of Lipshitz and Robinson. The work has applications for other Henselian fields.
Let \(L\) be a valued field with valuation ring \({\mathcal O}\), value group \(\Gamma\), maximal ideal \({\mathcal M}\), and residue field \(k\). Define \(\text{RV}:= L^*/(1+{\mathcal M})\). There is an exact sequence
\[
0\rightarrow k^*\rightarrow\text{RV}\rightarrow\Gamma\rightarrow0,
\]
so RV combines the structure of the value group and residue field.
For any complete theory \(T\), there is a naturally defined Grothendieck semiring of definable sets, considered up to definable bijections (in the paper and this review, `definable' means `definable without parameters'). The semiring is denoted \(K_+(T)\), or \(K_+(M)\), where \(M\models T\). Mostly the authors work with semigroups and semirings rather than groups and rings, to avoid collapse. The authors work over a base field \(L_0\), but in the model-theoretic setting of algebraically closed valued fields, with VF denoting the field sort. There is a natural map \({\mathbb L}\) taking definable sets in RV to definable sets in VF, which induces a surjective homomorphism of filtered semirings \(K_+\text{RES}[*]\otimes K_+\Gamma[*]\rightarrow K_+\text{VF}[*]\); the \(*\) indicates the filtration, which is via a notion of dimension, and RES denotes a residue field structure, basically the residue field equipped with a family of vector spaces on it, with the induced structure of `generalised algebraic varieties'. The kernel of \({\mathbb L}\) is identified as a certain congruence \(I_{\text{sp}}\). The inverse map is regarded as an Euler characteristic. The content of surjectivity is that, up to definable bijections, definable sets in VF come from definable sets in the value group and the residue field.
There are notions of `definable set with volume form' in the VF and RV categories, and corresponding Grothendieck rings, denoted \(K_+\mu\text{VF}\) and \(K_+\mu\text{RV}\), and \({\mathbb L}\) induces an isomorphism of semirings \(K_+\mu\text{RV}[*]\rightarrow K_+\mu\text{VF}[*]\), again with a precisely described kernel. Here \(K_+\mu\text{RV}[*]\) is a tensor product of corresponding semirings for \(\Gamma\) and the residue field structure RES. The inverse isomorphism \(K_+\mu\text{VF}[*]\rightarrow K_+\mu\text{RV}[*]/I_{\text{sp}}^{\mu}\) is viewed as a motivic integral.
The initial theory is developed in the setting of C-minimality, a slight specialisation of the theory developed in [\textit{D. Haskell} and \textit{D. Macpherson}, Ann. Pure Appl. Logic 66, No. 2, 113--162 (1994; Zbl 0790.03039)]. Essentially, a C-minimal structure is a set with a nested sequence of equivalence relations, indexed by a definable dense linear ordering, such that any 1-variable definable set is a Boolean combination of equivalence classes (`balls'). Algebraically closed valued fields, with the equivalence relation \(E_{\gamma}\) (\(\gamma\in\Gamma\)) defined by \(E_{\gamma} xy\Leftrightarrow v(x-y)\geq\gamma\) and with other equivalence relations corresponding to open balls, provide the motivating example. The authors develop the geometry of C-minimality (for example the structure of stable definable sets, non-interaction between open and closed balls). They work with a further specialisation, V-minimality, for which one requires that any definable chain of balls has non-empty intersection, plus other conditions. Under this hypothesis they prove piecewise continuity and differentiability results for definable functions (part of the content of Sections 3--5). The map \({\mathbb L}\) lifts definable sets from RV to VF, and there is a corresponding lift of definable functions, respected by \({\mathbb L}\), described in Section 6. Via a theory of `RV-blow-ups', the kernel \(I_{\text{sp}}\) is identified in Section 7. The main theorems about \({\mathbb L}\) are proved in Section 8, with the differentiability theory mediating in the volume form case.
The paper includes an analysis of Grothendieck semirings and rings for divisible ordered abelian groups, in work overlapping with that of J. Maříková. It is shown that for \(\text{DOAG}_A\) (the theory of divisible ordered abelian groups with constants for a fixed submodel \(A\)) the Grothendieck ring is \({\mathbb Z}^2\). In particular, this gives two independent Euler characteristics on \(\text{DOAG}_A\).
A theory of integration of definable functions into certain Grothendieck rings is developed in Section 11. Here the authors work with rings rather than semirings, avoiding collapse and loss of information by working with bounded sets. Some of the results here were obtained independently by Cluckers and Loeser. Among other results, it is shown that for sufficiently large \(p\), the \(p\)-adic Fourier transform of a rational polynomial is locally constant away from an exceptional subvariety.
As indicated above, the bulk of the work is in the setting of algebraically closed valued fields, but in Section 12 the authors work with more general Henselian fields, with a richer class of definable sets, under an assumption that quantifier elimination is obtained by adding relation symbols in the RV sort. Much of the theory extends to this setting. From this, for example, for the theory RCVF of real closed valued fields, they derive two Euler characteristics from \(K(\text{RCVF})\) to \({\mathbb Z}[t]\), extending a result of T. Mellor. Also, they re-prove rapidly the theorem of [\textit{R. Cluckers} and \textit{D. Haskell}, Bull. Symb. Log. 7, No. 2, 262--269 (2001; Zbl 0988.03058)] that the Grothendieck ring of each \(p\)-adic field is trivial. Some of the model theory of Henselian fields of residue characteristic 0, in particular QE in the Denef-Pas language, is also derived from these methods, thereby bypassing the machinery of pseudo-convergent sequences.
In the final section, the authors use the theory developed to answer a question of Kontsevich and Gromov (also answered by different methods by M. Larsen and V. A. Lunts). They show that if \(X,Y\) are smooth \(d\)-dimensional subvarieties of a smooth projective \(n\)-dimensional variety \(V\), with \(V\setminus X\) and \(V\setminus Y\) birationally isomorphic, then \(X\times{\mathbb A}^{n-d}\) and \(Y\times{\mathbb A}^{n-d}\) are birationally equivalent (and \(X\) and \(Y\) are birationally equivalent if they contain no rational curves). In particular, two elliptic curves with isomorphic complements in projective space are isomorphic. motivic integration; geometry of algebraically closed valued fields; V-minimal expansions; definable sets; Grothendieck semigroup Hrushovski, E., Kazhdan, D.: Integration in valued fields. In: Algebraic Geometry and Number Theory. In Honor of Vladimir Drinfeld's 50th Birthday, pp. 261-405. Birkhäuser, Basel (2006) Model-theoretic algebra, Other nonanalytic theory, Model theory of fields, Cycles and subschemes Integration in valued 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 We consider \(d\)-fold branched coverings of the projective plane \(\mathbb {RP}^2\) and show that the hypergeometric tau function of the BKP hierarchy of Kac and van de Leur is the generating function for weighted sums of the related Hurwitz numbers. In particular, we get the \(\mathbb {RP}^2\) analogues of the \(\mathbb {CP}^1\) generating functions proposed by \textit{A. Okounkov} [Math. Res. Lett. 7, No. 4, 447--453 (2000; Zbl 0969.37033)] and by \textit{I. P. Goulden} and \textit{D. M. Jackson} [Adv. Math. 219, No. 3, 932--951 (2008; Zbl 1158.37026); Proc. Am. Math. Soc. 125, No. 1, 51--60 (1997; Zbl 0861.05006)]. Other examples are Hurwitz numbers weighted by the Hall-Littlewood and by the Macdonald polynomials. We also consider integrals of tau functions which generate Hurwitz numbers related to base surfaces with arbitrary Euler characteristics \(e\), in particular projective Hurwitz numbers \(e=1\). Hurwitz numbers; tau functions; BKP; projective plane; Schur polynomials; Hall-Littlewood polynomials; hypergeometric functions; random partitions; random matrices S. M. Natanzon and A. Yu. Orlov, ''Hurwitz numbers and BKP hierarchy,'' arXiv:1407.8323v2 [nlin.SI] (2014). Exact enumeration problems, generating functions, Enumerative problems (combinatorial problems) in algebraic geometry, Applications of Lie algebras and superalgebras to integrable systems, Soliton equations, KdV equations (Korteweg-de Vries equations), NLS equations (nonlinear Schrödinger equations), Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions, Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with infinite-dimensional Lie algebras and other algebraic structures BKP and projective Hurwitz numbers | 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 studies the geometry of the Weil-Tate multiplicative reciprocity law in connection with Koszul complexes and with the Krichever map. If \(f\), \(g\) are two meromorphic functions on a projective algebraic curve \(X\) (holomorphic outside some smooth point \(a\in X\)), with dominant terms \(z^{-n}\) and \(z^{-m}\) with respect to some local parameter \(z\) at \(a\), then the reciprocity law reads:
\[
\prod _{x\in X\backslash \{ a\} }f(x)^{v_x(g)}=(-1)^{mn} \prod _{x\in X\backslash \{ a\} }g(x)^{v_x(f)}.\tag \(*\)
\]
A direct algebraic proof of \((*)\) is given based on the fact that the left hand-side is det\((f,A/gA)\) where \(A=H^0(X\backslash \{ a\} ,{\mathcal O}_X)\). This can be described as the determinant of the Koszul double complex for \(f\) and \(g\) acting on \(A\). This Koszul complex approach is related to the proof of \((*)\) given by \textit{E. Previato} [Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 2, 167-171 (1991; Zbl 0739.30034)] where a construction based on differential operators and due to Krichever is used. Weil-Tate reciprocity law; Koszul double complex; Krichever map; Nakayashiki-Mukai Fourier transform; differential operators Brylinski J.-L., Previato E.: Koszul Complexes, Differential Operators, and the Weil-Tate Reciprocity Law. J. Algebra 230, 89--100 (2000) Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Differentials on Riemann surfaces, Meromorphic functions of one complex variable (general theory) Koszul complexes, differential operators, and the Weil-Tate reciprocity law | 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 consider the generation of prime order elliptic curves (ECs) over a prime field \(\mathbb F_p\) using the Complex Multiplication (CM) method. A crucial step of this method is to compute the roots of a special type of class field polynomials with the most commonly used being the Hilbert and Weber ones, uniquely determined by the CM discriminant \(D\). In attempting to construct prime order ECs using Weber polynomials two difficulties arise (in addition to the necessary transformations of the roots of such polynomials to those of their Hilbert counterparts). The first one is that the requirement of prime order necessitates that \(D\equiv 3 \mod 8\), which gives Weber polynomials with degree three times larger than the degree of their corresponding Hilbert polynomials (a fact that could affect efficiency). The second difficulty is that these Weber polynomials do not have roots in \(\mathbb F_p\). In this paper we show how to overcome the above difficulties and provide efficient methods for generating ECs of prime order supported by a thorough experimental study. In particular, we show that such Weber polynomials have roots in \(\mathbb F_{p^3}\) and present a set of transformations for mapping roots of Weber polynomials in \(\mathbb F_{p^3}\) to roots of their corresponding Hilbert polynomials in \(\mathbb F_p\). We also show how a new class of polynomials, with degree equal to their corresponding Hilbert counterparts (and hence having roots in \(\mathbb F_p\)), can be used in the CM method to generate prime order ECs. Finally, we compare experimentally the efficiency of using this new class against the use of the aforementioned Weber polynomials. elliptic curve cryptosystems; generation of prime order elliptic curves; complex multiplication; class field polynomials Elisavet Konstantinou, Aristides Kontogeorgis, Yannis C. Stamatiou, and Christos Zaroliagis, Generating prime order elliptic curves: difficulties and efficiency considerations, Information security and cryptology --- ICISC 2004, Lecture Notes in Comput. Sci., vol. 3506, Springer, Berlin, 2005, pp. 261 -- 278. Complex multiplication and moduli of abelian varieties, Algebraic number theory computations, Class numbers, class groups, discriminants, Class field theory, Modular and automorphic functions, Cryptography, Applications to coding theory and cryptography of arithmetic geometry Generating prime order elliptic curves: difficulties and efficiency considerations | 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 pair \((B_{1},B_{2})\) of reduced complex projective plane curves is called a Zariski pair if \(B_{1}\) and \(B_{2}\) have the same degree \(m\), the same combinatoric data and \(\mathbb P^2 \setminus B_1\) is not homeomorphic to \(\mathbb P^2 \setminus B_2\). In the paper under review, the authors find two Zariski pairs of degree \(6\) with only simple singularities. Furthermore, they give the four sextic curves explicitly. If the simple singularities of plane curves are denoted by \(a_{k},d_{k},e_{k}\), the sextics the authors find have maximal Milnor index \(19\), where the Milnor index \(\mu(B)\) of a plane curve \(B\) with only simple singularities is definite as the sum of the subindices of the types of singularities on \(B\). In previous articles the authors have found examples of Zariski pairs with Milnor index less than \(19\). The authors give two different methods to prove that, for both the pairs, that \(\mathbb P^2 \setminus B_1\) is not homeomorphic to \(\mathbb P^2 \setminus B_2\); the first method consists in the computation of the Alexander polynomials of \(\mathbb P^2 \setminus B_i\), for \(i=1,2\); in the second method, they reduce the problem to compute the Mordell-Weil groups of the elliptic \(K3\) surfaces obtained as the minimal resolutions of the double coverings of \(\mathbb P^2\) branched along the sextics \(B_i\), \(i=1,2\). Zariski pair; Mordell-Weil group; K3 surfaces; complement of plane curves; Milnor index; Alexander polynomials E Artal Bartolo, H Tokunaga, Zariski pairs of index 19 and Mordell-Weil groups of \(K3\) surfaces, Proc. London Math. Soc. \((3)\) 80 (2000) 127 Plane and space curves, Coverings of curves, fundamental group, \(K3\) surfaces and Enriques surfaces Zariski pairs of index 19 and Mordell-Weil groups of \(K3\) 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 Connected components of a real \(n\)-dimensional non-singular algebraic variety \(A\) define homology classes in \(H_ n(A(\mathbb{C}),\mathbb{F}_ 2)\). The aim of this paper is to improve the author's previous upper bounds [see Math. USSR, Izv. 22, 247-275 (1984); translation from Izv. Akad. SSSR, Ser. Mat. 47, No. 2, 268-297 (1983; Zbl 0537.14035)] on the number \(k\) of relations between these classes. Here the author uses analogous computations with Galois-Grothendieck cohomology of \(A(\mathbb{C})\) over \(\mathbb{F}_ 2\) with respect to complex conjugation action. Among numerous upper bounds, generalizing the known ones \((k\leq 1\) for curves, \(k\leq 1+q\) for surfaces, where \(q\) is the irregularity) it should be underlined the estimate \(k\leq 1\) for complete intersections. components of a real algebraic variety; homology classes; Galois- Grothendieck cohomology; algebraic variety V. A. Krasnov, ''On homology classes defined by real points of a real algebraic variety,''Izv. Akad. Nauk SSSR Ser. Mat. [Math. USSR-Izv.],55, No. 2, 282--302 (1991). Topology of real algebraic varieties, Étale and other Grothendieck topologies and (co)homologies On homology classes determined by real points of a real algebraic 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 Let \(P_i(x)\) for \(1\leq i\leq n-1\) be the \(n\times n\)-matrix obtained from the \(n\times n\) identity matrix by placing the block \(\left( \begin{smallmatrix} x&1\\0&1\end{smallmatrix}\right)\) with \(x\) at the \((i,i)\)'th coordinate. Then the matrices \(P_i(x)\) satisfy the Coxeter relations \(P_i(x) P_j(y) =P_j(y) P_i(x)\) if \(|i-j|\geq 2\) and \(P_i(x) P_{i+1}(y) P_i(z) =P_{i+1}(z)P_i(y+xz) P_{i+1}(x)\). It is shown that, for any reduced decomposition \(i=(i_1, i_2, \dots , i_N)\) of a permutation \(w\) and any ring \(R\), there is a bijection \(P_i:(x_1,x_2, \dots , x_N) \to P_{i_1}(x_1) P_{i_2}(x_2) \cdots P_{i_N}(x_N)\) from \(\mathbb{R}^N\) to the Schubert cell of \(w\). Moreover, it is shown how to factor explicitly any element of the Schubert cell corresponding to \(w\) into a product of such matrices. Thus one obtains a parametrization of the Schubert cell.
The formulas use planar configurations naturally associated to reduced decompositions. It is shown that the linear parts of these parametrizations give exactly all injective balanced labelings of the diagram of \(w\) [as defined by \textit{S. Fomin, C. Greene, V. Reiner} and \textit{M. Shimozono}, Eur. J. Comb. 18, No. 4, 373-389 (1997; Zbl 0871.05059)], and that the quadratic part characterizes the commutation classes of reduced decompositions. elementary matrices; Coxeter relations; Schubert cells; reduced decompositions; labelings of diagrams; planar configurations; factorization of matrices; commutation classes C. Kassel, A. Lascoux, and C. Reutenauer, ''Factorizations in Schubert cells,'' Adv. Math. 150 (2000), no. 1, 1--35. Factorization of matrices, Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of representation theory, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Factorizations in 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 subgroup \(K=\mathrm{GL}_p \times \mathrm{GL}_q\) of \(\mathrm{GL}_{p+q}\) acts on the (complex) flag variety \(\mathrm{GL}_{p+q}/B\) with finitely many orbits. We introduce a family of polynomials specializing representatives for cohomology classes of the orbit closures in the Borel model. We define and study \(K\)-orbit determinantal ideals to support the geometric naturality of these representatives. Using a modification of these ideals, we describe an analogy between two local singularity measures: the \(H\)-polynomials and the Kazhdan-Lusztig-Vogan polynomials. flag variety; symmetric pair; cohomology class representative; Kazhdan-Luztig-Vogan polynomials Wyser, BJ; Yong, A, Polynomials for \(\text{GL}_p\times \text{ GL }_q\) orbit closures in the flag variety, Sel. Math., 20, 1083-1110, (2014) Group actions on combinatorial structures, Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds Polynomials for \(\mathrm{GL}_p\times \mathrm{GL}_q\) orbit closures in the flag 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 Let \(P(x_1,\ldots,x_{n})\) be a polynomial. A set in \(\mathbb R^{n}\) is called semi-algebraic if it has a representation as a finite union of intersections of sets of the form \(\{(x_1,\ldots,x_{n})|P(x_1,\ldots,x_{n})=0\}\) or \(\{(x_1,\ldots,x_{n})|P(x_1,\ldots,x_{n})<0\}\). The authors prove that if on a compact semi-algebraic set \(M\) a flow with isolated singular points \(x_{i}\) is given, then \(\chi(M)=\sum \text{ind} (x_{i})\), where \(\chi(M)\) is the Euler characteristic of \(M\); \(\text{ind} (x_{i})\) is the index of the singular point \(x_{i}\). index of isolated point; flow; real 2-dimensional semi-algebraic sets; polynomials; Euler characteristic Stratifications in topological manifolds, Semialgebraic sets and related spaces, Dynamics induced by flows and semiflows The index of isolated point of the flow on the real 2-dimensional semi-algebraic sets | 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 review Lie polynomials as a mathematical framework that underpins the structure of the so-called double copy relationship between gauge and gravity theories (and a network of other theories besides). We explain how Lie polynomials naturally arise in the geometry and cohomology of \(\mathcal{M}_{0,n}\), the moduli space of \(n\) points on the Riemann sphere up to Mobiüs transformation. We introduce a twistorial correspondence between the cotangent bundle \(T^{\ast}_D\mathcal{M}_{0,n}\), the bundle of forms with logarithmic singularities on the divisor \(D\) as the twistor space, and \(\mathcal{K}_n\) the space of momentum invariants of \(n\) massless particles subject to momentum conservation as the analogue of space-time. This gives a natural framework for Cachazo He and Yuan (CHY) and ambitwistor-string formulae for scattering amplitudes of gauge and gravity theories as being the corresponding Penrose transform. In particular, we show that it gives a natural correspondence between CHY half-integrands and scattering forms, certain \(n-3\)-forms on \(\mathcal{K}_n\), introduced by Arkani-Hamed, Bai, He and Yan (ABHY). We also give a generalization and more invariant description of the associahedral \(n-3\)-planes in \(\mathcal{K}_n\) introduced by ABHY. scattering amplitudes; Lie polynomials; twistor theory Yang-Mills and other gauge theories in quantum field theory, Gravitational interaction in quantum theory, \(2\)-body potential quantum scattering theory, Spinor and twistor methods applied to problems in quantum theory, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Other special orthogonal polynomials and functions, Scattering theory for PDEs Lie polynomials and a twistorial correspondence for amplitudes | 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 [Algebra Number Theory 2, No. 2, 135--155 (2008; Zbl 1158.14042)], \textit{E. Katz} and \textit{S. Payne} used localization to describe the restriction map from equivariant Chow cohomology to ordinary Chow cohomology for complete toric varieties in terms of piecewise polynomial functions and Minkowski weights.
In the paper under review, it is shown that the assignment introduced in the paper cited above agrees with the inductive intersection product of rational functions introduced in [\textit{L. Allermann} and \textit{J. Rau}, Math. Z. 264, No. 3, 633--670 (2010; Zbl 1193.14074)]. By Proposition \(3.12\), each piecewise polynomial on a tropical fan is a sum of products of rational functions; this is used to intersect cocycles with tropical cycles (Definition \(3.13\)). In Theorem \(3.29\), a Poincaré duality result is established for cocycles on the cycle \(\mathbb R^n\). Finally, in Theorem \(4.1\) the author shows that each subcycle of a matroid variety is cut out by a cocycle and, in Corollary \(4.9\), a Poincaré duality in codimension \(1\) and \(0\) for smooth tropical cycles is proven. tropical varieties; Chow groups; intersection theory; piecewise polynomials G. François, Cocycles on tropical varieties via piecewise polynomials. Proc. Am. Math. Soc. 141, 481-497 (2013) , Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, (Co)homology theory in algebraic geometry Cocycles on tropical varieties via piecewise 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 This paper should be seen as part of the study by I. M. Gelfand and his school of generalized hypergeometric functions. A rather detailed study of the notion of strata on compact homogeneous spaces of complex semisimple Lie groups is given. Several equivalent definitions are described, and the relations to matroids and to convex polytopes are explained. Schubert cells; Grassmannian; generalized hypergeometric functions; strata; compact homogeneous spaces; complex semisimple Lie groups Semisimple Lie groups and their representations, Harmonic analysis on homogeneous spaces, Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions), Grassmannians, Schubert varieties, flag manifolds Strata of a maximal torus in a compact homogeneous space | 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 present parametric equations for the curve \(x^n+y^n=1\) with even exponents, in terms of the double area of the sector bounded by the arc between \((1,0)\) and \((x,y)\) and the radius vectors of these points. We determine also the area enclosed by this curve. Weierstrass' gamma function; double area; sector Surfaces in Euclidean and related spaces, Plane and space curves, Area and volume (educational aspects) Some properties of the curves \(x^n+y^n=1\) with even exponents | 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 idea that algebras of observables in quantum mechanics should be interpreted as deformations of commutative algebras of functions on certain manifolds (phase spaces) has led to the concept of deformation quantization. This viewpoint was proposed in 1978 by \textit{F. Bayen}, \textit{M. Flato}, \textit{C. Frønsdal}, \textit{A. Lichnerowicz}, and \textit{D. Sternheimer} in their pioneering paper ``Deformation theory and quantization. I: Deformations of symplectic structures'' [Ann. Phys. 111, No. 1, 61-110 (1979; Zbl 0377.53024)]. In the program of deformation quantization initiated by these authors, one basic step is to construct an associative (but possibly non-commutative) multiplication law, a so-called ``star product'', on the vector space \(C^\infty(X)\) of functions on a Poisson manifold \(X\), which is compatible with gauge transformations.
A rigorous mathematical proof of the existence of a canonically defined gauge equivalence class of star products on any Poisson manifold \(X\) has been given only recently by the author of the paper under review. In his preprint ``Deformation quantization of Poisson manifolds. I'' [cf. \textit{M. Kontsevich}, \(q\)-alg/9709040] he established his so-called ``formality theorem'', from which he derived the existence theorem for star products mentioned above.
His formality theorem states that in a suitably defined homotopy category of differential graded Lie algebras, two objects are equivalent. The first object is the Hochschild complex of the algebra of functions on the manifold \(X\), and the second object is a certain graded Lie superalgebra of polyvector fields on \(X\). Via an interpretation in the framework of Feynman diagrams, Kontsevich's explicit isomorphism in the case \(X=\mathbb{R}^n\) provides a canonical way for deformation quantization.
Shortly after this break-through, Tamarkin gave another proof of the formality theorem for the case \(X=\mathbb{R}^n\) [cf. \textit{D. E. Tamarkin}, ``Another proof of M. Kontsevich formality theorem'', math.QA/9803025]. His approach is not only more general, but also makes the conjectured relation between the classifying space for deformation quantizations and the Grothendieck-Teichmüller group much more transparent.
The present paper is closely related to the author's (unpublished) talk delivered at the ICM-98 Congress in Berlin, and its purpose is to further extend and generalize Tamarkin's improvements of the author's earlier results. Using throughout the framework of operads, and of the homotopy theory for algebraic structures, the author discusses the present state of the recent developments in deformation quantization in a unified and systematic way, including some furthergoing conjectures and speculations.
Section 1 of the letter gives a motivating introduction to the subject. Section 2 is devoted to generalities on operads and algebras over operads. However, the main topic discussed here is Deligne's conjecture on operad actions on the Hochschild complex of an associative algebra, together with a generalization of it to higher dimensions. Section 3 provides a more general proof of Tamarkin's formality theorem, and a sketch of its application to the author's formality theorem in deformation quantization. Section 4 deals with the possible relations between the motivic Galois group, the Grothendieck-Teichmüller group, and various homogeneous spaces appearing in deformation quantization. In this context, the author explains five deep conjectures of his, and in the concluding Section 5 he adds two more conjectures concerning the link between higher-dimensional algebras, motives, and candidates for quantum field theories. Altogether, this letter is highly inspiring, tremendously rich of new ideas, and truly programmatic with respect to further research in this direction. little discs operads; Hochschild cohomology; algebras of observables; quantum mechanics; deformation quantization; gauge transformations; gauge equivalence class; star products; Poisson manifold; graded Lie superalgebra; Feynman diagrams; Grothendieck-Teichmüller group; operads; algebras over operads; formality theorem; motivic Galois group; higher-dimensional algebras; motives; quantum field theories Maxim Kontsevich, ``Operads and motives in deformation quantization'', Lett. Math. Phys.48 (1999) no. 1, p. 35-72 , Deformations of associative rings, Loop spaces, de Rham cohomology and algebraic geometry, Classical real and complex (co)homology in algebraic geometry, Supermanifolds and graded manifolds, Monoidal categories (= multiplicative categories) [See also 19D23], Quantum field theory; related classical field theories, Loop space machines and operads in algebraic topology Operads and motives in deformation quantization | 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 Two parameter families of plane conics are called nets of conics. There is a natural group action on the vector space of nets of conics, namely the product of the group reparametrizing the underlying plane, and the group reparametrizing the parameter space of the family. We calculate equivariant fundamental classes of orbit closures. Based on this calculation we develop the invariant theory of nets of conics. As an application we determine Thom polynomials of contact singularities of type (3,3). We also show how enumerative problems, in particular the intersection multiplicities of the determinant map from nets of conics to plane cubics, can be solved studying equivariant classes of orbit closures. nets of conics; Thom polynomials M. Domokos, L. M. Fehér, and R. Rimányi, Equivariant and invariant theory of nets of conics with an application to Thom polynomials, 2011. Topological invariants on manifolds, Families, moduli of curves (algebraic), Enumerative problems (combinatorial problems) in algebraic geometry Equivariant and invariant theory of nets of conics with an application to Thom 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 Let \(\pi\) :\(X\to Y\) be a proper birational morphism between surfaces with rational double points. We first show that \(\pi\) can be factored into a sequence of ''elementary'' morphisms, each of which contracts a smooth rational curve to a point. (This is a special case of a theorem of F. Sakai.) We then study conormal sheaves of smooth curves on surfaces with rational double points, and give necessary and sufficient conditions for an ''elementary'' contraction to exist in terms of conormal sheaves. This generalizes the classical case of smooth surfaces, in which the ''elementary'' morphism is the blow-up of a smooth point, and a smooth rational curve C can be contrasted to a smooth point if and only if its conormal sheaf is \({\mathcal O}_ C(1)\). The paper also contains an application of these results to the study of Gorenstein threefold singularities with small resolutions. decomposition of birational morphism; smooth curves on surfaces with rational double points; blow-up of a smooth point; Gorenstein threefold singularities D. Morrison: ''The birational geometry of surfaces with rational double points'', Math. Ann., Vol. 271, (1985), pp. 415--438. Global theory and resolution of singularities (algebro-geometric aspects), Rational and birational maps, Singularities of surfaces or higher-dimensional varieties, 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 The authors construct an example of a non-rational but unirational variety with zero Brauer cohomological invariant but non-zero higher cohomological invariants of Grothendieck. This is a first application of Grothendieck invariants to the problem of rationality. The example resembles the famous example of Artin-Mumford of a non-rational unirational 3-fold. It is a birationally quadric bundle over \({\mathbb{P}}^ 3\), whose general fibre is isomorphic to the quadric \(X^ 2_ 0+f_ 1X^ 2_ 1+f_ 2X^ 2_ 2+f_ 1f_ 2X^ 2_ 3+g_ 1g_ 2X^ 2_ 4=0\) with certain conditions on the elements \(f_ 1,f_ 2,g_ 1,g_ 2\) of the field \({\mathbb{C}}(x,y,z)\). Recall that the higher cohomological invariants of Grothendieck of a field of algebraic functions K over a field of constants k are defined as the intersection \(F_ n^{j,i}\) of all kernels of the homomorphisms \(H^ j(K,\mu_ n^{\otimes^ i})\to H^ j(k_ A,\mu_ n^{\otimes^{i-1}})\), where A is a discrete valuation ring of K with the residue field \(k_ A\) containing k. The Brauer invariants are the groups \(F_ n^{2,1}\). In the example, \(F_ 2^{3,3}\neq 0\) but all \(F_ n^{2,1}=0\). non-rational but unirational variety; Grothendieck invariants; rationality; Brauer invariants Colliot-Thélène, Jean-Louis; Ojanguren, Manuel, Variétés unirationnelles non rationnelles: au-delà de l'exemple d'Artin et Mumford, Invent. Math., 0020-9910, 97, 1, 141-158, (1989) Rational and unirational varieties, Rational points, Brauer groups of schemes, Applications of methods of algebraic \(K\)-theory in algebraic geometry Variétés unirationelles non rationelles: Au-delà de l'exemple d'Artin et Mumford. (Non rational unirational varieties: Beyond the Artin-Mumford example) | 0 |
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