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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 In this work, the authors revisit the determination of the boundary map coefficients for the cellular homology of real flag manifolds, a problem equivalent to finding the incidence coefficients of the differential map for the cohomology. They prove a new formula for these coefficients with respect to the height of certain roots.
A generalized flag manifold \(\mathbb F \) is a homogeneous space \(G/P\), where \(G\) is a real noncompact semisimple Lie group and \(P\) is a parabolic subgroup. It admits a cellular decomposition called Bruhat decomposition, where the cells are the Schubert cells and parametrized by the Weyl group \(W\). There is the Bruhat Chevalley order on elements of the Weyl group. In this case, there is a root \(\beta\) such that \(w = s_{\beta}\cdot w'\). In both [\textit{R. R. Kocherlakota}, Adv. Math. 110, No. 1, 1--46 (1995; Zbl 0832.22020)] and [\textit{L. Rabelo} and \textit{L. A. B. San Martin}, Indag. Math., New Ser. 30, No. 5, 745--772 (2019; Zbl 1426.57052)], the authors summarized how to compute the coefficient \(c(w,w')\).
The papers [\textit{L. Rabelo}, Adv. Geom. 16, No. 3, 361--379 (2016; Zbl 1414.57018)] and [\textit{J. Lambert} and \textit{L. Rabelo}, Australas. J. Comb. 75, Part 1, 73--95 (2019; Zbl 1429.05005)] apply this procedure in the context of the Isotropic Grassmannians and the results obtained (for instance, see [\textit{J. Lambert} and \textit{L. Rabelo}, ``Integral homology of real isotropic and odd orthogonal Grassmannians'', Preprint, \url{arXiv:1604.02177}, to appear in Osaka J. Math.], Theorem 3.12) suggest a formula for the coefficients in terms of the height of some root.
Overall they prove a new formula for the cellular homology coefficients of real flag manifolds in terms of the height of certain roots. For real flag manifolds of type \(A\), they get simple expressions for the coefficients that allow us to compute the first and second integral homology groups exhibiting their generators. real flag manifolds; symmetric group; root systems; Schubert cells; homology; height of roots; boundary coefficients Homology and cohomology of homogeneous spaces of Lie groups, Permutations, words, matrices, Grassmannians, Schubert varieties, flag manifolds, Root systems A correspondence between boundary coefficients of real flag manifolds and height of roots | 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 the polynomial defined by
\[
P(x,z)=z^d+\sum_{i=1}^da_i(x)z^{d-i}\; ,
\]
where each \(a_i\) is a real-analytic function on an open interval \(I\subset\mathbb{R}\). \textit{F. Rellich} [Math. Ann. 113, 600--619 (1936; Zbl 0016.06201)] proved that if, for any \(x\in I\), the polynomial \(x\mapsto P(x,z)\) is hyperbolic, i.e., it has only real roots, then the roots of \(P\) can be chosen analytically.
In this paper, the authors provide two multiparameter generalizations of Rellich's theory. For the first generalization, it is proved that the roots of \(P\) can be chosen locally in a Lipschitz way. For the second generalization, after a suitable blowup of the space of parameters, they write locally the roots of hyperbolic polynomials as analytic functions of the parameters.
In the second half of the paper, another result by \textit{F. Rellich} is generalized [Perturbation theory of eigenvalue problems. Notes on Mathematics and its Applications. New York-London-Paris: Gordon and Breach Science Publishers (1969; Zbl 0181.42002)] stating that a 1-parameter analytic family of symmetric matrices admits a uniform diagonalization. In the end, analytic families of antisymmetric matrices, depending on several parameters, are also considered. hyperbolic polynomials; perturbation theory; linear operators; eigenvalue; symmetric matrices; uniform diagonalization; antisymmetric matrices Kurdyka K., Paunescu L.: Hyperbolic polynomials and multiparameter real-analytic perturbation theory. Duke Math. J. 141(1), 123--149 (2008) Eigenvalues, singular values, and eigenvectors, Semi-analytic sets, subanalytic sets, and generalizations, Nash functions and manifolds Hyperbolic polynomials and multiparameter real-analytic perturbation theory | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We determine conditions on \(q\) for the nonexistence of deep holes of the standard Reed-Solomon code of dimension \(k\) over \(\mathbb F_q\) generated by polynomials of degree \(k+d\). Our conditions rely on the existence of \(q\)-rational points with nonzero, pairwise-distinct coordinates of a certain family of hypersurfaces defined over \(\mathbb F_q\). We show that the hypersurfaces under consideration are invariant under the action of the symmetric group of permutations of the coordinates. This allows us to obtain critical information concerning the singular locus of these hypersurfaces, from which the existence of \(q\)-rational points is established. finite fields; Reed-Solomon codes; deep holes; symmetric polynomials; singular hypersurfaces; rational points Cafure, A.; Matera, G.; Privitelli, M.: Singularities of symmetric hypersurfaces and Reed-Solomon codes. Adv. math. Commun. 6, No. 1, 69-94 (2012) Applications to coding theory and cryptography of arithmetic geometry, Symmetric functions and generalizations, Algebraic coding theory; cryptography (number-theoretic aspects), Combinatorial codes Singularities of symmetric hypersurfaces and Reed-Solomon codes | 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 give a characterization of arithmetically Gorenstein Schubert varieties in a minuscule generalized flag variety \(G/P\) over an algebraic closed field. Recall that \(G/P\) is minuscule if \(P\) is a maximal parabolic subgroup associated to a minuscule (fundamental) weights. If \(G=\mathrm{SL}_n\), then \(G/P\) is just a Grassmannian.
The minuscule weights plus zero are a set of ``minimal'' representatives for the cosets of the root lattice in the weight lattice.
All Schubert varieties (in a flag variety) are Cohen-Macaulay, but not all of them are smooth. The smooth ones are completely classified. The Gorenstein property is a geometric property between Cohen-Macaulayness and the smoothness property. The problem of classifying the Gorenstein Schubert varieties is an open problem. The minuscule varieties have a canonical closed embedding in a projective space (associated to the ample generator of the Picard group). A Schubert variety is arithmetically Gorenstein (with respect to this embedding) if the affine cone over it is Gorenstein; this property implies the Gorenstein property.
In [Adv. Math. 207, No. 1, 205--220 (2006; Zbl 1112.14058)], \textit{A. Woo} and \textit{A. Yong} have given a characterization of Gorenstein Schubert varieties in \(\mathrm{SL}_n/B\). In this case the Weyl group is isomorphic to \(S_n\) and the characterization can be given in terms of the Young diagram associated to the Schubert variety. This implies a combinatorial characterization of the Gorenstein Schubert varieties in the Grassmannian. The authors of this work give stronger results, namely, a characterization of the arithmetic Gorenstein property (for the Plücker embedding).
The first main result of this work is a generalization of this combinatorial characterization in the case of the orthogonal Grassmannian where \(G=\mathrm{SO}_n\) and \(P\) is associated to one of the right end roots. The Schubert varieties can be again be represented by a ``generalized'' Young diagram and the combinatorial condition is the same.
The idea of the proof is the following: the homogeneous coordinate ring of a Schubert variety \(X\) (with respect to the canonical embedding of \(G/P\)) is a Cohen-Macaulay and graded Hodge algebra with a set of generators indexed by the Bruhat poset of Schubert subvarieties of \(X\). This facts together with a result of Stanley give a characterization of the Gorenstein property for this algebra.
When \(P=P_1\) and \(G\) is \(\mathrm{SO}_{2m}\) or \(\mathrm{Sp}_{2m}\), the authors prove that all Schubert varieties are arithmetically Gorenstein. In particular, \(\mathrm{Sp}_{2m}/P_1\) is a projective space and the Schubert subvarieties are linear subspaces, in particular smooth.
The authors give also a list of the arithmetically Gorenstein Schubert varieties in the exceptional cases. Finally, they described the arithmetically Gorenstein Schubert varieties of the generalized flag varieties of \(\mathrm{SL}_n\), even if they are non minuscule. Schubert varieties; minuscule; Gorenstein Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus Arithmetically Gorenstein Schubert varieties in a minuscule \(G/P\) | 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 Although there had been scattered examples of duality before 1800, it was Poncelet who introduced the general idea, first for plane curves; if a conic \(Q(x,y,z)=0\) is given in \({\mathbb{P}}^ 2\), to a point (\(\xi\),\(\eta\),\(\zeta\)) in \({\mathbb{P}}^ 2\) corresponds its polar, the line \(\xi \frac{\partial Q}{\partial x}+\eta \frac{\partial Q}{\partial y}+\zeta \frac{\partial Q}{\partial z}=0;\) we now consider it as a point in the dual projective plane. To a curve C in \({\mathbb{P}}^ 2\) is thus associated its dual curve Č in \({\check{\mathbb{P}}}^ 2\); if C is an algebraic curve of degree \(n,\) \(\check{C}\) is algebraic and its degree is called the class of C; it is the number of distinct tangents to C from a ``general'' point (\(\xi\),\(\eta\),\(\zeta\)); Poncelet showed that the points of contact of these tangents are on a curve of degree \( n-1,\) the polar of (\(\xi\),\(\eta\),\(\zeta\)) with respect to C; its equation is \(\xi \frac{\partial F}{\partial x}+\eta \frac{\partial F}{\partial y}+\zeta \frac{\partial F}{\partial z}=0\) if \(F(x,y,z)=0\) is the equation of C; so the class m of C is at most \(n(n-1).\) But C is the dual curve to \(\check{C}\), and it is not always possible that \(n=m(m-1),\) for instance when \(m=n(n-1),\) which is the case if C is smooth. The relation between n and m thus necessarily relied on a study of the singular points of C and of \(\check{C}\). The famous Plücker formulas solved the problem for the simplest singularities (double points with distinct tangents and cusps of the first kind). But new ideas were necessary to master the general case of arbitrary singularities; they were the concept of local intersection number of two curves at a common singular point, and later the concept of ``infinitely near points''; both were based on the Puiseux expansions in the neighbourhood of a singular point. Using the intersection numbers for a curve and its polar, H. J. S. Smith was able to obtain the general expression of the class of a curve.
Poncelet and his immediate successors had already thought of the extension of duality to surfaces in \({\mathbb{P}}^ 3\), and Cayley and Salmon generalized it to projective varieties in \({\mathbb{P}}^ N\). The modern developments start around 1930 with Severi and his idea of defining equivalence of cycles of any dimension. This was implemented by J. A. Todd, using a bold generalization of polar varieties, based on the concept of Schubert cycle (it should be mentioned that some of the results of Todd were independently discovered by Eger).
This paper is the first part of a lecture; the second part discussed the evolution of duality from Todd to the present day. local intersection number of two curves; infinitely near points; intersection numbers; curve and its polar; duality; Schubert cycle Teissier, B.: Quelques points de l'histoire des variétés polaires, de Poncelet à nos jours, (1988) History of geometry, History of mathematics in the 19th century, History of mathematics in the 20th century, Questions of classical algebraic geometry, History of algebraic geometry Quelques points de l'histoire des variétés polaires, de Poncelet à nos jours. (Some points of the history of polarized varieties, from Poncelet till our days) | 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 communication considers when a prevariety of locally convex spaces is a Grothendieck ideal space. It is shown that the prevariety P\({\mathfrak M}\) is Groth(\({\mathfrak A})\) for an injective ideal \({\mathfrak A}\) iff, for any cardinal \({\mathfrak m}\) \((={\mathfrak m^{c}})\), the universal generator F(\({\mathfrak m})\) has a ``stabilized base'' of unit neighbourhoods. No proofs are included. prevariety of locally convex spaces; Grothendieck ideal space; universal generator; stabilized base Methods of algebraic topology in functional analysis (cohomology, sheaf and bundle theory, etc.), General theory of locally convex spaces, Foundations of algebraic geometry Grothendieck prevarieties of locally convex 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 The paper generalizes the concept of partition varieties introduced in a preceding paper by the same author. For this purpose, \(\gamma\)-compatible partitions \(\lambda\) are introduced, where \(\gamma\) is a composition of some integers. It is shown that a certain quotient space associated to such composition \(\gamma\) and partition \(\lambda\) is a projective variety. Moreover, this generalized partition variety is a CW-complex which can be described combinatorially in terms of \(\gamma\)-compatible rook placements of the Ferrers board of \(\lambda\). Finally, the Poincaré polynomial \(P_{\lambda,\gamma}(q)\) for the cohomology equals \(RL_{\lambda,\gamma}(q^2)\), where \(RL_{\lambda,\gamma}\) is the \(\gamma\)-rook length polynomial. The combinatorial description of generalized partition varieties applies to the homology and cohomology of flag manifolds and Grassmann manifolds. rook length polynomial; projective variety; partition variety; Grassmannian manifold; flag manifold; Schubert cells; Bruhat order Ding, K., Rook placements and generalized partition varieties, Discrete Math., 176, 1-3, 63-95, (1997) Algebraic combinatorics, Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of partitions of integers, Permutations, words, matrices Rook placements and generalized partition varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We prove a rigid analytic analogue of the Artin-Grothendieck vanishing theorem. Precisely, we prove (under mild hypotheses) that the geometric étale cohomology of any Zariski-constructible sheaf on any affinoid rigid space \(X\) vanishes in all degrees above the dimension of \(X\). Along the way, we show that branched covers of normal rigid spaces can often be extended across closed analytic subsets, in analogy with a classical result for complex analytic spaces. We also prove some new comparison theorems relating the étale cohomology of schemes and rigid analytic varieties, and give some applications of them. In particular, we prove a structure theorem for Zariski-constructible sheaves on characteristic-zero affinoid spaces. rigid analytic spaces; étale cohomology; Artin-Grothendieck vanishing; comparison theorems Rigid analytic geometry, Vanishing theorems in algebraic geometry, Étale and other Grothendieck topologies and (co)homologies Vanishing and comparison theorems in rigid analytic geometry | 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 normalized Schur polynomials are strongly log-concave. As a consequence, we obtain Okounkov's log-concavity conjecture for Littlewood-Richardson coefficients in the special case of Kostka numbers. Schur polynomials; log-concavity; Lorentzian polynomials; weight multiplicities Combinatorial aspects of representation theory, Determinantal varieties, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights) Logarithmic concavity of Schur and related 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 We establish the formula for multiplication by the class of a special Schubert variety in the integral cohomology ring of the flag manifold. This formula also describes the multiplication of a Schubert polynomial by either an elementary or a complete symmetric polynomial. Thus, we generalize the classical Pieri formula for Schur polynomials (associated to Grassmann varieties) to Schubert polynomials (associated to flag manifolds). Our primary technique is an explicit geometric description of certain intersections of Schubert varieties. This method allows us to compute additional structure constants for the cohomology ring, some of which we express in terms of paths in the Bruhat order on the symmetric group, which in turn yields an enumerative result about the Bruhat order. multiplication by the class of a special Schubert variety; integral cohomology ring of the flag manifold; Pieri formual; Bruhat order Frank Sottile, Pieri's formula for flag manifolds and Schubert polynomials, Ann. Inst. Fourier (Grenoble) 46 (1996), no. 1, 89-110 (English, with English and French summaries). Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations Pieri's formula for flag manifolds and Schubert polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a complex semisimple, simply-connected algebraic group and let \(P\) be a parabolic subgroup. Consider the flag variety \(X:= G/P\) and the Schubert subvarieties \(X_w:= \overline{BwP/P}\subset G/P\) for any \(w\in W/W_P\), where \(W\) is the Weyl group of \(G\), \(W_P\) is the Weyl group of \(P\) and \(B\) is a Borel subgroup of \(G\) contained in \(P\). Then, any subvariety \(V\) of \(X\) is rationally equivalent to a linear combination of Schubert cycles \([X_w]\) with uniquely determined nonnegative integral coefficients. Then, Brion calls \(V\) multiplicity free if these coefficients are 0 or 1. Examples of multiplicity free \(V\) include the Schubert varieties \(X_w\) themselves, \(G\)-stable (irreducible) subvarieties of \(X\times X\) (under the diagonal action of \(G\)), irreducible hyperplane sections of \(X\) in its smallest projective embedding and the irreducible hyperplane sections of Schubert varieties in Grassmannians embedded by the Plücker embedding.
The main theorem of the paper under review asserts that any multiplicity-free subvariety \(V\subset X\) is normal and Cohen-Macaulay. Further, \(V\) admits a flag degeneration inside \(X\) to a reduced Cohen-Macaulay union of Schubert varieties. Hence, for any globally generated line bundle \(G\) on \(X\), the restriction map \(H^0(X,{\mathcal L})\to H^0(V,{\mathcal L})\) is surjective and \(H^i(V,{\mathcal L})= 0\) for all \(i\geq 1\). If \(L\) is ample, then \(H^i(V, {\mathcal L}^{-1})= 0\) for any \(i<\dim V\). Thus, \(V\) is arithmetically normal and Cohen-Macaulay in the projective embedding given by any ample \({\mathcal L}\). Schubert varieties; Cohen-Macaulay; arithmetically normal; projective embedding Brion, M.: Multiplicity-free subvarieties of flag varieties. Commutative algebra (Grenoble/Lyon, 2001), 13-23, Contemp. Math., \textbf{331}, Amer. Math. Soc., Providence, RI, 2003 Grassmannians, Schubert varieties, flag manifolds, Group actions on varieties or schemes (quotients), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) Multiplicity-free subvarieties of 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 Let \(G\) be a complex semisimple algebraic group with maximal torus \(T\) and Borel subgroup \(B\) (corresponding to the positive roots), and let \(X\) denote the character group of \(T\). Let \(m_{\lambda}^{\mu}\) denote the multiplicity of the weight \(\mu\) in the simple \(G\)-module of highest weight \(\lambda\). \textit{G. Lusztig} [Astérisque 101--102, 208--229 (1983; Zbl 0561.22013)] introduced certain \(q\)-analogues \(m_{\lambda}^{\mu}(q)\) of weight multiplicity (also known as Kostka-Foulkes polymomials). These were based on Kostant's partition function that counts the multiplicity of a weight in a symmetric power \(S^j({\mathfrak u})\) where \({\mathfrak u}\) is the Lie algebra of the unipotent radical of \(B\). Let \(P\) be a parabolic subgroup of \(G\), and let \(N\) be a \(P\)-stable subspace of a finite-dimensional rational \(G\)-module such that the \(T\)-weights of \(N\) lie in an open half-space of \(X\otimes_{\mathbb Z}{\mathbb Q}\). The \(T\)-weights of \(N\) form a finite multiset \(\Psi\) in \(X\). Associated to \(\Psi\), the author introduces a generalization of Lusztig's \(q\)-polynomials. Here one makes use of a generalized Kostant partition function that counts the multiplicity of a weight in a symmetric power \(S^j(N)\).
By obtaining a vanishing theorem on line bundle cohomology for \(G\times_{P} N\), the author determines some conditions under which the coefficients of the generalized \(m_{\lambda}^{\mu}(q)\) are non-negative. Next, the author focuses on the special case that \(\Psi\) is the set of short positive roots (in a non-simply laced root system). Here the author obtains a precise determination of when the coefficients are non-negative. This extends work of \textit{A. Broer} [Invent. Math. 113, 1--20 (1993; Zbl 0807.14043)].
Lastly, the author introduces the notion of ``short'' Hall-Littlewood polynomials \(P_{\lambda}(q)\) (for a dominant weight \(\lambda\)) and deduces a number of basic properties. Considering all roots to be short in the simply-laced case, the results generalize the work of \textit{R. Gupta} [J. Lond. Math. Soc., II. Ser. 36, No. 1--2, 68--76 (1987; Zbl 0649.17009)], [Bull. Am. Math. Soc., New Ser. 16, 287--291 (1987; Zbl 0648.22011)]. In particular, let \(\chi_{\lambda}\) denote the character of the simple \(G\)-module with highest weight \(\lambda\). Then, it is shown that \(\chi_{\lambda} = \sum m_{\lambda}^{\mu}(q)P_{\mu}(q)\) where the sum runs over all dominant weights \(\mu\) (and the \(m_{\lambda}^{\mu}(q)\) correspond to \(\Psi\) being the set of short positive roots). semisimple Lie algebra; weight multiplicity; \(q\)-analogue; Hall-Littlewood polynomials; Kostant partition function Panyushev, D. I., Generalised Kostka-Foulkes polynomials and cohomology of line bundles on homogeneous vector bundles, Selecta Math. (N.S.), 16, 315-342, (2010) Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Vanishing theorems in algebraic geometry, Homogeneous spaces and generalizations, Representation theory for linear algebraic groups Generalised Kostka-Foulkes polynomials and cohomology of line bundles on homogeneous vector bundles | 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 theory of Newton-Okounkov polytopes is a generalization of that of Newton polytopes for toric varieties, and gives a systematic method of constructing toric degenerations of projective varieties. In the case of Schubert varieties, their Newton-Okounkov polytopes are deeply connected with representation theory. Indeed, Littelmann's string polytopes and Nakashima-Zelevinsky's polyhedral realizations are obtained as Newton-Okounkov polytopes of Schubert varieties. In this paper, we apply the folding procedure to a Newton-Okounkov polytope of a Schubert variety, which relates Newton-Okounkov polytopes of Schubert varieties of different types. As an application, we obtain a new interpretation of Kashiwara's similarity of crystal bases. crystal basis; fixed point Lie subalgebra; Newton-Okounkov body; orbit Lie algebra; Schubert variety Quantum groups (quantized enveloping algebras) and related deformations, Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds, Toric varieties, Newton polyhedra, Okounkov bodies Folding procedure for Newton-Okounkov polytopes of 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 The author proves an effective formula giving the minimal number of polynomials needed to write generically a given algebraically constructible function as a sum of signs. Recall that the algebraically constructible functions on a real algebraic set are sums of signs of polynomials on this set. Such algebraically constructible functions on a real algebraic set were introduced by \textit{C. G. McCrory} and \textit{A. Parusiński} in order to study the topology of real algebraic sets [Ann. Sci. Éc. Norm. Supér., IV Sér. 30, No. 4, 527--552 (1997; Zbl 0913.14018)]. More precisely, let \(V\) be an algebraic set in some \({\mathbb R}^n\). Originally, an algebraically constructible functions on \(V\) were defined as linear combinations, with integer coefficients, of Euler characteristics of fibres of proper regular morphisms. Later, it was proved that these functions on \(V\) are just the sums of signs of polynomials on \(V\). Recall that for a polynomial function \(P\) on \(V\), the sign of \(P\) on \(V\) is the function \(\text{ sgn}\,P\) from \(V\) to \(\mathbb Z\) such that for all \(x\in V, \text{sgn}\,P(x)=1, -1\), or 0 according as \(P(x)>0,\;P(x)<0\), or \(P(x)=0\).
As we have pointed out, the author of this article obtains a formula giving the minimal number of polynomials needed to express generically a given algebraically constructible function as a sum of signs of polynomials. As another main result in this paper, the author gives an effective characterization of the polynomials appearing in such a generic presentation of an algebraically constructible function as a sum of signs of polynomials with the minimal number of polynomials. generical description; minimal number of polynomials; sums of signs Bonnard, I., Description of algebraically constructible functions, Adv. Geom., 3, 145-161, (2003) Real algebraic sets, Topology of real algebraic varieties Description of algebraically constructible 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 Fix a split connected reductive group \(G\) over a field \(k\), and a positive integer \(r\). For any \(r\)-tuple of dominant coweights \(\mu_i\) of \(G\), we consider the restriction \(m_{\mu_\bullet}\) of the \(r\)-fold convolution morphism of Mirkovic-Vilonen to the twisted product of affine Schubert varieties corresponding to \(\mu_\bullet\). We show that if all the coweights \(\mu_i\) are minuscule, then the fibers of \(m_{\mu_\bullet}\) are equidimensional varieties, with dimension the largest allowed by the semi-smallness of \(m_{\mu_\bullet}\). We derive various consequences: the equivalence of the non-vanishing of Hecke and representation ring structure constants, and a saturation property for these structure constants, when the coweights \(\mu_i\) are sums of minuscule coweights. This complements the saturation results of Knutson-Tao and Kapovich-Leeb-Millson. We give a new proof of the P-R-V conjecture in the ``sums of minuscules'' setting. Finally, we generalize and reprove a result of Spaltenstein pertaining to equidimensionality of certain partial Springer resolutions of the nilpotent cone for \(\text{GL}_n\). split connected reductive groups; dominant coweights; affine Schubert varieties; equidimensional varieties; Springer resolutions; affine Grassmannians; Hecke algebras; structure constants; saturation Grothendieck, A., Diéudonné, J.: Éléments de Géométrie Algeébrique, IV: Étude locale ded schémas e des morphismes de schémas, Seconde partie, Inst.~ Hautes Études Sci.~Publ.~Math.~\textbf{24} (1965) Linear algebraic groups over local fields and their integers, Group actions on varieties or schemes (quotients), Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Hecke algebras and their representations, Representation theory for linear algebraic groups Equidimensionality of convolution morphisms and applications to saturation problems. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Based on the method from [\textit{P. Cassou-Noguès}, Abh. Math. Semin. Univ. Hamb. 58, 103--123 (1988; Zbl 0685.32010)], the authors compute the roots of the Bernstein polynomial associated with an irreducible plane curve singularity which has two Puiseux pairs and whose algebraic monodromy has distinct eigenvalues. In fact, they exploit classical techniques of calculus of meromorphic integrals and residues. Finally, taking into account that in the case under consideration the \(b\)-exponents coincide with the opposite of the roots of the corresponding Bernstein polynomial, the authors also conclude that their computations support the well-known conjecture due to \textit{T. Yano} [Sci. Rep. Saitama Univ., Ser. A 10, No. 2, 21--28 (1982; Zbl 0519.14022)]. plane curve singularities; \(\mu\)-constant stratum; \(b\)-exponents; Bernstein polynomials; improper integrals; residues E. Artal Bartolo, Pi. Cassou-Nogu`es, I. Luengo, and A. Melle-Hern'andez, Yano's conjecture for 2Puiseux pairs irreducible plane curve singularities, Publ. RIMS Kyoto Univ. 53 (2017), no. 1, 211--239. DOI: 10.4171/PRIMS/53-1-7 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Local complex singularities, Other generalizations of function theory of one complex variable Yano's conjecture for two-Puiseux-pair irreducible plane curve 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 The results of this paper are a generalization of those in the authors' paper [Special functions, Proc. Hayashibara Forum, Okayama/Jap. 1990, ICM-90 Satell. Conf. Proc., 149-168 (1991)]. Let \(k_ q[G]\) be the quantum algebra of functions on a semisimple algebraic group \(G\) of rank \(\ell\) (in [loc. cit.], the considered \(G= \text{SL}(N)\)). Let \(B\) be a Borel subgroup of \(G\) and let \(P\supseteq B\) be a maximal parabolic subgroup of \(G\). Let \(k_ q[B]\) be the quantum Hopf algebra of functions on \(B\). Let \(w\) be an element of the Weyl group and let \(X(w) \subset G/B\) be the corresponding Schubert variety. The authors define the quantum algebras \(k_ q[G/P]\), \(k_ q[G/B]\), \(k_ q[X(w)]\); the first two are subcomodules of \(k_ q[G]\), the last is a quotient of \(k_ q[G/B]\). Each of these algebras has, in the classical case, a basis consisting of standard monomials---compatible with canonical \(\mathbb{Z}\) or \(\mathbb{Z}^ \ell\)-gradations. The authors prove the existence of such basis and gradations in the quantum case and give a presentation for \(k_ q[G/B]\). quantum algebra of functions; semisimple algebraic group; Schubert variety; basis; gradations V. Lakshmibai and N. Reshetikhin. ''Quantum flag and Schubert schemes''. Deformation Theory and Quantum Groups with Applications to Mathematical Physics. Contemp. Math., Vol. 134. American Mathematical Society, 1992, pp. 145--181. Quantum groups (quantized enveloping algebras) and related deformations, Grassmannians, Schubert varieties, flag manifolds Quantum flag and Schubert schemes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \({\mathcal A}_ 5\) be the moduli space of 5-dimensional, principally polarized abelian varieties and let \({\mathcal R}_ 6\) be the Prym moduli space parametrizing double covers of curves of genus 6. By \textit{W. Wirtinger} [''Untersuchungen über Thetafunctionen'' Teubner, Leipzig 1895), there is the Prym map \(p : {\mathcal R}_ 6\to {\mathcal A}_ 5\) assigning to a double cover \(\pi : \tilde C\to C\) the neutral component of \(Ker(\pi_* : J(\tilde C)\to J(C)),\) where \(J(\;)\) denotes the Jacobian variety, and p is known to be dominant. A main result of the present article is to show that \({\mathcal R}_ 6\) and \({\mathcal A}_ 5\) are unirational. The idea is to use the theory of nets of quadrics. A net of quadrics in \({\mathfrak P}^ 6\) is a family of quadrics \(\{\) A(t);t\(\in \Pi \}\) in \({\mathfrak P}^ 6\) parametrized by a projective plane \(\Pi\). Such a net is invertible if a general member is nondegenerate and if every member has rank\(\geq 6\). Associated to such a net, we have a pair (C,L) of the discriminant locus \(C=\{t\in \Pi;\;A(t) \text{is degenerate}\}\), i.e., the locus of singular quadrics, and a non-vanishing theta characteristic L. Let \(\bar N_ 0\) be the space of projective equivalence classes of invertible nets whose discriminant locus is \(S\cup \ell\), where S is a sextic in \({\mathfrak P}^ 2\) with 4 double points and \(\ell\) is a line. Then there is a dominant map \(f : \bar N_ 0\to {\mathcal R}_ 6\) via the discriminant map \(\{A(t);t\in \Pi \}\mapsto (S\cup \ell,L)\mapsto (\nu (S),\pi),\) where \(\nu\) (S) is the normalization of S and \(\pi\) is an étale double cover of \(\nu\) (S) given by a line bundle \(L\otimes {\mathcal O}(-2)\). More precisely, f is dominant on each component of \(\bar N_ 0\). By a detailed analysis of \(\bar N_ 0\), the author shows that \({\mathcal R}_ 6\) and \({\mathcal A}_ 5\) are unirational. We note that the unirationality of \({\mathcal A}_ 4\) was proved by H. Clemens. unirationality of moduli space of 5-dimensional principally; polarized abelian varieties; Prym moduli space parametrizing double covers of curves of; genus 6; nets of quadrics; theta characteristic; Prym moduli space parametrizing double covers of curves of genus 6 R. Donagi, The unirationality of
\[
\mathcal{A}_{5}
\]
. Ann. Math. 119, 269--307 (1984) Rational and unirational varieties, Algebraic moduli of abelian varieties, classification, Pencils, nets, webs in algebraic geometry, Theta functions and abelian varieties, Families, moduli of curves (algebraic) The unirationality of \({\mathcal A}_ 5\). | 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 Theta-vexillary signed permutations are elements in the hyperoctahedral group that index certain classes of degeneracy loci of type B and C. These permutations are described using triples of \(s\)-tuples of integers subject to specific conditions. The objective of this work is to present different characterizations of theta-vexillary signed permutations, describing them in terms of corners in the Rothe diagram and pattern avoidance. permutations; Schubert varieties Permutations, words, matrices, Grassmannians, Schubert varieties, flag manifolds Theta-vexillary signed permutations | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(X\) be a smooth complex quasi-projective variety and \(\text{Hilb}^n (X)\) its Hilbert scheme of zero dimensional subschemes of length \(n\). The author expresses the virtual Hodge polynomials of \(\text{Hilb}^n (X)\) -- defined by cohomology with compact support -- in terms of those of \(X\) and the Hilbert scheme of subschemes of length \(n\) supported at a point of \(X\). -- The proof proceeds by comparison with the \(n\)-fold symmetric product of \(X\) and related spaces and uses a lemma on point Hilbert schemes from \(L\). Göttsche's 1991 Bonn thesis [see \textit{L. Göttsche}, ``Hilbertschemata nulldimensionaler Unterschemata glatter Varietäten'', Bonner Math. Schr. 243 (1991; Zbl 0846.14002)]. The key properties of the virtual Hodge polynomial used in the proof are its additivity over stratifications and multiplicativity for fibrations. The results extend those found for the Poincaré and Hodge polynomials of surfaces in a paper by \textit{L. Göttsche} [Math. Ann. 286, No. 1-3, 193-207 (1990; Zbl 0679.14007)]. Hilbert scheme; virtual Hodge polynomials; symmetric product J. Cheah, ''On the Cohomology of Hilbert Schemes of Points,'' J. Algebr. Geom. 5, 479--511 (1996). Parametrization (Chow and Hilbert schemes), Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) On the cohomology of Hilbert schemes of points | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(Q\) be a quiver with no oriented cycles. For dimension vectors \(\alpha\) and \(\beta\), define \(N(\beta,\alpha)\) as the number of \(\beta\)-dimensional subrepresentations of a general \(\alpha\)-dimensional representation of \(Q\). If \(\langle\beta,\alpha-\beta\rangle=0\) (here \(\langle\cdot,\cdot\rangle\) denotes the Ringel bilinear form), then \(N(\beta,\alpha)\) is finite. Denote by \(M(\beta,\alpha)\) the dimension of the space of semi-invariant polynomial functions with weight \(\langle\beta,\cdot\rangle\) on the space of \((\alpha-\beta)\)-dimensional representations of \(Q\) (note that any non-zero semi-invariant on \(\text{Rep}(Q,\alpha-\beta)\) has weight of such special form).
The main result of this paper is that \(M(\beta,\alpha)=N(\beta,\alpha)\) when \(\langle\beta,\alpha-\beta\rangle=0\). The proof is that the number \(M(\beta,\alpha)\) can be expressed via a Littlewood-Richardson calculation, which is then compared by the authors with the expression of \(N(\beta,\alpha)\) given by \textit{W. Crawley-Boevey} [Bull. Lond. Math. Soc. 28, No. 4, 363-366 (1996; Zbl 0863.16008)] using intersection theory.
Applying results of \textit{A. Schofield} [J. Lond. Math. Soc., II. Ser. 43, No. 3, 383-395 (1991; Zbl 0779.16005)], a basis of the corresponding space of semi-invariants is obtained. The result is generalized to covariants as follows: the cohomology class of the variety of \(\beta\)-dimensional subrepresentations of an \(\alpha\)-dimensional representation in general position can be expressed in terms of multiplicities in isotypic components of the coordinate ring of \(\text{Rep}(Q,\beta-\alpha)\). semi-invariants of quivers; Schubert classes; Littlewood-Richardson coefficients; representations of quivers; Ringel bilinear forms Derksen, H., Schofield, A., Weyman, J.: On the number of subrepresentations of a general quiver representation. J. Lond. Math. Soc. (2) 76(1), 135--147 (2007) Representations of quivers and partially ordered sets, Actions of groups on commutative rings; invariant theory, Grassmannians, Schubert varieties, flag manifolds On the number of subrepresentations of a general quiver representation. | 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 characteristic map for the symmetric group is an isomorphism relating the representation theory of the symmetric group to symmetric functions. An analogous isomorphism is constructed for the symmetric space of symplectic forms over a finite field, with the spherical functions being sent to Macdonald polynomials with parameters \((q,q^2)\). An analogue of parabolic induction is interpreted as a certain multiplication of symmetric functions. Applications to Schur-positivity of skew Macdonald polynomials with parameters \((q,q^2)\) are given. characteristic map; symmetric space; Macdonald polynomials Symmetric functions and generalizations, Combinatorial aspects of representation theory, Classical groups (algebro-geometric aspects), Compactifications; symmetric and spherical varieties A characteristic map for the symmetric space of symplectic forms over a finite field | 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 derive necessary and sufficient conditions for periodic and for elliptic periodic trajectories of billiards within an ellipse in the Minkowski plane in terms of an underlining elliptic curve. We provide several examples of periodic and elliptic periodic trajectories with small periods. We observe a relationship between Cayley-type conditions and discriminantly separable and factorizable polynomials. Equivalent conditions for periodicity and elliptic periodicity are derived in terms of polynomial-functional equations as well. The corresponding polynomials are related to the classical extremal polynomials. In particular, the light-like periodic trajectories are related to the classical Chebyshev polynomials. Similarities and differences with respect to the previously studied Euclidean case are highlighted. Minkowski plane; relativistic conics; elliptic billiards; periodic trajectories; extremal polynomials; Chebyshev polynomials; Akhiezer polynomials; discriminantly separable polynomials Dynamical systems with singularities (billiards, etc.), Relations of finite-dimensional Hamiltonian and Lagrangian systems with algebraic geometry, complex analysis, special functions, Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests, Relationships between algebraic curves and integrable systems, Approximation by polynomials, Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics Periodic billiards within conics in the Minkowski plane and Akhiezer 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 The authors consider ``generic'' arrangements of high-dimensional Schubert cells in the real flag variety \(\mathbb{P} T^*\mathbb{P}^ n\) of all flags in \(\mathbb{P}^ n\) consisting of a hyperplane and a distinguished point in it. It is shown that the sum of the Betti numbers (with coefficients \(\mathbb{Z}/2\mathbb{Z})\) of those arrangements coincides with the sum of the Betti numbers of their complexifications \((M\)-property). In general, the \(M\)-property does not hold: an example is given, namely arrangements in \(G_{2,4}\). Finally, for some class of configuration spaces, the Mayer-Vietoris spectral sequence is shown to degenerate in the \(E_ 1\) term. flag varieties; Schubert cell decomposition; \(M\)-property; configuration spaces; Mayer-Vietoris spectral sequence Shapiro B. Z., Topology Appl. 43 (1) pp 65-- (1992) Real-analytic manifolds, real-analytic spaces, Complex spaces, Grassmannians, Schubert varieties, flag manifolds, Spectral sequences in algebraic topology The M-property of 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 In the paper under review a class of authentication codes with secrecy is studied. The authors, extending some results of \textit{E. Çakçak} and \textit{E. Özbudak} [Finite Fields Appl. 14, No.~1, 209--220 (2008; Zbl 1128.14013)], compute the number of rational places of a certain class of algebraic function fields and use it for the computation of the maximum success probabilities of the impersonation and the substitution attacks on these codes and the level of secrecy. Furthermore, all of the parameters of these authentication codes are at least as good as those in [\textit{C. Ding, A. Salomaa, P. Solé} and \textit{X. Tian}, J. Pure Appl. Algebra 196, No. 2--3, 149--168 (2005; Zbl 1068.94021)]. authentication codes with secrecy; algebraic function fields; linearized polynomials Özbudak, E.K.; Özbudak, F.; Saygı, Z., A class of authentication codes with secrecy, Des. codes cryptogr., 59, 287-318, (2011) Cryptography, Authentication, digital signatures and secret sharing, Applications to coding theory and cryptography of arithmetic geometry A class of authentication codes with secrecy | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Motivated by the intersection theory of moduli spaces of curves, we introduce psi classes in matroid Chow rings and prove a number of properties that naturally generalize properties of psi classes in Chow rings of Losev-Manin spaces. We use these properties of matroid psi classes to give new proofs of (1) a Chow-theoretic interpretation for the coefficients of the reduced characteristic polynomials of matroids, (2) explicit formulas for the volume polynomials of matroids, and (3) Poincaré duality for matroid Chow rings. intersection theory of moduli spaces of curves; matroid Chow rings; polynomials of matroids Combinatorial aspects of matroids and geometric lattices, Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects) Matroid psi classes | 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 large-structure tools of cohomology including toposes and derived categories stay close to arithmetic in practice, yet published foundations for them go beyond ZFC in logical strength. We reduce the gap by founding all the theorems of Grothendieck's SGA, plus derived categories, at the level of Finite-Order Arithmetic, far below ZFC. This is the weakest possible foundation for the large-structure tools because one elementary topos of sets with infinity is already this strong. Grothendieck universe Axiomatics of classical set theory and its fragments, Foundations of classical theories (including reverse mathematics), Foundations of algebraic geometry, Foundations, relations to logic and deductive systems, Topoi, Derived categories, triangulated categories The large structures of Grothendieck founded on finite-order arithmetic | 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 articles of this volume will be reviewed individually. For Vol. 2 see Zbl 0868.00040 below. Geometric Galois actions; Grothendieck's esquisse d'un programme; Proceedings; Conference; Geometry; Arithmetic; Moduli spaces Leila Schneps and Pierre Lochak , Geometric Galois actions. 1, London Mathematical Society Lecture Note Series, vol. 242, Cambridge University Press, Cambridge, 1997. Around Grothendieck's ''Esquisse d'un programme''. Proceedings of conferences of miscellaneous specific interest, Proceedings, conferences, collections, etc. pertaining to number theory, Proceedings, conferences, collections, etc. pertaining to algebraic geometry, Proceedings, conferences, collections, etc. pertaining to field theory Geometric Galois actions. 1. Around Grothendieck's esquisse d'un programme. Proceedings of the conference on geometry and arithmetic of moduli spaces, Luminy, France, August 1995 | 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 \(f_t:({\mathbb{R}}^n,0)\to ({\mathbb{R}}^p,0)\) be a family of weighted homogeneous polynomial maps, where \(t\in J\) (an open interval of \({\mathbb{R}}\)). We define \(F:({\mathbb{R}}^n\times J,\{0\}\times J) \to ({\mathbb{R}}^p,0)\) by \(F(x,t)=f_t(x)\) and we suppose that it is also polynomial.
The main result of this paper is that if \(f_t^{-1}(0)\cap \Sigma f_t=\{0\}\) for any \(t\in J\), then \(({\mathbb{R}}^n\times J,F^{-1}(0))\) admits a \(\pi_\alpha\)-modified Nash trivialization. This means that there exists a \(t\)-level preserving Nash diffeomorphism \(\phi:(E\times J,E_0\times J)\to (E\times J,E_0\times J)\) which induces a \(t\)-level preserving homeomorphism \(\widetilde\phi:({\mathbb{R}}^n\times J,\{0\}\times J)\to ({\mathbb{R}}^n\times J,\{0\}\times J)\) such that
\[
\widetilde\phi({\mathbb{R}}^n\times J,F^{-1}(0))= ({\mathbb{R}}^n\times J,f_{t_0}^{-1}(0)\times J).
\]
Here, \(E\) is a Nash manifold, \(E_0\) is a Nash submanifold and \(\pi_\alpha:(E,E_0)\to({\mathbb{R}}^n,0)\) is a finite modification. As a consequence, it follows not only the well-known fact that the family is topologically trivial, but that it is modified analytically trivial in the sense of \textit{T.-C. Kuo} [J. Math. Soc. Japan 32, 605-614 (1980; Zbl 0509.58007)]. It is also shown that in the homogeneous case (that is, when the weights are equal to 1), it also implies strong \(C^0\) triviality. topological triviality; modified Nash triviality; weighted homogeneous polynomials Koike, S., Modified Nash triviality theorem for a family of zero-sets of weighted homogeneous polynomial mappings, J. Math. Soc. Japan, 49, 617-631, (1997) Differentiable maps on manifolds, Theory of singularities and catastrophe theory, Singularities of differentiable mappings in differential topology, Deformations of singularities Modified Nash triviality theorem for a family of zero-sets of weighted homogeneous polynomial mappings | 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 Although the Bruhat order on a Weyl group is closely related to the singularities of the Schubert varieties for the corresponding Kac-Moody group, it can be difficult to use this information to prove general theorems. This paper uses the action of the affine Weyl group of type \(\tilde{A}_2\) on a Euclidean space \(V \cong \mathbb{R}^2\) to study the Bruhat order on \(W\). We believe that these methods can be used to study the Bruhat order on arbitrary affine Weyl groups. Our motivation for this study was to extend the lookup conjecture of \textit{B. D. Boe} and \textit{W. Graham} [Am. J. Math. 125, No. 2, 317--356 (2003; Zbl 1074.14045)] (which is a conjectural simplification of the Carrell-Peterson criterion (see [\textit{J. B. Carrell}, Proc. Symp. Pure Math. 56, 53--61 (1994; Zbl 0818.14020)]) for rational smoothness) to type \(\tilde{A}_2\). Computational evidence suggests that the only Schubert varieties in type \(\tilde{A}_2\) where the ``nontrivial'' case of the lookup conjecture occurs are the spiral Schubert varieties, and as a step towards the lookup conjecture, we prove it for a spiral Schubert variety \(X ( w )\) of type \(\tilde{A}_2\). The proof uses descriptions we obtain of the elements \(x \leq w\) and of the rationally smooth locus of \(X ( w )\) in terms of the \(W\)-action on \(V\). As a consequence we describe the maximal nonrationally smooth points of \(X ( w )\). The results of this paper are used in a sequel to describe the smooth locus of \(X ( w )\), which is different from the rationally smooth locus. Schubert variety; rationally smooth; lookup conjecture Combinatorial aspects of algebraic geometry, Grassmannians, Schubert varieties, flag manifolds The Bruhat order, the lookup conjecture and spiral Schubert varieties of type \(\tilde{A}_2\) | 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 Für eine allgemeine Curve \(4^{\text{ter}}\) Ordnung giebt es bekanntlich 63 Systeme von 4 doppelt berührenden Kegelschnitten, indem die vier Punkte, in welchen die Curve von einem solchen Kegelschnitte berührt wird, ein Büschel von Kegelschnitten bestimmen, von denen jeder wieder die Curve in den 4 Berührungspunkten eines 4-doppeltberührenden Kegelschnittes schneidet. Zu solchen Kegelschnitten gehören auch ein Paar von Doppeltangenten; und vier, solche deren Berührungspunkte auf demselben Kegelschnitte liegen, bilden drei zu demselben Systeme gehörige Linienpaare. Die Aufgabe, die gestellt wird, ist für die von Herrn Zeuthen aufgestellte sogenannte dreiseitige oder vierseitige Curve (siehe F. d. M. VI. 367, JFM 06.0367.01) zu ermitteln, wie die Doppeltangenten mit reeller Berührung sich auf die verschiedenen Systeme von Kegelschnitten vertheilen. Es wird dabei von besonderer Wichtigkeit zu entscheiden, wann die Berührungpunkte von vier Doppeltangenten auf demselben Kegelschnitte liegen können. Von den von Zeuthen gefundenen Resultaten ausgehend entwickelt der Verfasser in dieser Rücksicht eine Reihe von hübschen Sätzen, indem er zwischen Doppeltangenten erster Art d. h. solchen, die denselben Zweig der Curve zweimal berühren, und Doppeltangenten zweiter Art, welche zwei verschiedene Zweige berühren, unterscheidet. Zwei Doppeltangenten, welche die selben beiden Zweige berühren, werden ferner gleichartig oder ungleichartig genannt, je nachdem die beiden Zweige in demselben Paare von Scheitelwinkeln liegen oder nicht. Durch diese Sonderung wird eine Classification der verschiedenen Tangentenpaare und demnächst der Kegelschnittssysteme möglich. Auf die speciellen Curvenformen angewendet, ergeben die aufgestellten Sätze besonders für die vierseitige Curve die interessanten Resultate, dass die 16 reellen Doppeltangenten derselben sich alle auf die 30 der Kegelschnittssysteme vertheilen, während die übrigen 33 kein reelles Tangentenpaar enthalten. Wie aus einer späteren (p. 190-192) beigefügten Note des Herrn Zeuthen hervorgeht, haben die erlangten Resultate eine allgemeine Gültigkeit, indem sie auch noch für reelle Doppeltangenten mit imaginärer Berührung gelten, und sodann auf alle Curven \(4^{\text{ter}}\) Ordnung mit 4 oder 3 reellen Zweigen anwendbar werden, und sich ferner auf Curven mit 2, 1 oder keinem reellen Zweige oder ringförmige Curven erweitern lassen. Ferner macht Herr Zeuthen auf den Zusammenhang des gestellten Problems mit der Untersuchung der Geraden auf einer Fläche \(3^{\text{ter}}\) Ordnung aufmerksam. double tangent; fourth order curves; conics; pencil of conics Plane and space curves, Projective techniques in algebraic geometry, Cubic and quartic Diophantine equations On the distribution of double tangents in the various systems of quadruply tangent conics for some curves of the \(4^{text{th}}\) order. | 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 first edition of this extraordinary textbook on modern abstract algebra and Galois theories was published more than 25 years ago, back then in two separate volumes (Zbl 0428.12018 and Zbl 0428.30034). While Volume I (Chapters I, II, III) provided, in a dense Bourbakian style, a wealth of basic material from set theory, category theory, and commutative algebra, the subsequent Volume II (Chapters IV, V, VI) was devoted to the authors' pioneering main goal, namely to develop both algebraic Galois theory of field extensions and topological Galois theory of coverings in a parallel fashion, very much so in the spirit of A. Grothendieck's and N. Bourbaki's way of thinking in mathematics.
The book under review is the second, revised and enlarged edition of this unique, meanwhile classic text, this time appearing in one single volume. Apart from several re-arrangements, insertion, and methodological improvements, the authors have added a new chapter (Chapter 7) on \textit{A. Grothendieck's} theory of ``dessins d'enfants'' [Esquisse d'une programme, Schneps, Leila (ed.) et al., Geometric Galois actions. 1. Around Grothendieck's ``Esquisse d'un programme''. Proceedings of the conference on geometry and arithmetic of moduli spaces, Luminy, France, August 1995. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 242, 5--48; English translation: 243--283 (1997; Zbl 0901.14001)], together with an up-dating of the relevant bibliography. On the other hand, the main body of the well-established original text has been left entirely intact.
Thus, in this new edition, the material is arranged in seven chapters, each of which is subdivided into up to ten sections.
Chapter 1 provides the basics from set theory and general topology as far as needed in the sequel. This includes the axiom of choice, choice functions and the Hilbert-Bourbaki tau symbol, Zorn's lemma and its applications, Noetherian orderings, topological ultrafilters, and Tikhonov's theorem on products of compact topological spaces.
Chapter 2 introduces categories, functors and their morphisms, representable functors, projective and injective limits, adjoint functors, and -- with a view toward infinite Galois theory treated in Chapter 5 -- pro-finite spaces and profinite groups.
Chapter 3 is devoted to the fundamentals of commutative and (multi-)linear algebra. In the ten sections of this chapter, the authors discuss ring and ideal theory, various classes of rings, the linear algebra of free modules over a ring, the structure of finitely generated modules over a principal ideal domain, Noetherian rings, algebras of polynomials, tensor products of modules and algebras, graded modules, projective and injective modules, complexes, and resolutions of modules.
Chapter 4 turns to the topological aspects pursued in the sequel. Coverings, universal coverings, Galois coverings, the fundamental group, Van Kampen's theorem, graphs, and loops are the main topics of this chapter, where the interplay between the algebraic and the topological viewpoint is strongly emphasized. For instance, it is shown that the algebraic definition of the fundamental group (as the automorphism group of a certain functor on coverings) coincides with the topological definition of the Poincaré group (via loops) for locally simply connected spaces, which may be regarded as just one typical indication for both the modern style and the rather sophisticated level of the text.
Chapter 5 explains classical algebraic Galois theory in the general categorical context which has been prepared for in the foregoing chapters. Finite algebras over a field, especially diagonal algebras and étale algebras, are here the central objects of study, and the entire framework of classical algebraic Galois theory (finite and infinite) is presented in the language of those algebras and their functorial transformations. At the end, algebraic Galois theory appears as an anti-equivalence of certain categories, thereby revealing its true and very general nature in a striking way.
The analoguous theory for ramified coverings of Riemann surfaces is developed in Chapter 6. This chapter begins with generalities on Riemann surfaces and their ramified coverings, and turns then to the study of finite ramified analytic coverings by means of étale algebras. The analogy with algebraic Galois theory is established by the fundamental theorem stating that there is an anti-equivalence between the category of finite ramified analytic coverings of a connected compact Riemann surface \(B\) on the one hand, and the category of étale algebras over the meromorphic function field of \(B\) on the other. In the sequel, the authors demonstrate the power of this analogy by showing how the two Galois theories, the algebraic and the analytic-topological one, indeed clarify and enhance each other, be it by determining certain algebraic Galois groups via transcendental methods, or by investigating special automorphism groups of Riemann surfaces algebraically. In addition, this chapter also discusses triangulations of Riemann surfaces, their simplicial homology, the Riemann-Hurwitz formula, some uniformization theory, the hyperbolic geometry (or Poincaré geometry) of the plane, and pavements of the disk.
The new Chapter 7 enriches the original text by providing an introduction to \textit{G. V. Belyĭ's} theorem [On Galois extensions of a maximal cyclotomic field, Math. USSR, Izv. 14, 247--256 (1980); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 43, 267--276 (1979; Zbl 0409.12012)] and to A. Grothendieck's program ``dessins d'enfants'' [\textit{J. Oesterlé}, ``Dessins d'enfants'', Bourbaki Seminar, Volume 2001/2002, Exposés 894--908, Astérisque 290, 285--305 (2003; Zbl 1076.14040)]. Belyĭ' s theorem states that a compact Riemann surface is arithmetic, i.e. definable over an algebraic number field, if and only if it can be obtained as a covering of the Riemann sphere ramified only over the locus \(\{0,1,\infty\}\), and as such it is the starting point of Grothendieck's program, which amounts to translate, via the theory of ramified coverings of Riemann surfaces, the problem of classifying algebraic number fields into purely combinatorial problems visualized by simple drawings. In the vein of the previous chapters of the book, this approach is used to describe equivalences between various categories which, a priori, comprise objects of totally different nature: algebraic, topological, complex-analytic, and combinatorial. Finally, the authors illustrate the advanced theoretical content of this additional chapter by two instructive examples of concrete Belyĭ polynomials and actions of the mysterious profinite group \(\Aut_{\mathbb{Q}}(\mathbb{Q})\) on certain trees.
Now as before, this outstanding text presents a wealth of material from algebra, topology, complex-analytic geometry, arithmetic, and combinatorics in a unique manner. Each chapter comes with a rich amount of exercises, grouped according to the single sections of the respective chapter, and these exercises also reflect the Bourbakian style of the book. In fact, they are plentiful, far-ranging, theoretically supplementing, and often extremely challenging. Like throughout the whole text, special attention is paid to analytic examples and applications, thereby demonstrating the unity of mathematics also in its abstract categorical setting.
No doubt, this text presents mathematics at its finest. Written in a highly abstract, sophisticated, concise, rigorous and elegant style, this book is a treasury for advanced readers, who will find it extremely enlightening and inspiring. However, it appears to be less suitable as a primer for beginners in the field, who might be overstrained by just these unique features characterizing it. At any rate, this second, revised and significantly enlarged edition, of ``Algebra and Galois Theories'' by Régine and Adrien Douady is a great source for researchers, teachers, and seasoned graduate students in the field, and a highly valuable complement to the more elementary textbooks on the subject, likewise. It remains to be desired that an English edition of this outstanding book will follow in the not too far future, as this would do justice to its global significance for the mathematical community worldwide. Galois theory; rings; modules; algebras; coverings; Riemann surfaces; categories; functors; Belyi polynomials; dessins d'enfants Régine Douady and Adrien Douady, Algèbre et théories galoisiennes. 1, CEDIC, Paris, 1977 (French). Algèbre. [Algebra]. Régine Douady and Adrien Douady, Algèbre et théories galoisiennes. 2, CEDIC, Paris, 1979 (French). Théories galoisiennes. [Galois theories]. Separable extensions, Galois theory, Mathematics in general, Compact Riemann surfaces and uniformization, Transcendental field extensions, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to field theory, Special algebraic curves and curves of low genus, Coverings of curves, fundamental group, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory) Algebra and Galois theories. | 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 define a special type of hypersurface varieties inside \(\mathbb{P}^{n-1}_k\) arising from connected planar graphs and then find their equivalence classes inside the Grothendieck ring of projective varieties. Then we find a characterization for graphs in order to define irreducible hypersurfaces in general. Grothendieck ring; banana graphs; flower graphs Applications of methods of algebraic \(K\)-theory in algebraic geometry, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Graph labelling (graceful graphs, bandwidth, etc.), Structural characterization of families of graphs, Grothendieck groups and \(K_0\) Grothendieck ring class of banana and flower graphs | 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 0682.00009.]
Let C be an irreducible smooth curve of genus 3, defined over \({\mathbb{C}}\) of general moduli. To each non zero halfperiod \(\sigma\) of the jacobian of C is associated an elliptic-hyperelliptic (e.h.) curve \(\tilde X=W_ 2(C)\cdot (W_ 2(C)+\sigma)\) and the fixed point free involution \(i_{\sigma}: x\to x+\sigma\). The authors prove that \(\tilde X\) is irreducible, smooth of genus 7 and define a rational map \(\rho: R_ 3\to R_ 4^{eh}\), where \(R_ 3\) and \(R_ 4^{eh}\) are the moduli spaces of étale double covers of genus 3 curves and étale e.h. double covers of genus 4 curves.
In this paper the authors prove that \(\rho\) is birational and construct explicitly the inverse map. These results are intimately related to the question of whether \(R_ 3\) is rational or not. The paper contains also interesting remarks on the projective geometry of e.h. double covers of a curve of genus \( 4\). genus 3 curves; genus 4 curves; elliptic-hyperelliptic curve; Kummer variety; moduli spaces; étale double covers Families, moduli of curves (algebraic), \(K3\) surfaces and Enriques surfaces On a property of the Kummer variety and a relation between two moduli spaces of 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 \(V\) be a Schubert variety in some Grassmannian \(G=G(k,n)\), over the complex field. The aim of the author is to determine effective methods for computing the singular Chern classes of \(V\). These classes were introduced by MacPherson, in the homology of singular varieties \(X\), by means of Nash blows up of \(X\). An effective computation of singular Chern classes of Schubert varieties, using the definition, is difficult because the Nash blows up of \(V\) are not easy to describe.
The author uses instead the (small) resolution of singularities of Schubert varieties \(\pi: Z\to V\), introduced by Zelevinsky. Indeed, for such a small resolution, the author proves that the Chern-Mather class of \(V\) is equal to the push forward \(\pi_*(c(TZ)\cap[Z])\). \(Z\) is a smooth subvariety of a product of Grassmannian, and its class can be computed. The author then detemines two ways of computing the push forward, going through Zelevinsky's construction. Applications of the resulting formulas for understanding the positivity of singular Chern classes of Schubert varieties, are provided. Schubert varieties Jones, B. F., \textit{singular Chern classes of Schubert varieties via small resolution}, Int. Math. Res. Not. IMRN, 2010, 1371-1416, (2010) Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Classical problems, Schubert calculus Singular Chern classes of Schubert varieties via small resolution | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(f\in \mathbb{Q}[y]\) be a polynomial of degree \(n\) over the rationals. Assume \(f\) is indecomposable and consider the splitting field \(\Omega_f\) of \(f(y)-x\) over \(\mathbb{Q}(x)\). Denote the constants of \(\Omega_f\) by \(\widehat\mathbb{Q}_f\). Then, \(\widehat \mathbb{Q}_f \subset \mathbb{Q} (\zeta_n)\) where \(\zeta_n\) is a primitive \(n\)th root of 1. When \(n=p\), a prime, and \(f= x^p\) (cyclic polynomial), \(\widehat \mathbb{Q}_f= \mathbb{Q} (\zeta_p)\). When \(f=T_p\), the \(p\)th Chebyshev polynomial, \(\widehat \mathbb{Q}_f= \mathbb{Q} (\zeta_p+ \zeta_p^{-1})\). Cohen raised the following question. If \(\widehat\mathbb{Q}_f\) is nontrivial \((f\) has nontrivial extension of constants), it is then true that \(f\) is linearly equivalent over \(\overline\mathbb{Q}\) to a cyclic or Chebyshev polynomial? We show this is false for each non-square odd integer \(n\). This uses elementary group theory and the branch cycle argument. Such \(f\) also give counterexamples to a conjecture of Chowla and Zassenhaus: For all sufficiently large \(p\) (dependent on the degree of \(f)\), \(f(x)-b\) is irreducible for some \(b\in \mathbb{F}_p\). That is, we show for these particular \(f\)'s, for infinitely many \(p\), there is no \(b\in \mathbb{F}_p\) so that \(f(x)-b\) is irreducible over \(\mathbb{F}_p\). Also, for these \(p\), there is no \(b\in \mathbb{F}_p\) so that \(f(x)-b\) splits completely over \(\mathbb{Z}/p\). Further, using Müller's classification of geometric monodromy groups of polynomials we show \(n\) must be odd for such counterexamples. These are \((A_n,S_n)\) realizations by polynomials over \(\mathbb{Q}\). More delicate examples require rigidity applied to non-Galois covers. These contrast the arithmetic of covers with and without using braid operations on branch cycle descriptions. Braid operations describe four families of covers that include the renowned Davenport polynomials of degree 7. Chowla-Zassenhaus conjecture; cyclic polynomial; \(p\)th Chebyshev polynomial; extension of constants; branch cycle; Davenport polynomials Fried M D. Extension of Constants, Rigidity, and the Chowla-Zassenhaus Conjecture. Finite Fields Appl, 1995, 1: 326--359 Galois theory, Separable extensions, Galois theory, Coverings in algebraic geometry, Arithmetic theory of algebraic function fields Extension of constants, rigidity, and the Chowla-Zassenhaus conjecture | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We describe the possible linear actions of a cyclic group \(G=\mathbb{Z}/n\) on a complex torus, using the cyclotomic exact sequence for the group algebra \(\mathbb{Z}[G]\). The main application is devoted to a structure theorem for Bagnera-De Franchis Manifolds (these are the quotients of a complex torus by a free action of a cyclic group), but we also give an application to hypergeometric integrals, namely, we describe the intersection product and Hodge structures for the homology of fully ramified cyclic coverings of the projective line. complex tori; hyperelliptic manifolds; Bagnera-De Franchis manifolds; Hodge structures; cyclic coverings; group algebra; factorial rings; cyclotomic rings; resultants of cyclotomic polynomials; fundamental groups Kähler manifolds, Automorphism groups of \(\mathbb{C}^n\) and affine manifolds, Classification theorems for complex manifolds, Abelian varieties and schemes, Families, fibrations in algebraic geometry, Congruences; primitive roots; residue systems, Cyclotomic extensions, Structure, classification theorems for modules and ideals in commutative rings Cyclic symmetry on complex tori and Bagnera-De Franchis 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 We give a new proof to V. B. Mehta and A. Ramanathan's theorem that the Schubert subschemes in a flag scheme are all simultaneously compatible split, using the representation theory of infinitesimal algebraic groups. In particular, the present proof dispenses with the Bott-Samelson schemes. Schubert subschemes; flag scheme; representation theory of infinitesimal algebraic groups Kaneda, M., On the Frobenius morphism of flag schemes, Pacific J. Math., 163, 315-336, (1994) Grassmannians, Schubert varieties, flag manifolds, Other algebraic groups (geometric aspects) On the Frobenius morphism of flag schemes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors define a category of planar diagrams whose Grothendieck group contains an integral version of the infinite rank Heisenberg algebra, thus yielding a categorification of this algebra. This category, which is a \(q\)-deformation of one defined by Khovanov, acts naturally on the categories of modules for Hecke algebras of type \(A\) and finite general linear groups. In this way, they obtain a categorification of the bosonic Fock space. They also develop the theory of parabolic induction and restriction functors for finite groups and prove general results on biadjointness and cyclicity in this setting. Heisenberg algebras; Hecke algebras; planar diagrammatics; finite general linear groups; Grothendieck groups; categorification A. Licata & A. Savage, ``Hecke algebras, finite general linear groups, and Heisenberg categorification'', Quantum Topol.4 (2013) no. 2, p. 125-185 Hecke algebras and their representations, Infinite-dimensional Lie (super)algebras, Module categories in associative algebras, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Deformations of associative rings Hecke algebras, finite general linear groups, and Heisenberg 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 Based on Pieri's formula on Schubert varieties [see \textit{F. Sottile}, Can. J. Math. 49, 1281-1298 (1997; Zbl 0933.14031)], the Pieri homotopy algorithm was first proposed by \textit{B. Huber, F. Sottile}, and \textit{B. Sturmfels} [J. Symb. Comput. 26, 767-788 (1998; Zbl 1064.14508)]\ for numerical Schubert calculus to enumerate all \(p\)-planes in \({\mathbb C}^{m+p}\) that meet \(n\) given planes in general position. The algorithm has been improved by \textit{B. Huber} and \textit{J. Verschelde} [SIAM J. Control Optim. 38, 1265-1287 (2000; Zbl 0955.14038)]\ to be more intuitive and more suitable for computer implementations.
A different approach of employing the Pieri homotopy algorithm for numerical Schubert calculus is presented in this paper. A major advantage of our method is that the polynomial equations in the process are all square systems admitting the same number of equations and unknowns. Moreover, the degree of each polynomial equation is always 2, which warrants much better numerical stability when the solutions are being solved. Numerical results for a big variety of examples illustrate that a considerable advance in speed as well as much smaller storage requirements have been achieved by the resulting algorithm. enumerative geometry; Schubert variety; Pieri formula; Pieri homotopy algorithm; Pieri poset; algorithm Li D, Qi L, Zhou S (2002) Descent directions of quasi-Newton methods for symmetric nonlinear equations. SIAM J Numer Anal 40(5): 1763--1774 Grassmannians, Schubert varieties, flag manifolds, Computational aspects of higher-dimensional varieties, Numerical computation of solutions to systems of equations, Symbolic computation and algebraic computation, Enumerative problems (combinatorial problems) in algebraic geometry Numerical Schubert calculus by the Pieri homotopy algorithm | 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 Grothendieck local residue symbol
\[
\text{Res}_0 \left({gdx_0 \wedge \cdots \wedge dx_n \over f_0\dots f_n} \right)= {1\over (2\pi i)^{n+1}} \int_{| f_i| = \varepsilon} {gdx_0 \wedge\dots \wedge dx_n \over f_0\dots f_n}
\]
[see \textit{P. Griffiths} and \textit{J. Harris}, ``Principles of algebraic geometry'' (1978; Zbl 0408.14001 or 1994; Zbl 0836.14001); chapter 5] is defined whenever \(g,f_0,\dots,f_n\) are holomorphic in a neighborhood of \(0\in \mathbb{C}^{n+1}\) and \(f_0, \dots, f_n\) do not vanish simultaneously except at 0. \textit{C. Peters} and \textit{J. Steenbrink} [in: Classification of algebraic and analytic varieties, Proc. Symp., Katata 1982, Prog. Math. 39, 399-463 (1983; Zbl 0523.14009)] observed that when \(f_0, \dots, f_n\) are homogeneous of degree \(d\) and \(g\) is homogeneous of degree \(\rho= (n+1) (d-1)\), the residue symbol has the following nice properties:
Quotient property. The map \(g \mapsto\text{Res}_0 \left( {g dx_0 \wedge\dots \wedge dx_n \over f_0\dots f_n} \right)\) induces an isomorphism
\[
\mathbb{C} [x_0, \dots, x_n]_\rho/ \langle f_0,\dots, f_n\rangle_\rho \simeq \mathbb{C}
\]
(the subscript refers to the graded piece in degree \(\rho)\) uniquely characterized by the fact that the Jacobian determinant \(J=\text{det} (\partial f_i/ \partial x_j)\) maps to \(d^{n+1}\).
Trace property. Čech cohomology gives a cohomology class \([\omega_g] \in H^n (\mathbb{P}^n, \Omega^n_{\mathbb{P}^n})\) such that under the trace map \(\text{Tr}_{\mathbb{P}^n}: H^n (\mathbb{P}^n, \Omega^n_{\mathbb{P}^n}) \simeq \mathbb{C}\), we have \(\text{Res}_0 \left({g dx_0 \wedge \dots \wedge dx_n \over f_0\dots f_n} \right)= \text{Tr}_{\mathbb{P}^n} ([\omega_g])\).
In this paper, we will show how these properties of residues can be generalized to an arbitrary projective toric variety. The paper is organized into six sections as follows. In \(\S 1\), we define the cohomology class \([\omega_g]\in H^n(\mathbb{P}^n, \Omega^n_{\mathbb{P}^n})\), and then \(\S 2\) generalizes this to define toric residues in terms of a toric analog of the trace property. We recall some commutative algebra associated with toric varieties in \(\S 3\), and \(\S 4\) introduces a toric version of the Jacobian. In \(\S 5\), we show that the toric residue is uniquely characterized using a toric analog of the quotient property. Then \(\S 6\) explores different ways of representing the toric residue as an integral, and an appendix discusses the relation between the trace map and the Dolbeault isomorphism. Grothendieck local residue symbol; toric variety; cohomology class; Dolbeault isomorphism D. Cox, ''Toric Residues,'' Ark. Mat. 34(1), 73--96 (1996). Toric varieties, Newton polyhedra, Okounkov bodies, Residues for several complex variables, Classical real and complex (co)homology in algebraic geometry Toric 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 Étant donné \(g,k\geq0\) et une partition \(\mu=(m_{1},\dots,m_{n})\) de \(k(2g-2)\), on définit l'espace \(\mathcal{H}_{g}^{k}(\mu)\subset \mathcal{M}_{g,n}\) paramètrant les \((C;p_{1},\dots,p_{n})\) tels que \(\mathcal{O}_{C}(\sum m_{i}p_{i})\) est le fibré canonique de \(C\). \textit{G. Farkas} et \textit{R. Pandharipande} [J. Inst. Math. Jussieu 17, No. 3, 615--672 (2018; Zbl 1455.14056)] ont introduit pour \(k=1\) un espace des modules propre paramétrisant les diviseurs twistés canoniques contenant \(\mathcal{H}_{g}^{1}(\mu)\). Cet article introduit et étudie la généralisation naturelle de ces espaces \(\tilde{\mathcal{H}}_{g}^{k}(\mu)\) à tout \(k\geq 0\).
La dimension des composantes irréductibles de \(\tilde{\mathcal{H}}_{g}^{k}(\mu)\) est obtenue en calculant la dimension des composantes connexes de \(\mathcal{H}_{g}^{k}(\mu)\), c.f. théorème 1.1 et proposition 1.2. Notons que cette question à été abordé par de nombreux auteurs à différents niveaux de généralité et on pourra consulter la remarque 1.3 pour un panorama sur cette question.
Cet article donne une formule conjecturale entre la classe fondamentale de \(\tilde{\mathcal{H}}_{g}^{k}(\mu)\) et un élément explicite de l'anneau tautologique de la compactification de Deligne-Mumford \(\bar{\mathcal{M}}_{g,n}\), voir conjectures A et A'. Cette conjecture est prouvée dans quelques cas en petit genre.
Enfin, l'auteur montre que les pluridifférentielles stables dont la restriction à chaque composante est non nulle est lissable sans changer les ordres des zéros et des pôles de ces restrictions. Notons que c'est un cas particulier du résultat principal de l'article de \textit{M. Bainbridge} et al. [``Strata of \(k\)-differentials'', Preprint, \url{arXiv:1610.09238}]. strata of \(k\)-differentials; deformation theory; tautological classes; double ramification cycles Schmitt, J., Dimension theory of the moduli space of twisted \textit{k}-differentials Families, moduli of curves (algebraic), Differentials on Riemann surfaces Dimension theory of the moduli space of twisted \(k\)-differentials | 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 Fano threefold \(X\) is a quartic double solid if it is a double cover of the \(3\)-dimensional projective space, branched over a nodal quartic surface. The problem of the rationality of these threefolds, depending on the number and the nature of their singularities, has been considered by many authors. It is known that, if \(X\) is smooth or has at most \(4\) nodes, then it is irrational. Other results of irrationality, for up to \(7\) nodes, have been proved only under generality assumptions. On the other hand, it is known that if \(X\) has \(15\) or \(16\) nodes, then it is rational.
In this interesting and well written article, the aim of the authors is to remove the generality assumptions. Their main results are the following. Let \(X\) be any nodal quartic double solid with \(k\) nodes. If \(k\leq 6\), then \(X\) is irrational. If \(k\geq 11\), then \(X\) is rational.
Their approach is based on a study of the property of \(\mathbb Q\)-factoriality of \(X\), that in this case is equivalent to the coincidence of Weil and Cartier divisors. They exhibit an explicit example of a nodal quartic surface \(S\) with \(6\) nodes, and they prove that if \(X\) is a double cover of \(\mathbb P^3\) branched over \(S\), then \(X\) is not \(\mathbb Q\)-factorial, and it is irrational if and only if it has exactly \(6\) nodes. They prove that this is the only example of a non \(\mathbb Q\)-factorial irrational quartic double solid. Their main result is obtained thanks to the fact, due to Clemens, that \(X\) is \(\mathbb Q\)-factorial if and only if the nodes of the surface \(S\) impose independent conditions to the quadrics.
The authors also propose the following conjecture: Every \(\mathbb Q\)-factorial nodal quartic solid is irrational. double cover; quartic surface; intermediate Jacobian; rational threefold; conic bundle Rationality questions in algebraic geometry, Rational and unirational varieties, \(3\)-folds Which quartic double solids are 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 Consider two families of cubic surfaces with homogeneous coordinates \((x : y : z : w)\) in \(\mathbb{P}^3\). The first is given by the equation
\[
x^3 - 2yz^2 - y^2w + x(s_1 w^2 + s_2zw + s_3 z^2) + s_4 w^3 + s_5 zw^2 + s_6 z^2w = 0
\]
over the six- dimensional parameter space \(S\) with coordinates \(s_i\), \(i = 1, \ldots, 6\). It is not difficult to see that the restriction of the family to the hyperplane \(\{w = 1\}\) is nothing but the minimal versal deformation of the simple \(E_6\)-singularity whose normal form \(f = x^3 + y^4 + z^2\) can be rewritten also as \(f' = x^3 - 2yz^2 - y^2\) by means of an elementary change of variables. -- The second family is defined by the equation
\[
\begin{multlined} \rho w \{\lambda x^2 + \mu y^2 + \nu z^2 + (\rho - 1)^2 (\lambda \mu \nu \rho - 1)^2 w^2 +\\ + (\mu \nu + 1) yz + (\lambda \nu + 1) xz + (\lambda \mu + 1) xy -\\ - (\rho - 1) (\lambda \mu \nu \rho - 1) w [(\lambda + 1) x + (\mu + 1) y + (\nu + 1) z]\} + xyz = 0\end{multlined}
\]
over the four-dimensional parameter space \(\Lambda\) with coordinates \(\lambda, \mu, \nu, \rho\). In fact, this family is a modified Cayley family [see \textit{I. Naruki} and the author, Proc. Jap. Acad., Ser. A 56, 122-125 (1980; Zbl 0472.14018)] containing all nonsingular cubics in \(\mathbb{P}^3\). Since the moduli space of the cubic surfaces is also four-dimensional, there is a map \(\Psi : S \to \Lambda\) that corresponds to transformations of the first equation to the form of the second one. Of course, the map \(\Psi\) is multivalued. The author explicitly constructs a covering space \(\widetilde S \to S\) admitting a linear \(W(E_6)\)-action, where \(W(E_6)\) is the Weyl group of type \(E_6\). He then proves that the induced map \(\Phi : \widetilde S \to \Lambda \) is \(W(E_6)\)-equivariant. Hence \(\Phi\) defines a single-valued map to \(\Lambda\). The construction is based on observations due to \textit{M. Yoshida} and \textit{B. Hunt} concerning the realization of \(W(E_6)\) as a group of birational transformations of the configuration space of 6 points in \(\mathbb{P}^2\) as well as the standard affine space \(\mathbb{C}^4\). Although the essential part of the proof has computational character and is partially performed with the help of a computer the author writes out all resulting formulas in a highly compact form. This enables him to give a clear interpretation of several earlier results by \textit{I. Naruki} [Proc. Lond. Math. Soc., III. Ser. 45, 1-30 (1982; Zbl 0508.14005)] and \textit{T. Terada} [J. Math. Soc. Japan 35, 451-475 (1983; Zbl 0506.33001)], and some others. cross ratio variety; rational double point; Weyl groups; Appell- Lauricella hypergeometric function; deformation of \(E_ 6\)-singularity; cubic surface; Cayley family J. Sekiguchi: The versal deformation of the \(E_6\)-singularity and a family of cubic surfaces , J. Math. Soc. Japan 46 (1994), 355--383. Families, moduli, classification: algebraic theory, Deformations of singularities, Appell, Horn and Lauricella functions, Singularities of curves, local rings, Complex surface and hypersurface singularities The versal deformation of the \(E_ 6\)-singularity and a family of cubic surfaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(\gamma: C \to X\) be a double cover of smooth curves of respective genera \(\pi\) and \(g\). Then every pencil of degree \(d \leq \pi - 2g\) on \(C\) is induced from a pencil on \(X\) by pull-back, thanks to the Castelnuovo-Severi inequality [\textit{E. Kani}, J. Reine Angew. Math. 352, 24--70 (1984; Zbl 0536.14016)]. In the paper under review, the author fixes a smooth curve \(X\) of genus \(g\), \(\pi \geq 3g\), \(d > \pi-2g\) and constructs a double cover \(C \to X\) as above and a basepoint-free linear system on \(C\) which is not a pull-back from \(X\). Previously this was known only when the stronger inequality \(\pi \geq 4g+5\) holds [\textit{E. Ballico, C. Keem} and \textit{S. Park}, Proc. Am. Math. Soc. 132, 3153--3158 (2004; Zbl 1056.14041)]. The range \(2g < \pi \leq 3g\) and \(d \geq \pi-2g+1\) remains open.
Given \(X\), \(\pi \geq 3g\) and \(d>\pi-2g\) as above, the author draws on his work with \textit{M. Pedreira} [Note Mat. 24, 25--63 (2005; Zbl 1150.14008); Math. Nachr. 278, 240--257 (2005; Zbl 1067.14030)] to construct the curve \(C\) on a decomposable ruled surface \(S \to X\) with prescribed branch points. The pencils constructed are restrictions of pencils on \(S\). The paper closes with inequalities on the Clifford index and gonality of a double covers \(C \to X\): \({\roman {gon}} (C) \geq \min\{ 2{\roman {gon}}(X), \pi -2g + 1\}\) and \({\roman {Cliff}} (C) \geq \min\{2 {\roman {gon}}(X) -2, \pi-2g+1\}\). double covers; basepoint-free pencils Luis Fuentes García, Pencils on double coverings of curves, Arch. Math. (Basel) 92 (2009), no. 1, 35 -- 43. Special divisors on curves (gonality, Brill-Noether theory), Coverings of curves, fundamental group, Rational and ruled surfaces Pencils on double coverings of 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 A conjecture of \textit{M. Finkelberg} and \textit{A. Ionov} [Bull. Inst. Math., Acad. Sin. (N.S.) 13, No. 1, 31--42 (2018; Zbl 1397.05203)] is proved on the basis of a generalization of the Springer resolution and the Grauert-Riemenschneider vanishing theorem. As a corollary, it is proved that the coefficients of the multivariable version of Kostka functions introduced by Finkelberg and Ionov are nonnegative. Kostka-Shoji polynomials; cohomology vanishing; quivers; Lusztig convolution diagram Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Symmetric functions and generalizations, Representations of quivers and partially ordered sets, Vanishing theorems in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds Higher cohomology vanishing of line bundles on generalized Springer resolution | 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 a two-variable invertible polynomial, we show that its category of maximally-graded matrix factorisations is quasi-equivalent to the Fukaya-Seidel category of its Berglund-Hübsch transpose. This was previously shown for Brieskorn-Pham and \(D\)-type singularities by Futaki-Ueda. The proof involves explicit construction of a tilting object on the B-side, and comparison with a specific basis of Lefschetz thimbles on the A-side. Fukaya-Seidel category; pretriangulated \(A_\infty\)-categories; derived category of singularities; Brieskorn-Pham polynomials Symplectic aspects of mirror symmetry, homological mirror symmetry, and Fukaya category, Mirror symmetry (algebro-geometric aspects) Homological Berglund-Hübsch mirror symmetry for curve 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 In the torus equivariant \(K\)-theory ring of a Grassmannian the classes of the structure sheaves of Schubert varieties form a natural, geometric basis. Understanding the structure constants with respect to this basis is equivalent to describing the ring structure. In an earlier work the authors presented the structure constants as the cardinalities of certain combinatorially defined tableaux. In the paper under review the authors show a bijection from those tableaux to certain puzzles (fillings of a triangle with certain permitted puzzle pieces). As a result they obtain a (mild modification of the) puzzle rule for the structure constants as the number of puzzles, originally conjectured by Knutson and Vakil.
For part I, see [\textit{O. Pechenik} and \textit{A. Yong}, Forum Math. Pi 5, Article ID e3, 128 p. (2017; Zbl 1369.14060)]. Grassmannians; equivariant \(K\)-theory; puzzle; Schubert calculus; Littlewood-Richardson rule O. Pechenik and A. Yong. ''Equivariant K-theory of Grassmannians II: The Knutson-Vakil conjecture''. 2015. arXiv:1508.00446. Grassmannians, Schubert varieties, flag manifolds, Symmetric functions and generalizations, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] Equivariant \(K\)-theory of Grassmannians. II: The Knutson-Vakil conjecture. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The \(q\)-Kostka polynomials \(K_{\lambda\mu}(q)\) expressing the Schur function \(S_{\lambda}(\mathbf{x})\) as a linear combination of Hall-Littlewood polynomials \(P_{\mu}(\mathbf{x},q)\) have been an object of intensive study in the last two decades at the crossroads of combinatorics, algebra and geometry. The topic of the paper under review comes from the work of Lascoux and Schützenberger presenting the variant polynomial \(\widetilde K_{\lambda\mu}(q)=q^{n(\mu)}K_{\lambda\mu}(1/q)\) as a sum of the atom polynomials \(R_{\lambda\nu}(q)\), \(\nu\geq\mu\), where \(R_{\nu\mu}(q)\) themselves have non-negative coefficients. The main purpose of the paper is to give a geometric interpretation of the atomic decomposition in the language of scheme-theoretic intersections of nilpotent orbit varieties introduced by Kraft and De Concini-Procesi in the early 80's. In particular, involving a recent result of Broer, and as a consequence of their approach, the authors obtain a new proof of the atomic decomposition of the \(q\)-Kostka polynomials. Kostka polynomials; atomic decomposition; nilpotent conjugacy classes; nilpotent orbit varieties William Brockman and Mark Haiman, Nilpotent orbit varieties and the atomic decomposition of the \?-Kostka polynomials, Canad. J. Math. 50 (1998), no. 3, 525 -- 537. Combinatorial aspects of representation theory, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Special varieties, Representation theory for linear algebraic groups Nilpotent orbit varieties and the atomic decomposition of the \(q\)-Kostka 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 studies relation between equivariant \(K\)-theories of the framed \(A_n\)-type quiver variety \(X_n\) and of the rank \(r\) ADHM moduli space \(M_r= \bigsqcup_{k \ge 0}M_{r,k}\), where \(k\) denotes the instanton number. The \(K\)-theories considered are the equivariant quantum \(K\)-theory \(H_n:=K_{T^{n+2}}(QM(\mathbb{P}^1,X_n))\) and the equivariant \(K\)-theory \(K_{r,k}:=K_{T^2}(M_{r,k})\), where \(T\) denotes the one-dimensional torus.
The main results are Theorem 3.3 and Theorem 4.6, where an embedding \(\bigoplus_{l=0}^k K_{r,l} \hookrightarrow H_{n r}\) is constructed. The embedding is designed to map the \(T^2\)-fixed point classes in \(K_{r,l}\) to the coefficients of vertex functions in \(H_{n r}\). The construction is done by explicit calculations using Macdonald symmetric polynomials.
The paper also proposes an interesting Conjecture 5.4 which relates the eigenvalues of quantum multiplication operators in \(K\)-theory of \(M_r\) with those of the elliptic Ruijsenaars-Schneider model. quiver varieties; ADHM moduli; equivariant \(K\)-theory; quantum equivariant \(K\)-theory; Macdonald symmetric polynomials Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), String and superstring theories; other extended objects (e.g., branes) in quantum field theory, Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.) A-type quiver varieties and ADHM moduli 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 Let \(\pi:X \to S\) be a finite flat morphism of schemes over a field of characteristic \(p\). In this paper, the author gives necessary and sufficient conditions for this morphism to be dominated by a map \(g:Y \to S\); i.e., there exist maps \(g:Y \to S\) and \(h:Y \to X\) such that \(g=\pi h\) and \(g\) and \(h\) are finite Galois covers with groups \(G\) and \(H\) respectively. One has to assume that the morphism \(\pi\) is finite differentially homogeneous (fdh), as studied by \textit{P. J. Sancho de Salas} [J. Algebra 221, No. 1, 279--292 (1999; Zbl 0971.14004)]. These are twisted forms of certain finite \(\mathbb{F}_p\)-schemes in the flat topology by Proposition 1.12. The main theorem of the paper, Theorem 2.11, states that an fdh morphism \(\pi\) as above can be dominated by a torsor \(g:Y \to S\) if and only if it is \(F\)-constant; i.e., if the pull-back of \(\pi\) over some iterate of the absolute Frobenius becomes isomorphic to \(\Sigma^\nu\) for a tuple \(\nu=(\nu_1, \ldots, \nu_r)\) of nonnegative integers, which is defined as the prime spectrum of \(\mathbb{F}_p[t_1, \ldots, t_r]/(t_1^{p^{\nu_1}}, \ldots, t_r^{p^{\nu_r}})\). In section 3, the author proves that under suitable conditions, \(F\)-constant morphisms are the same as essentially finite morphisms as studied by \textit{M. Antei} and \textit{M. Emsalem} [J. Pure Appl. Algebra 215, No. 11, 2567--2585 (2011; Zbl 1264.14024)]. Section 4 includes applications to fundamental group schemes. torsors; fundamental groups; Grothendieck topologies Group schemes, Étale and other Grothendieck topologies and (co)homologies On the ``Galois closure'' for finite morphisms | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We construct for a given arbitrary skew diagram \({\mathcal{A}}\) all partitions \(v\) with maximal principal hook lengths among all partitions with \([v]\) appearing in \([{\mathcal{A}}]\). Furthermore, we show that these are also partitions with minimal Durfee size. We use this to give the maximal Durfee size for \([v]\) appearing in \([{\mathcal{A}}]\) for the cases when \({\mathcal{A}}\) decays into two partitions and for some special cases of \({\mathcal{A}}\). We also deduce necessary conditions for two skew diagrams to represent the same skew character. principal hook lengths; Durfee size; skew characters; symmetric group; skew Schur functions; Schubert Calculus Gutschwager, C.: On principal hook length partitions and durfee sizes in skew characters. Annals Comb. (to appear). arXiv:0802.0417v2 Symmetric functions and generalizations, Combinatorial aspects of representation theory, Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Representations of finite symmetric groups On principal hook length partitions and Durfee sizes in skew characters | 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 define and study odd analogues of classical geometric and combinatorial objects associated to permutations, namely odd Schubert varieties, odd diagrams, and odd inversion sets. We show that there is a bijection between odd inversion sets of permutations and acyclic orientations of the Turán graph, that the dimension of the odd Schubert variety associated to a permutation is the odd length of the permutation, and give several necessary conditions for a subset of \([ n ] \times [ n ]\) to be the odd diagram of a permutation. We also study the sign-twisted generating function of the odd length over descent classes of the symmetric groups. permutation; generating function; descent class; diagram; odd length; Schubert variety Permutations, words, matrices, Exact enumeration problems, generating functions, Reflection and Coxeter groups (group-theoretic aspects), Grassmannians, Schubert varieties, flag manifolds, Symmetric groups Odd length: odd diagrams and descent classes | 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 is arranged in the following way. In \S1 we introduce notations for Schubert cells and we will show what cells and their neighbourhoods give rise to a uniquely constructed Schubert stratification.
In \S2 we show how a flattening corresponds to a Schubert cell.
In \S3 we introduce the concept of stable equivalence of flattenings, which allows us to compare cascades that generally consist of a different number of curves lying in spaces of different dimensions.
The relation of equivalence of flattenings is constructed in \S4.
In \S5 we give a classification of the flattenings occurring in generic three-parameter families of cascades; relative to this equivalence, we study the singularities of their bifurcation diagrams, and give results of V. I. Arnol'd and O. P. Shcherbak on the connection of these singularities with the geometry of the swallowtail, tangential singularities, and the singularities of projections.
In \S6 we give lists of the simple flattenings of curves, and also of cascades, corresponding to complete flags. The methods used in the proof of the classification theorems of \S\S5 and 6 are validated in \S\S7-9. In \S7 we prove a generalization of the Frobenius theorem on integrable distributions to the case of distributions with singularities. In \S8 this result is carried over to the case of the infinite-dimensional space of germs of cascades. Using these results, we prove the finite determinacy and versality theorems in \S9.
Section 10 and 11 are devoted to applications of the theory of flattenings to the study of oscillatory properties of linear differential equations and to the decomposition of Weierstrass points of algebraic curves, respectively. Grassmannians; flag manifolds; Schubert cells; Schubert stratification; flattening; equivalence; singularities; bifurcation; Weierstrass points; algebraic curves DOI: 10.1070/RM1991v046n05ABEH002844 Deformations of complex singularities; vanishing cycles, Formal methods and deformations in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Differentiable maps on manifolds, Theory of singularities and catastrophe theory Flattenings of projective curves, singularities of Schubert stratifications of Grassmannians and flag varieties, and bifurcations of Weierstrass points of 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 Let \(M\) be a compact oriented 4-manifold. If \(\omega \) is a symplectic form on \(M\), one might well ask whether the space of such forms is connected. In fact, it is not difficult to construct examples where the answer is negative. A more subtle question, however, is whether the group of orientation-preserving diffeomorphisms \(M\to M\) acts transitively on the set of connected components of the orientation-compatible symplectic structures. As was recently pointed out by \textit{C. T. McMullen} and \textit{C. H. Taubes} [Math. Res. Lett. 6, 681-696 (1999; Zbl 0964.53051)], there are 4-manifolds \(M\) for which this subtler question also has a negative answer. The purpose of the present note is to point out many examples of this interesting phenomenon arise from certain complex surfaces with Kodaira fibrations.
A Kodaira fibration is by definition a holomorphic submersion \(f:M\to B\) from a compact complex surface to a compact complex curve, with base \(B\) and fiber both of genus \(\geq 2\). Let \(\psi\) denote Kähler form on \(M\). Let \(\varphi\) be any area form on \(B\), compatible with its complex orientation, and for \(\varepsilon >0\) consider the closed 2-form \(\omega=\varepsilon \psi -f^{*}\varphi\). Thus, for sufficiently small \(\varepsilon >0\), \(\omega\) is a symplectic form for the reverse-oriented 4-manifold \(\overline{M}\).
Definition. Let \(M\) be a complex surface equipped with two Kodaira fibrations \(f_j:M\to B_j\), \(j=1,2\). Let \(g_j\) denote the genus of \(B_j\), and suppose that the induced map \(f_1\times f_2: M\to B_1\times B_2\) has degree \(r>0\). We will then say that \((f_1,f_2)\) is a Kodaira double-fibration of \(M\) if the signature \(\tau(M)\neq 0\) and \((g_2-1)\nmid r(g_1-1)\).
Given a Kodaira double-fibered surface \((M,f_1,f_2)\) we now have two different symplectic structures \(\omega_1\) and \(\omega_2\) on \(\overline{M}\).
Theorem. Let \((M,f_1,f_2)\) be any Kodaira double-fibered complex surface. Then for any diffeomorphism \(\Phi:M\to M\), the symplectic structures \(\omega_1\) and \(\pm\Phi^{*}\omega_2\) are deformation inequivalent. symplectic manifold; complex surface; Seiberg-Witten invariant; Kodaira double-fibration C LeBrun, Diffeomorphisms, symplectic forms and Kodaira fibrations, Geom. Topol. 4 (2000) 451 Global theory of symplectic and contact manifolds, Applications of global analysis to structures on manifolds, Surfaces of general type Diffeomorphisms, symplectic forms and Kodaira fibrations | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials \textit{P. van Wamelen} [Math. Comput. 68, No. 225, 307--320 (1999; Zbl 0906.14025)] lists 19 curves of genus two over \(\mathbb{Q}\) with complex multiplication (CM). However, for each curve, the CM-field turns out to be cyclic Galois over \(\mathbb{Q}\), and the generic case of a non-Galois quartic CM-field did not feature in this list. The reason is that the field of definition in that case always contains the real quadratic subfield of the reflex field.
We extend Van Wamelen's list to include curves of genus two defined over this real quadratic field. Our list therefore contains the smallest `generic' examples of CM curves of genus two.
We explain our methods for obtaining this list, including a new height-reduction algorithm for arbitrary hyperelliptic curves over totally real number fields. Unlike van Wamelen, we also give a proof of our list, which is made possible by our implementation of denominator bounds of \textit{K. Lauter} and \textit{B. Viray} [Am. J. Math. 137, No. 2, 497--533 (2015; Zbl 1392.11033)] for Igusa class polynomials. reflex field; CM curves; Igusa class polynomials Bouyer, F., Streng, M.: Examples of CM curves of genus two defined over the reflex field. LMS J. Comput. Math. (to appear). arXiv:1307.0486 Complex multiplication and moduli of abelian varieties, Elliptic curves over global fields, Complex multiplication and abelian varieties, Elliptic curves over local fields Examples of CM curves of genus two defined over the reflex field | 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 part I see \textit{A. Szenes}, ibid. 587-597 (1993; see the preceding review).]
Let \({\mathcal M}^ C_ 0\) be the moduli space of semi-stable rank 2 vector bundles with fixed even degree determinant on a curve \(C\), and denote by \(L_ 0\) the generator of \(\text{Pic} {\mathcal M}^ C_ 0\). The authors prove the following formula
\[
\dim H^ 0 ({\mathcal M}^ C_ 0, L_ 0^{k-2}) = \sum^{k-1}_{J = +} (k/1 - \cos 2j \pi/k)^{g-1}.
\]
To prove this formula the authors use the Hecke correspondence between the two moduli spaces \({\mathcal M}^ C_ 0\) and \({\mathcal M}^ C_ 1\) to transfer the calculation from \({\mathcal M}^ C_ 0\) to \({\mathcal M}^ C_ 1\). Hilbert polynomials; moduli space of semi-stable rank 2 vector; Hecke correspondence A. Bertram and A. Szenes, ''Hilbert polynomials of moduli spaces of rank-\(2\). Vector bundles. II,'' Topology, vol. 32, iss. 3, pp. 599-609, 1993. Algebraic moduli problems, moduli of vector bundles, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series Hilbert polynomials of moduli spaces of rank 2 vector bundles. II | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials If the \(\ell\)-adic cohomology of a projective smooth variety, defined over a \(\mathfrak p\)-adic field \(K\) with finite residue field \(k\), is supported in codimension \(\geq 1\), then any model over the ring of integers of \(K\) has a \(k\)-rational point. \(\ell\)-adic cohomology; Grothendieck topologies; \(\mathfrak p\)-adic fields Hélène Esnault, Coniveau over \?-adic fields and points over finite fields, C. R. Math. Acad. Sci. Paris 345 (2007), no. 2, 73 -- 76 (English, with English and French summaries). Étale and other Grothendieck topologies and (co)homologies, Rational points, Finite ground fields in algebraic geometry Coniveau over \(\mathfrak p\)-adic fields and points over finite 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 prove that the Lam-Shimozono ``down operator'' on the affine Weyl group induces a derivation of the affine Fomin-Stanley subalgebra. We use this to verify a conjecture of \textit{C. Berg} et al. [Electron. J. Comb. 19, No. 2, Research Paper P55, 20 p., electronic only (2012; Zbl 1253.05138)] describing the expansion of non-commutative \(k\)-Schur functions of ``near rectangles'' in the affine nilCoxeter algebra. Consequently, we obtain a combinatorial interpretation of the corresponding \(k\)-Littlewood-Richardson coefficients. symmetric functions; \(k\)-Schur functions; affine Schubert calculus; dual graded graphs; affine nilCoxeter algebra; \(k\)-cores Berg, Chris; Saliola, Franco; Serrano, Luis, The down operator and expansions of near rectangular \(k\)-Schur functions, J. Combin. Theory Ser. A, 120, 3, 623-636, (2013) Symmetric functions and generalizations, Classical problems, Schubert calculus, Reflection and Coxeter groups (group-theoretic aspects) The down operator and expansions of near rectangular \(k\)-Schur 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 We consider a polynomial \(f(x) = x^{2^{2r} - 2^r + 1} + g(x)\) over \(\mathbb {F}_q\) of Kasami degree where degree \(d\) of \(g(x)\) is \(< 2^{2r } - 2^r + 1\). We prove that if \(d \equiv 3 \bmod {4}\), then the function \(f(x)\) is not APN on infinitely many extensions of \(\mathbb F _q\). We also obtain obtain partial results in the case where \(d = 2^it\) with \(t \equiv 1 \bmod {4}\). APN functions; finite fields; absolute irreducible polynomials Polynomials over finite fields, Polynomials in general fields (irreducibility, etc.), Computational aspects of algebraic surfaces, Algebraic coding theory; cryptography (number-theoretic aspects) A infinite class of Kasami functions that are not APN infinitely often | 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 prove the polynomiality of the cycle valued twisted Gromov-Witten invariants for root of orbifold line bundles. They produce a formula for DR-cycles with orbifold targets. The authors apply their results to study the relation between relative and absolute orbifold Gromov-Witten invariants. root of orbifold line bundles; polynomiality; double ramification cycle; absolute/relative orbifold Gromov-Witten theory Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Gromov-Witten invariants, quantum cohomology, Frobenius manifolds Double ramification cycles with orbifold targets | 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 algebraic group with Weyl group \((W, S)\) and let \(w \in W\). We consider the descent set \(D(w) = \{ s \in S \mid l(ws) < l(w)\}\). This has been generalized to the situation of the Bruhat poset \(W^J\), where \(J \subset S\). To do this one identifies a certain subset \(S^J \subset W^J\) that plays the role of \(S \subset W\) in the well known case \(J = \emptyset\). One ends up with the descent system \((W^J, S^J)\). On the other hand, each subset \(J \subset S\) determines a projective, simple \(G \times G\)-embedding \(\mathbb{P}(J)\) of \(G\). The case where \(J = \emptyset\) is closely related to the wonderful embedding. One obtains a complete list of all subsets \(J \subset S\) such that \(\mathbb{P}(J)\) is a rationally smooth algebraic variety. In such cases, we determine the Betti numbers of \(\mathbb{P}(J)\) in terms of \((W^J, S^J)\). It turns out that \(\mathbb{P}(J)\) can be decomposed into a union of ``rational'' cells. The descent system is used here to help record the dimension of each cell. Betti numbers; descent systems; H-polynomials; rationally smooth Compactifications; symmetric and spherical varieties, Algebraic monoids, Group actions on varieties or schemes (quotients), Representation theory for linear algebraic groups The Betti numbers of simple embeddings | 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 the double conformal/Darboux cyclide geometric algebra (DCGA), based in the \(\mathcal {G}_{8, 2}\) Clifford geometric algebra. DCGA is an extension of CGA and has entities representing points and general (quartic) Darboux cyclide surfaces in Euclidean 3D space, including circular tori and all quadrics, and all surfaces formed by their inversions in spheres. Dupin cyclides are quartic surfaces formed by inversions in spheres of torus, cylinder, and cone surfaces. Parabolic cyclides are cubic surfaces formed by inversions in spheres that are centered on points of other surfaces. All DCGA entities can be transformed by versors, and reflected in spheres and planes. conformal geometric algebra; Darboux Dupin cyclide; quadric surface; double conformal/Darboux cyclide geometric algebra; Clifford geometric algebra Easter, R.B., Hitzer, E.: \textit{Double conformal space-time algebra}, AIP Conference Proceedings, vol. 1798, AIP Publishing, p. 020066 (2017) Clifford algebras, spinors, Conformal differential geometry, Surfaces in Euclidean and related spaces, Euclidean analytic geometry, General theory of distance geometry, Rational and ruled surfaces Double conformal geometric 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 This interesting paper is devoted to study of a canonical pairing on a proper geometrically normal and geometrically connected scheme \(X_K\) over fractional field \(K\) of the complete discrete valuation ring \(R\) with algebraically closed residue field \(k\).
In variance with the intersection multiplicities on the Néron model \(A\) over \(R\) of an abelian variety \(A_K\) by \textit{A. Néron} [Ann. Math. (2) 82, 249--331 (1965; Zbl 0163.15205)] and intersection theory on a proper flat regular \(R-\) model \(X\) of a curve \(X_K\) by \textit{B. H. Gross} [in: Arithmetic geometry, Pap. Conf., Storrs/Conn. 1984, 327--339 (1986; Zbl 0605.14027)] and by \textit{P. Hriljac} [Am. J. Math. 107, 23--38 (1985; Zbl 0593.14004)] the author of the paper under review defines by intersection computations a pairing on an arbitrary scheme \(X_K\) as above. In this situation he proves that intersection computations are valid for arbitrary proper smooth and geometrically connected scheme over \(K\) ``and arbitrary \(0\)-cycles of degree zero, by using a proper flat normal and semifactorial model \(X\) of \(X_K\) over \(R\)``. The proof is based on the paper by the author [Math. Ann. 355, No. 1, 147--185 (2013; Zbl 1263.14046)].
Let \(\overline{A}\) be a normal semifactorial compactification of \( A \). The final two sections give ``an interpretation of Grothendieck`s duality for the Néron model \(A\), in terms of the Picard functor of \(\overline{A}\) over \(R\)'' and ``an explicit description of Grothendieck's duality pairing when \(A_K\) is the Jacobian of a curve of index one''. Néron symbol; Picard functor; Néron model; duality; Grothendieck pairing Pépin, C., Néron's pairing and relative algebraic equivalence, Algebra Number Theory, 6, 1315-1348, (2012) Picard schemes, higher Jacobians, Arithmetic varieties and schemes; Arakelov theory; heights, Arithmetic ground fields for abelian varieties, Abelian varieties of dimension \(> 1\) Néron's pairing and relative algebraic 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 The aim of this paper is to factorize hyperbolic polynomials with quasianalytic coefficients. The author generalizes some results by \textit{K. Kurdyka} and \textit{L. Paunescu} [Duke Math. J. 141, No. 1, 123--149 (2008; Zbl 1140.15006)] on perturbation theory of families of symmetric matrices to the quasianalytic setting. quasianalytic perturbation; hyperbolic polynomials; quasianalytic and arc-quasianalytic functions; polynomially bounded structures; eigenvalues; eigenspaces; symmetric and antisymmetric matrices; spectral theorem; quasianalytic diagonalization Krzysztof Jan Nowak, Quasianalytic perturbation of multi-parameter hyperbolic polynomials and symmetric matrices, Ann. Polon. Math. 101 (2011), no. 3, 275 -- 291. Real-analytic and semi-analytic sets, Semi-analytic sets, subanalytic sets, and generalizations, Eigenvalues, singular values, and eigenvectors, \(C^\infty\)-functions, quasi-analytic functions Quasianalytic perturbation of multi-parameter hyperbolic polynomials and symmetric matrices | 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 \(M(n,k)\) be the \(n\times n\)-matrices with entries in an algebraically closed field \(k\) and let \(V\) be a vector space over \(k\) of dimension \(n\). \textit{N. Spaltenstein} [Nederl. Akad. Wet., Proc., Ser. A 79, 452--456 (1976; Zbl 0343.20029)] established a bijection between the irreducible components of the space of full flags \(\mathcal F_x\) fixed by a nilpotent element \(x\in M(n,k)\) and the standard tableaux associated to the Young diagram of \(x\).
The main result of the present article is to determine, when \(x\) is of hook type, for each irreducible component \(X\) of \(\mathcal F_x\), the unique Schubert cell \(\mathcal C_X\) of the full flag manifold \(\mathcal F(V)\), such that \(\mathcal C_X\cap X\) is a dense subspace of \(X\). flag manifolds; nilpotent element; standard tableaux; Young diagram; Schubert cell; hook type; irreducible components Pagnon, NGJ, On the spaltenstein correspondence, Indag. Mathem., 15, 101-114, (2004) Grassmannians, Schubert varieties, flag manifolds, Linear algebraic groups over arbitrary fields On the Spaltenstein correspondence. | 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 provides a classification of the simple integrable modules of double affine Hecke algebras via perverse sheaves. Let \(\underline G\) be a simple connected simply connected linear algebraic group. Let \(\underline{\text{Lie}}\,\underline G\) denote the Lie algebra of \(\underline G\), let \(\underline{\text{Lie}}\,\underline H\subset\underline{\text{Lie}}\,\underline G\) be a Cartan subalgebra and let \(\underline{\text{Lie}}\,\underline B\subset\underline{\text{Lie}}\,\underline G\) be a Borel subalgebra containing \(\underline{\text{Lie}}\,\underline H\). Let \(\underline\Phi\) be the root system of \(\underline G\) and let \(\Phi^\vee\) be the root system dual to \(\underline\Phi\). Let \(\{\alpha_i:i\in\underline I\}\), \(\{\alpha_i^\vee:i\in\underline I\}\) be the set of simple roots and of simple coroots, respectively. Let \(I:=\underline I\sqcup\{0\}\). Let \(\underline W\), \(W\) be the Weyl group and the affine Weyl group of \(\underline G\). We identify \(\underline I\) (resp. \(I\)) with the set of simple reflections in \(\underline W\) (resp. \(W\)). Let \(s_i\in W\) be the simple reflection corresponding to \(i\in I\). For all \(i,j\in I\), let \(m_{ij}\) denote the order of the element \(s_is_j\) in \(W\). Let \(\underline X\) be the weight lattice of \(\underline\Phi\) and let \(Y^\vee\) be the root lattice of \(\underline\Phi v\). Let \(\{\omega_i:i\in\underline I\}\) be the set of fundamental weights. Consider the lattices \(Y=\bigoplus_{i\in I}\mathbb{Z}\alpha_i\subset X=\mathbb{Z}\delta\oplus\bigoplus_{i\in I}\mathbb{Z}\omega_i\), \(Y^\vee=\bigoplus_{i\in I}\mathbb{Z}\alpha_i^\vee\), where \(\delta\) is a new variable. There is unique pairing \(X\times Y^\vee\to\mathbb{Z}\) such that \((\omega_i:\alpha_j^\vee)=\delta_{ij}\) and \((\delta:\alpha_j^\vee)=0\).
The double affine Hecke algebra \(\mathbf H\) is the unital associative \(\mathbb{C}[q,q^{-1},t,t^{-1}]\)-algebra generated by \(\{t_i,x_\lambda:i\in I\), \(\lambda\in X\}\) modulo the following defining relations:
\[
x_\delta=t,\quad x_\lambda x_\mu=x_{\lambda+\mu}(t_i-q)(t_i+1)=0,
\]
\[
t_it_jt_i\cdots=t_jt_it_j\cdots\text{ if }i\neq j\;(m_{ij}\text{ factors in both products),}
\]
\[
t_ix_\lambda-x_\lambda t_i=0\text{ if }(\lambda:\alpha_i^\vee)=0,\quad t_ix_\lambda-x_{s_i(\lambda)}t_i=(q-1)x_\lambda\text{ if }(\lambda:\alpha_i^\vee)=1,
\]
for all \(i,j\in I\), \(\lambda,\mu\in X\).
One important step of the proof is the construction of a ring homomorphism from \(\mathbf H\) to a ring defined via the equivariant \(K\)-theory of an affine analogue \(\mathcal Z\) of the Steinberg variety. \(\mathcal Z\) is an ind-scheme of ind-infinite type. It comes with a filtration by subsets \({\mathcal Z}_{\leq y}\) with \(y\) in the affine Weyl group \(W\). The subsets \({\mathcal Z}_{\leq y}\) are reduced separated schemes of infinite type, and the inclusions \({\mathcal Z}_{\leq y'}\subset{\mathcal Z}_{\leq y}\) with \(y'\leq y\) are closed immersions. The set \(\mathcal Z\) is endowed with an action of a torus \(A\) which preserves each term of the filtration. For a well-chosen element \(a\in A\), the fixed point set \({\mathcal Z}^a\subset{\mathcal Z}\) is a scheme locally of finite type. Hence there is a convolution ring \(\mathbf K^A({\mathcal Z}^a)\): it is the inductive limit of the system of \({\mathbf R}_A\)-modules \(\mathbf K^A(({\mathcal Z}_{\leq y})^a)\) with \(y\in W\). (Here \({\mathbf R}_A\) means \({\mathbf K}_A(\text{point})\).) The author defines a ring homomorphism \(\Psi_a\colon{\mathbf H}\to\mathbf K^A({\mathcal Z}^a)_a\), where the subscript \(a\) means specialization at the maximal ideal \(J_a\subset{\mathbf R}_A\) associated to \(a\). The map \(\Psi_a\) becomes surjective after a suitable completion of \(\mathbf H\). It is certainly not injective. Using \(\Psi_a\), a standard sheaf-theoretic construction, due to Ginzburg in the case of affine Hecke algebras, provides a collection of simple \(\mathbf H\)-modules. These are precisely the simple integrable modules. -- The paper also give some estimates for the Jordan-Hölder multiplicities of induced modules. simple integrable modules; double affine Hecke algebras; perverse sheaves; linear algebraic groups; Lie algebras; Cartan subalgebras; Borel subalgebras; root systems; affine Weyl groups; simple reflections; pairings; equivariant \(K\)-theory; Jordan-Hölder multiplicities of induced modules Vasserot, Eric, Induced and simple modules of double affine Hecke algebras, Duke Math. J., 126, 2, 251-323, (2005) Hecke algebras and their representations, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Grassmannians, Schubert varieties, flag manifolds, Lie algebras of linear algebraic groups, Grothendieck groups, \(K\)-theory, etc., Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras Induced and simple modules of double affine Hecke 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 The main object of the paper under review is a smooth double cover \(X\) of a geometrically ruled surface \(S\) defined over a separably closed field \(k\) of characteristic different from 2. The authors compute the 2-torsion in the Brauer group of \(X\). This is done in an explicit way, which allows one in several important cases to exhibit the unramified central simple algebras representing the generators. This plays an important role in eventual applications to arithmetic problems. The techniques used by authors are based on the earlier works by \textit{O. Wittenberg} [Progr. Math. 226, 259--267 (2004; Zbl 1173.11336 )] and \textit{E. Ieronymou} [J. Inst. Math. Jussieu 9, 769--798 (2010; Zbl 1263.14023)], as well as on their earlier paper [Manuscr. Math. 147, 139--167 (2015; Zbl 1327.14158)]. ruled surface; double cover; Brauer group Creutz, B., Viray, B.: On Brauer groups of double covers of ruled surfaces. Math. Ann. (2014). 10.1007/s00208-014-1153-0 Brauer groups of schemes, Rational and ruled surfaces, Rational points On Brauer groups of double covers of ruled 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 We study the Grothendieck conjecture for the moduli spaces of hyperbolic curves of genus one. A consequence of the main results is that the isomorphism class of a certain moduli space of hyperbolic curves of genus one over a sub-\(p\)-adic field is completely determined by the isomorphism class of the étale fundamental group of the moduli space over the absolute Galois group of the sub-\(p\)-adic field. We also prove related results in absolute anabelian geometry. anabelian geometry; Grothendieck conjecture; moduli space; hyperbolic curve; configuration space Families, moduli of curves (algebraic), Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Fundamental groups and their automorphisms (group-theoretic aspects), Coverings of curves, fundamental group The Grothendieck conjecture for the moduli spaces of hyperbolic curves of genus one | 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, simply-connected, semisimple linear algebraic group over an algebraically closed field \(k\), \(B\) a Borel subgroup containing a maximal torus \(T\), \(W = N_G(T)/T\) the Weyl group, \(S\subset W\) the set of simple reflections with respect to \(B\). A simple reflection \(s\in S\) determines a minimal parabolic subgroup \(P_s = BsB\cup B\supset B\). Consider, also, the maximal parabolic subgroup \(Q_s\) which is the union of the double cosets \(BwB\) with \(w\) in the subgroup of \(W\) generated by \(S\setminus \{s\}\). If \({\mathfrak w}=(s_1,\dots ,s_n)\) is a sequence of simple reflections and \(1\leq i \leq n\), one denotes by \({\mathfrak w}[i]\) the truncated sequence \((s_1,\dots ,s_{n-i})\) and by \({\mathfrak w}(i)\) the subsequence of \({\mathfrak w}\) obtained by deleting \(s_i\).
The Bott-Samelson variety \(Z_{\mathfrak w}\) is defined as follows : \(B^n\) acts from the right on \(P_{s_1}\times \dots \times P_{s_n}\) by : \((p_1,\dots ,p_n)(b_1,\dots b_n) = (p_1b_1,b_1^{-1}p_2b_2,\dots , b_{n-1}^{-1}p_nb_n)\) and \(Z_{\mathfrak w} := P_{s_1}\times \dots \times P_{s_n}/B^n\). Since \(P_s/B\simeq {\mathbb P}^1\), \(\forall s\in S\), one has a tower of \({\mathbb P}^1\)-bundles : \(Z_{\mathfrak w}\rightarrow Z_{{\mathfrak w}[1]}\rightarrow \cdots \rightarrow Z_{{\mathfrak w}[n-1]}=P_{s_1}/B\simeq {\mathbb P}^1.\) For \(1\leq i\leq n\), the natural projection \({\pi}_i : Z_{\mathfrak w}\rightarrow Z_{{\mathfrak w}[i]}\) admits a section and \(Z_{{\mathfrak w}(i)}\) embeds as a divisor into \(Z_{\mathfrak w}\). The multiplication of \(G\) induces a morphism \(Z_{\mathfrak w}\rightarrow G/B\) and if the word \({\mathfrak w}\) is reduced then \(Z_{\mathfrak w}\) is a Demazure-Hansen desingularisation of the Schubert variety \(X(s_1\dots s_n)\) which is the Zariski closure in \(G/B\) of the (left) \(B\)-orbit of \(s_1\dots s_nB\).
Let \({\mathcal O}_{\mathfrak w}(1)\) denote the pull-back on \(Z_{\mathfrak w}\) of the ample generator of \(\text{Pic}(G/Q_{s_n})\simeq {\mathbb Z}\) (when \(G = \text{SL}(n)\), \(G/Q_{s_n}\) is a Grassmannian) by the composite map \(Z_{\mathfrak w}\rightarrow G/B\rightarrow G/Q_{s_n}\). One sees easily that \({\pi}_i^{\ast}{\mathcal O}_{{\mathfrak w}[i]}(1)\), \(i=0,\dots ,n-1\), is a \({\mathbb Z}\)-basis of \(\text{Pic}Z_{\mathfrak w}\). The authors show, firstly, that \({\mathcal O}_{\mathfrak w}(m_1)\otimes {\pi}_1^{\ast}{\mathcal O}_{{\mathfrak w}[1]}(m_2)\otimes \dots \otimes {\pi}_{n-1}^{\ast}{\mathcal O}_{{\mathfrak w}[n-1]}(m_n)\) is very ample (resp., globally generated) on \(Z_{\mathfrak w}\) iff \(m_i>0\) (resp., \(\geq 0\)) for \(i=1,\dots n\). Moreover, if \({\mathfrak w}\) is reduced then \({\mathcal O}(\mathop{\sum}_{i=1}^nm_iZ_{{\mathfrak w}(i)})\) is effective iff \(m_i\geq 0\) for \(i=1,\dots ,n\).
Then, using the Frobenius splitting technique of \textit{V. Mehta} and \textit{A. Ramanathan} [Ann. Math. (2) 122, 27--40 (1985; Zbl 0601.14043)], the authors show that if, for some \(1\leq t\leq q\leq n\), the subsequence \((s_t,\dots ,s_q)\) is reduced then \(\text{H}^i(Z_{\mathfrak w},{\mathcal L}\otimes {\mathcal O}(-\mathop{\sum}_{j=t}^q Z_{{\mathfrak w}(j)}))=0\), for all \(i>0\) and all globally generated line bundles \({\mathcal L}\) on \(Z_{\mathfrak w}\). This generalizes a result of \textit{S. Kumar} [Invent. Math. 89, 395--423 (1987; Zbl 0635.14023)] who assumed that \(\text{char}k = 0\) and that \({\mathcal L}\) is the line bundle associated to a dominant weight. Schubert varieties; cohomology of line bundles; semisimple algebraic groups; Frobenius splitting Lauritzen, N.; Thomsen, J. F., Line bundles on Bott-Samelson varieties, J. Algebraic Geom., 13, 461-473, (2004) Grassmannians, Schubert varieties, flag manifolds, Divisors, linear systems, invertible sheaves, Classical groups (algebro-geometric aspects), Vanishing theorems in algebraic geometry Line bundles on Bott-Samelson varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This Dover edition is the first publication in book form of the author's popular course text originally titled ``Abstract Algebra: The Basic Graduate Year'', which since several years is available from his website: \url{http://www.math.uiuc.edu/~r-ash/}, together with his various other course notes from the University of Illinois at Urbana-Champaign, USA.
Designed for upper-level undergraduate and graduate students, this text provides a modern and thorough introduction to the basic concepts and methods of abstract algebra, ranging from the study of fundamental algebraic structures up to their applications in such advanced areas as algebraic number theory, algebraic geometry, noncommutative algebra, and homology theory.
The present book also includes the author's introductory essay ``Remarks on Expository Writing in Mathematics'', which is available from the same website and explains his general experiences and principles in writing modern, research-oriented, efficient, and user-friendly mathematical textbooks. Placed in front of the main text of the book at hand, this didactic essay offers some concrete expert advice in this regard, which then, in fact, is reflected by both the masterly style and the unique features of this outstanding textbook under review. The author's didactic keynotes are summarized (by himself) as follows:
(1) Adopt a straight style of writing, without lengthy detours, digressions or thematic interruptions.
(2) Discuss the intuitive content of abstract results and formal proofs, as this leads to a greater understanding than a purely abstract reasoning.
(3) Prefer algorithmic procedures to abstract arguments wherever possible, as this has more impact on beginners.
(4) Use concrete examples with all the features of the respective general case, as this helps instruct the reader on how to find and write a formal proof.
(5) Avoid serious gaps in the reasoning, and never omit any steps that are essential for completely understanding the proof of an important result.
(6) Provide solutions to exercises that are used in the course of the main text, thereby making sure that the learning reader gets nowhere seriously blocked or discouraged.
(7) Try to communicate the intrinsic beauty of the subject, too, and include brief informal surveys of related large areas for further illustration, motivation, and stimulation.
All these guiding principles are strictly complied with, in the textbook under review, and this makes it not only a highly valuable enhancement of the existing literature in the field, but also a fairly specific and unique primer of basic and advanced abstract algebra. As for the contents, the main text consists of eleven chapters and a supplement.
Chapter 0 briefly recalls the necessary prerequisites from elementary number theory, set theory, and linear algebra. Chapter 1 developes the very fundamentals of group theory up to the isomorphism theorems and direct products of groups.
Chapter 2 is devoted to the basics of ring theory, including ideals, polynomial rings, factorial rings, principal ideal domains, localizations, and irreducibility in polynomial rings.
Chapter 3 treats field extensions, whereas Chapter 4 completes the introduction to the basic properties of fundamental algebraic structures by discussing the elements of the theory of modules and algebras.
Apart from the elementary structure theorems for modules over special rings, this chapter also contains applications to normal forms of matrices (Smith normal form) and introduces the technique of exact sequences and diagram chasing.
As Chapters 1--4 may be regarded as an idealized undergraduate course in abstract algebra, which however is written in the style of a graduate text, the author has added an extra section providing advice, explanations, and additional examples for each section in the foregoing chapters, mainly geared toward novices in abstract algebra.
Chapter 5 turns to some basic methods in the more advanced theory of groups, including group actions, the Sylow theorems, composition series, solvable and nilpotent groups, some deeper structure theory, and the description of groups by generators and relations.
In Chapter 6, this is combined with the theory of field extensions in order to develop elementary Galois theory and its applications to algebraic equations.
Chapter 7 gives a first introduction to algebraic number theory, along which some of the fundamental concepts of commutative algebra (integral ring extensions, noetherian and artinian modules and rings, Dedekind rings) are treated as well. This chapter concludes with a merely informal introduction to \(p\)-adic numbers and valuations.
Chapter 8 provides more material from commutative algebra, largely in the language of affine algebraic geometry. Affine varieties, Hilbert's Basis Theorem, Hilbert's Null\-stel\-len\-satz, the localization principle, the Lasker-Noether primary decomposition of ideals in noetherian rings, and general tensor products are the main topics of this chapter.
Chapter 9 begins the study of noncommutative rings and their modules, focusing on the basic theory of simple and semisimple rings and modules, the allied theory of group representations, and the relation between noetherian and artinian rings in general.
Chapter 10 is devoted to the fundamental concepts and methods of homological algebra for modules. Introducing and adopting the language of categories and functors, the author discusses exactness properties of functors, protective modules, injective modules, flat modules, embeddings into an infective module, and the construction of direct and inverse limits.
In a supplement, which is virtually an additional chapter, the author leads the reader much farther into homological algebra than is usual in basic algebra courses or introductory textbooks. The sections of this supplement briefly explain chain complexes, the crucial ``Snake Lemma'', the long exact homology sequence, protective and injective resolutions of modules, derived functors, the functors Ext and Tor, as well as the base change by tensorizing, thereby developing some of the basic techniques in algebraic topology and algebraic geometry.
The entire text comes with a wealth of instructive examples and carefully selected exercises. According to the author's afore-mentioned didactic (and utmost laudable) philosophy, complete solutions to the latter ones are provided at the end of the book, and that on more than 60 pages!
A rich bibliography of textbooks related to the different chapters, a list of used symbols, and a carefully compiled index of notions help the reader work efficiently with this overall excellent primer. No doubt, this is one of the best and most useful textbooks on basic abstract algebra I have ever seen. Without trying to be encyclopedic, the author has presented a text of great profundity and manageable size. Emphasizing the crucial role of intuitive thinking, without allowing any lack of formal rigor, and leading the reader to an advanced level as quickly and efficiently as possible, this book is a great source for both students and teachers in the field. instructional exposition; textbooks; group theory; field theory and polynomials; commutative rings and algebras; noncommutative rings; algebraic geometry; homological algebra Ash, R. B., Basic abstract algebra. for graduate students and advanced undergraduates, (2007), Dover Publications, Inc. Mineola, NY Mathematics in general, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to field theory, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to associative rings and algebras, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to category theory Basic abstract algebra. For graduate students and advanced undergraduates | 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 idempotent kernel endofunctors \(\sigma\) and \(\tau\) of a Grothendieck category \({\mathcal C}\) are said to be compatible if \(\sigma Q_\tau =Q_\tau \sigma\) where \(Q_\tau\) is the left adjoint of the inclusion functor of the full subcategory of all \(\tau\)-closed objects. For an adjoint pair \(F:{\mathcal C} \to {\mathcal D}\), \(G:{\mathcal D} \to {\mathcal C}\) with \(\alpha\) as the unit of adjunction and an idempotent kernel endofunctor \(\tau\) of \({\mathcal D}\) the idempotent kernel endofunctor \(\tau^e\) defined by \(\tau^e (M)= \alpha^{-1}_M (Q_\tau F(M))\) is called the extension of \(\tau\). The endofunctor \(M\vdash a(M) =\text{ker} \alpha_M\) is also an idempotent kernel functor. If \(FG=1\) and \(\sigma\) is an idempotent kernel endofunctor of \({\mathcal C}\) compatible with \(a\), then the endofunctor \(\sigma^r =F \sigma G\) of \({\mathcal D}\) is also an idempotent kernel functor, called the restriction of \(\sigma\). It is proved that under some mild conditions both the restriction and the extension of torsion theories preserve the compatibility. Furthermore, two idempotent kernel endofunctors of \({\mathcal D}\) are compatible iff their extensions are compatible in \({\mathcal C}\). If both \({\mathcal C}\) and \({\mathcal D}\) are symmetric closed Grothendieck categories, then some relations between exact sequences of Picard and Brauer groups for the extension and restriction of torsion theories are obtained. Brauer groups; Picard groups; idempotent kernel endofunctors; Grothendieck category; restriction; extension; compatibility; torsion theories Torsion theories, radicals, Torsion theories; radicals on module categories (associative algebraic aspects), Closed categories (closed monoidal and Cartesian closed categories, etc.), Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.), Brauer groups of schemes, Grothendieck categories, Torsion theory for commutative rings Extension, restriction and Picard-Brauer exact sequences for torsion theories | 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 short announcement (containing hardly any proof) of results concerning the birational structure of double covers of the projective plane. It also contains an application to the problem of the existence of plane curves with assigned singularities. birational structure of double covers of the projective plane Yoshihara, H.: Double coverings of P2. Proc. Japan Acad.66, 233--236 (1990) Coverings in algebraic geometry, Surfaces and higher-dimensional varieties, Projective techniques in algebraic geometry Double coverings of \(\mathbb{P}^ 2\) | 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 Although being one of the comparatively recent creations in mathematics, abstract commutative algebra has a long and fascinating genesis. Its development into a beautiful, deep and widely applied mathematical discipline, in its own right, must be understood as a function of that of algebraic number theory and algebraic geometry, both of which essentially gave birth to it.
In the second half of the 19th century, two concrete classes of commutative rings (and their ideal theory) marked the beginning of what is now called commutative algebra: rings of integers of algebraic number fields, on the one hand, and polynomial rings occurring in classical algebraic geometry and invariant theory, on the other hand. In the first half of the 20th century, after the basics of abstract algebra had been established, commutative algebra grew into an independent subject, mainly under the influence of E. Noether, E. Artin, W. Krull, B. L. van der Waerden, and others. In the 1940s this abstract and general framework was applied, in turn, to give classical algebraic geometry both a completely new footing and a new toolkit for further investigations. In this context, the epoch-making innovations by Chevalley, Zariski, and Weil not only created a revolution in algebraic geometry, but also had a very strong impact on the quickened growing of commutative algebra itself in the following decades.
The 1950s and 1960s saw the development of the structural theory of local rings, the foundations of algebraic multiplicity theory, Nagata's counter-examples to Hilbert's 14th problem, the introduction of homological methods into commutative algebra, and other pioneering achievements. However, the indisputably most characteristic mark of this period was A. Grothendieck's creation of the theory of schemes, the (till now) ultimate revolution of algebraic geometry. His foundational work culminated in a far-reaching alliance of commutative algebra and algebraic geometry, which also made is possible, in turn, to apply geometric methods as a tool in commutative algebra.
In the following decades, geometric and homological methods have substantially engraved the vigorous research activities in commutative algebra. At present, commutative algebra is an independent, abstract, deep, and smoothly polished subject for its own sake, on the one hand, and an indispensible conceptual, methodical, and technical resource for modern algebraic and complex analytic geometry, on the other hand. Studying or actively pursuing research in geometry or number theory requires today a profound knowledge of commutative algebra; however, most textbooks on algebraic or complex analytic geometry usually assume such a knowledge from the beginning, often refer to the (undoubtedly excellent) great standard texts on abstract commutative algebra, or survey a minimal account of the basic results, mostly in a series of appendices. The reason for that is quite clear, for the fusion of the two subjects is close to such an extent that the necessary prerequisites from commutative algebra would occupy a considerable part of the text, thereby possibly discouraging the reader who is mainly interested in the geometric aspects of the subject. Conversely, most textbooks on commutative algebra present the material in its purely algebraic and perfectly polished abstract form, with at most a few elementary hints and applications to the related geometry. This situation really creates a dilemma for both students and teachers of algebraic geometry or commutative algebra.
The present book under review aims at toning down this traditional, nevertheless somewhat artificial discrepancy. The author, in person one of the leading experts in both fields, has tried to write on commutative algebra in a way that makes the heritage, the geometric character, and the geometric applications of the subject as apparent as possible. A first attempt in this direction, namely to offer a textbook that mixes algebra and geometry in an organic manner, has been successfully carried out by \textit{E. Kunz} in 1980. His textbook ``Einführung in die kommutative Algebra und algebraische Geometrie.'' Braunschweig etc.: Friedr. Vieweg (1980; Zbl 0432.13001) and the English translation ``Introduction to commutative algebra and algebraic geometry.'' Boston etc.: Birkhäuser (1985; Zbl 0563.13001), (reprint 2013; Zbl 1263.13001) provided an exposition of the basic definitions and results in both commutative algebra and algebraic geometry, centered around the (at this time) recent solution of Serre's problem on projective modules over polynomial rings or, respectively, Kronecker's longstanding problem on complete intersections in projective spaces. Along this road, E. Kunz gave a very natural introduction to commutative algebra and algebraic geometry, especially emphasizing the concrete elementary nature of the objects which were at the beginning of both subjects.
The book under review aims at the same goal; however, it does so under much wider aspects. In fact, the author strives for a rather complete and up-to-date exposition of the present state of commutative algebra, with all its old and new links to modern algebraic geometry. As he says in the preface, his precise goal has been, from the beginning, to cover at least all the material that graduate students in algebraic geometry should have at their disposal, in particular those studying \textit{R. Hartshorne}'s matchless modern textbook ``Algebraic geometry'' [New York etc.: Springer-Verlag (1977; Zbl 0367.14001)] and, perhaps subsequently, \textit{A. Grothendieck}'s and \textit{J. Dieudonné}'s ``Éléments de géométrie algébrique'' [Publ. Math., Inst. Haut. Étud. Sci. I--IV (1960- 1967; Zbl 0118.36206; Zbl 0122.16102; Zbl 0136.15901; Zbl 0135.39701; Zbl 0144.19904; Zbl 0153.02202)].
According to this strategy, the text is subdivided into three major parts and six appendices.
After a beautiful introduction, providing readers of different background knowledge or expertise with instructions for both self-teaching and teaching, and after a brief synopsis of the elementary definitions concerning rings, ideals and modules, part I of the book under review discusses the ``Basic constructions'' in commutative algebra. This first part consists of seven separate chapters: Chapter 1 is still introductory and surveys some of the history of commutative algebra in number theory, algebraic curve theory, one-dimensional complex analysis, and invariant theory. It also explains the dictionary ``Commutative algebra -- projective algebraic geometry'', including Hilbert's basis theorem, Hilbert's syzygy theorem, Hilbert's Nullstellensatz, graded rings, and the Hilbert polynomials.
Chapter 2 deals with the localization principle in commutative algebra, with an analysis of zero-dimensional rings, and chapter 3 turns to associated prime ideals and the primary decomposition in noetherian rings. Chapter 4 discusses Bourbaki's proof of Hilbert's Nullstellensatz, Nakayama's lemma, integral dependence, and the normalization process, whereas chapter 5 goes back to graded rings and modules, including the construction of the blow-up algebra, Krull's intersection theorem, and their geometric interpretation. Chapter 6 introduces the concept of flatness, the Tor-functor, and derives the most important flatness criteria. Completions of rings and Hensel's lemma are presented in chapter 7, together with their geometric significance, and Cohen's structure theorems are also found here. -- Each chapter comes with a large number of well-prepared exercises, most of which concern additional theoretical material and results. The same holds for the following chapters, whereby hints and solutions for selected exercises are given at the end of the book. By this method, the author manages to cover even more interesting and recent material, at least in outlines.
Part II of the book is entitled ``Dimension theory''. It consists of nine more chapters. Chapter 8 illustrates the history of dimension theory in its topological, geometric, and algebraic aspects. In this complexity, it is much more advanced than the following ones, and (as the author says) it is actually meant to be read for motivation and ``for culture only''. Chapter 9 gives then the fundamental definitions of algebraic dimension theory (Krull dimension of a ring), with special emphasis on the case of dimension zero. Chapter 10 covers Krull's principal ideal theorem, regular sequences, parameter systems, regular local rings, and their algebro-geometric meanings. Chapter 11 is entitled ``Dimension and codimension one'' and treats normal rings, discrete valuation rings, Serre's criterion, Dedekind domains, and the ideal class group. Hilbert-Samuel functions and polynomials, together with their natural appearance in multiplicity theory are discussed in chapter 12. The geometric aspects of dimension theory, above all Noether normalization and the finiteness of the integral closure of an affine ring, are the subject of chapter 13, and the following chapter 14 is devoted to both classic and modern elimination theory. -- Chapter 15 gives, for the first time in a textbook on commutative algebra, an account of the fast growing computational part of the subject. Gröbner bases, initial ideals, and their applications to constructive module theory and projective algebraic geometry are the principal items here. The rather mathematical approach, relative to the usual more computational ones, is particularly convenient for algebraic geometers, and a set of seven computer algebra projects, at the end of the chapter, shows how the computational possibilities of this approach lead to new (at least conjectural) insights. -- Chapter 16 is concerned with the differential calculus in commutative algebra, that is with modules of differentials, tangent and cotangent bundles, smoothness and generic smoothness, the Jacobi criterion, infinitesimal automorphisms, and some deformation theory.
Part III of the book is devoted to the homological methods in commutative algebra. Chapter 17 deals with regular sequences by means of the Koszul complex, and with applications of the Koszul complex to the study of the cotangent bundle of \(\mathbb{P}^n\). Chapter 18 discusses the notion of depth and the Cohen-Macaulay property. The significance of the Cohen- Macaulay property is illustrated from various viewpoints, ranging from Hartshorne's theorem on connectedness in codimension one to the theorem on flatness over a regular base to primeness criteria using Serre's characterization of normality. -- The homological theory of regular local rings occupies chapter 19. This chapter contains, apart from the standard material on projective dimension, minimal resolutions, global dimension, and the Auslander-Buchsbaum formula, also an application to the factoriality of local rings via stably free modules. Chapter 20 concerns free resolutions and their role in algebra and algebraic geometry. The author, who has contributed to this topic by a good deal of his own research in the past, presents here various criteria of exactness, mainly based on the approach via Fitting ideals and Fitting invariants, and he gives some very instructive applications, e.g., the Hilbert-Burch theorem characterizing ideals of projective dimension one, and, at the end, an algebraic treatment of Castelnuovo-Mumford regularity. The concluding chapter 21 gives an account of duality theory for local Cohen-Macaulay rings, and some parts of the theory of Gorenstein rings. This includes the discussion of the canonical module and its properties, maximal Cohen- Macaulay modules and their duality theory, and the theory of linkage à la Peskine and Szpiro from the algebraic point of view. An interesting feature in this chapter, among many others in the text, is the treatment of the canonical module via reduction to the case of an Artinian ring. This makes the whole topic pleasantly concrete and lucid, at least for the beginner.
The main text is followed by seven appendices, in which the author provides both some more technical material from algebra, as it is needed in the course of the text, and some furthergoing topics related to it. In brevity, these appendices are the following:
1. Field theory (transcendency degree, separability, \(p\)-bases);
2. Multilinear algebra (including divided powers and Schur functors);
3. Homological algebra (projective modules, injective modules, complexes, homology, derived functors, Ext and Tor, double complexes, spectral sequences, and derived categories);
4. Local cohomology (local and global cohomology, local duality, depth and dimension via local cohomology);
5. Category theory (categories, functors, natural transformations, adjoint functors, limits, representable functors, and Yoneda's lemma);
6. Limits and colimits (flat modules as limits of free modules);
7. Where next? (hints for further reading).
The book ends with a section containing hints and solutions for more than one hundred selected exercises spread over the entire text. The bibliography is extremely rich and carefully selected, being a true help for both the reader and the interested expert.
Altogether, the book under review has filled a longstanding need for a text on commutative algebra which thoroughly reflects the naturally grown relations to algebraic geometry. Containing numerous novel results and presentations, the book is still fairly self-contained, accessible for beginners, and a treasure for teachers and researchers in both fields. The consequent mixing of algebra and geometry, from the beginning to the end, has made it impossible to present commutative algebra in its most systematic and abstract perfection, as it has been done, for example, in the great standard textbooks of \textit{O. Zariski} and \textit{P. Samuel} [cf. ``Commutative algebra'', Vol. I. Princeton etc.: D. van Nostrand Company (1958; Zbl 0081.26501) and II (1960; Zbl 0121.27801); reprints, respectively, New York: Springer-Verlag (1975; Zbl 0313.13001) and 1976 (Zbl 0322.13001)], \textit{N. Bourbaki} [``Commutative algebra'', Chapters 1--7. Paris: Hermann (1972; Zbl 0279.13001), 2nd printing Berlin etc.: Springer-Verlag (1989; Zbl 0666.13001)], and \textit{H. Matsumura} [``Commutative algebra.'' New York: W. A. Benjamin (1970; Zbl 0211.06501); 2nd edition (1980; Zbl 0441.13001)]. However, the book under review, apart from having the compensating advantage of combining algebra and geometry in a natural manner, at least touches upon numerous recent development and results not yet contained in any other textbook. In this sense, it should be regarded as an unique and excellent enrichment of the existing literature in commutative algebra and algebraic geometry, just as a new standard text among the celebrated others, and as a highly welcome supplement to them. bibliography; Hilbert's basis theorem; dictionary: commutative algebra-projective algebraic geometry; Hilbert's syzygy theorem; Hilbert's Nullstellensatz; Hilbert polynomials; dimension theory; Dedekind domains; Hilbert-Samuel functions; elimination theory; computer algebra; modules of differentials; homological methods; Koszul complex; Cohen-Macaulay property; duality theory; linkage Eisenbud D, \textit{Commutative Algebra: With a View Toward Algebraic Geometry}, 150, Springer New York, 1995. Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra, Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry, History of commutative algebra, General commutative ring theory, Theory of modules and ideals in commutative rings, Actions of groups on commutative rings; invariant theory, Dimension theory, depth, related commutative rings (catenary, etc.) Commutative algebra. With a view toward algebraic geometry | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This article can be regarded as a report on the progress in Hodge theory since 1975.
In the first section, we review Deligne's construction of mixed Hodge structures on the cohomology of all algebraic varieties, and simplicial varieties, over \(\mathbb{C}\), for it sets the tone of much of the work that followed [cf. \textit{P. Deligne}, Théorie de Hodge. I--III, Actes Congr. internat. Math. 1970, Part I, 425-430 (1971; Zbl 0219.14006), Publ. Math., Inst. Hautes Étud. Sci. 40(1971), 5-57 (1972; Zbl 0219.14007) and ibid. 44(1974), 5-77 (1975; Zbl 0237.14003)]. -- Much of section 2 also contains review material, namely the theorem of \textit{W. Schmid} [Invent. Math. 22, 211-319 (1973; Zbl 0278.14003)] on degeneration of Hodge structures in the abstract, i.e., in the absence of any hypothesis that the variation of Hodge structure arises from a family of smooth projective varieties. These are the nilpotent orbit theorem and the SL(2)-orbit theorem. -- Section 3 is about \(L_ 2\)-cohomology. It is an integral part of the subject. There is a general relation between \(L_ 2\)-cohomology and harmonic forms, which gives a key motivation for its introduction: it was the most familiar way to establish the existence of useful Hodge decompositions.
Ironically, it was not by \(L_ 2\)-cohomology that Hodge structures for the intersection homology groups of singular projective varieties were finally produced. Instead, it came out of work in algebraic analysis (i.e., \(\mathbb{D}\)-modules), a subject that also has blossomed during the past 15 years, as we report in section 4. The starting point is the so- called Riemann-Hilbert correspondence, which asserts that taking the de Rham complex sets up an equivalence of categories between holonomic \(\mathbb{D}\)-modules with regular singularities and perverse sheaves (and likewise for their derived categories). The idea is to equip such \(\mathbb{D}\)-modules with Hodge (and weight) filtrations, so that they induce (mixed) Hodge structures on hypercohomology. -- Section 5 is devoted to Deligne cohomology and to its generalization, by Beilinson, to noncompact varieties. -- Several approaches have been used to put mixed Hodge structures on homotopy groups of algebraic varieties, discussed in section 6. The method of \textit{J. W. Morgan} [Publ. Math., Inst. Hautes Étud. Sci. 48, 137-204 (1978; Zbl 0401.14003) and ibid. 64, 185 (1986; Zbl 0617.14013)] is based on Sullivan's theory of minimal models for differential graded algebras. The method of \textit{R. M. Hain} [\(K\)-Theory 1, 271-324 (1987; Zbl 0637.55006) and ibid. 1, No. 5, 481-497 (1987; Zbl 0657.14004)] is based on Chen's method of iterated integrals. Alternative approaches to mixed Hodge theory on homotopy groups, due Deligne and to Navarro Aznar, are only briefly mentioned.
The notion of a variation of mixed Hodge structure is a very natural generalization of that of a variation of Hodge structure. The idea, naturally, is that a variation of mixed Hodge structure on \(X\) must at least yield a filtered local system \((\mathbb{V},W)\) on \(X\), and that the graded local systems \(\text{Gr}^ W_ k \mathbb{V}\) should be honest polarized variations of Hodge structure. In section 7, we present such a notion, that of admissible variations of mixed Hodge structure. The definition is due to Steenbrink and Zucker in the case of curves, and to Kashiwara in the general case.
Section 8 is devoted to the question of determining which local systems on a given quasi-projective manifold underlie a variation of Hodge structure. We primarily discuss very recent work of \textit{C. Simpson}, which, in our opinion, promises to have profound repercussions in Hodge theory. He uses the nonlinear P.D.E. methods of differential geometry to obtain a correspondence between irreducible vector bundles on a compact Kähler manifold and stable Higgs bundles with vanishing total Chern class (see theorem 8.9). A Higgs bundle is a vector bundle \({\mathcal E}\), together with an operator-valued one-form \(\theta\) on \({\mathcal E}\) of square 0. \(\mathbb{C}^*\) acts on Higgs bundles in the obvious way, by dilating \(\theta\). The remarkable fact is that the fixed points of this \(\mathbb{C}^*\)-action correspond exactly to complex variations of Hodge structure (corollary 8.10), and real variations correspond to fixed points which give self-dual Higgs bundles (see proposition 8.12). Some striking applications are presented. Bibliography; \(L_ 2\)-cohomology; \(\mathbb{D}\)-modules; Beilinson cohomology; regulator maps; Grothendieck motives; degeneration of Hodge structures; intersection homology; Riemann-Hilbert correspondence; Deligne cohomology; variation of mixed Hodge structure; Higgs bundles Brylinski, J.-L., Zucker, S.: An overview of recent advances in Hodge theory. In: Several Complex Variables VI. Encyclopedia Math. Sci., vol. 69, pp. 39--142. Springer, Berlin (1990) Transcendental methods, Hodge theory (algebro-geometric aspects), Research exposition (monographs, survey articles) pertaining to algebraic geometry, Variation of Hodge structures (algebro-geometric aspects), Mixed Hodge theory of singular varieties (complex-analytic aspects), History of algebraic geometry, Monodromy; relations with differential equations and \(D\)-modules (complex-analytic aspects), Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), History of mathematics in the 20th century An overview of recent advances in Hodge theory | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Following work of \textit{Y. Bugeaud} et al. [Math. Z. 243, No. 1, 79--84 (2003; Zbl 1021.11001)] for integers, \textit{N. Ailon} and \textit{Z. Rudnick} [Acta Arith. 113, No. 1, 31--38 (2004; Zbl 1057.11018)] prove that for any multiplicatively independent polynomials, \(a,b\in\mathbb C[x]\), there is a polynomial \(h\) such that for all \(n\), we have
\[
\mathrm{gcd}(a^n-1, b^n-1)\mid h
\]
We prove a compositional analog of this theorem, namely that if \(f,g\in\mathbb C[x]\) are compositionally independent polynomials and \(c(x)\in\mathbb C[x]\), then there are at most finitely many \(\lambda\) with the property that there is an \(n\) such that \((x-\lambda)\) divides \(\mathrm{gcd}(f^{\circ n}(x) -c(x),g^{\circ n}(x)-c(x))\). heights; equidistribution; iteration of polynomials 10.2140/ant.2017.11.1437 Arithmetic and non-Archimedean dynamical systems involving polynomial and rational maps, Global ground fields in algebraic geometry, Diophantine inequalities, Recurrences Greatest common divisors of iterates of 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 Any group G of automorphisms of an algebraic variety X acts in a natural way on the module \(\Omega_ X\) of its differentials, \(\Omega_ X\) thus becoming a right kG-module. A well known result due to \textit{C. Chevalley} and \textit{A. Weil} [Abh. Math. Semin. Hamb. Univ. 10, 358-361 (1934; Zbl 0009.16001)], \textit{A. Weil}, Oeuvres Scientifiques, vol. I (1979; Zbl 0424.01027), pp. 68-71] describes this kG-module structure on \(\Omega_ X\) for the case when k is algebraically closed of characteristic zero and X is a curve over k.
The paper under review contains a strong generalization of the result for the case when the field k is algebraically closed (char(k) being arbitrary) and the factorization \(p: X\to Y=X/G\) is tamely ramified. - The result may be elegantly stated in terms of the Grothendieck group theory and reads: \([\Omega_ X]=[k\oplus (kG)^{\oplus (g_ Y-1)}\oplus \tilde R^*_ G]\), where \([\quad]\) denotes the class in the Grothendieck group, \(g_ Y\) is the genus of Y, and \(R_ G\) the (tame) ramification module. module of differentials of a curve; group of automorphisms; Grothendieck group; ramification module Kani, Ernst, The {G}alois-module structure of the space of holomorphic differentials of a curve, Journal für die Reine und Angewandte Mathematik. [Crelle's Journal], 367, 187-206, (1986) Coverings of curves, fundamental group, Group actions on varieties or schemes (quotients), Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Grothendieck groups (category-theoretic aspects) The Galois-module structure of the space of holomorphic differentials of a 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 Let \(X \longrightarrow \text{Spec}(R)\) be the minimal resolution of a rational double point defined over an algebraically closed field. Let \(E\) be the exceptional divisor and \(S_X=\Theta_X(-\text{log }E)\) the sheaf of logarithmic derivations. It is proved that the canonical morphism \(H^1(S_X) \longrightarrow H^1(X \setminus E,S_X)\) is an inclusion. The dimension of \(H^1_E(S_X\otimes \mathcal O_X(E))\) is computed. It is proved that \(H^0(X\setminus E,\Theta_X)/H^0(X,\Theta_X)\) is isomorphic to \(H^1_E(S_X)\) and the dimension is computed. rational double point; sheaf of logarithmic derivations Singularities of surfaces or higher-dimensional varieties, Complex surface and hypersurface singularities Further evaluation of Wahl vanishing theorems for surface singularities in characteristic \(p\) | 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 This article deals with the quantum cohomology of the classical flag manifold \(\text{GL}(n,\mathbb C)/B\), where \(B\) is the subgroup of upper triangular nonsingular matrices. The cohomology ring can be obtained as a factor ring of the polynomials in \(n\) variables modulo the ideal generated by the elementary symmetric polynomials. A distinguished basis for the classical cohomology ring is given by the Schubert polynomials. The quantum cohomology ring again is a factor ring of the polynomial ring but now with \(n-1\) additional variables, the deformation parameters, modulo the ideal generated by the ``quantum elementary polynomials''. The author relates the quantum Schubert polynomial and some other related basis with problems in algebraic combinatorics. For the proofs and further details he mainly refers to the following publications of the author [Adv. Math. 136, No. 2, 224-250 (1998; Zbl 0920.05069) and ``On algebraic and combinatorial properties of Schur and Schubert polynomials'' (Bayreuther Math. Schr. 59) (2000; Zbl 0958.05001)]. quantum cohomology; flag manifold; Schubert polynomial; elementary symmetric polynomial; standard elementary monomial Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Enumerative problems (combinatorial problems) in algebraic geometry From quantum cohomology to algebraic combinatorics: The example of flag manifolds | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(L\) be any standard Levi subgroup which acts by left multiplication on a Schubert variety \(X(w)\) in the Grassmannian. We give a complete classification of the pairs \(L\) and \(X(w)\), where \(X(w)\) is a spherical variety for the action of \(L\). Schubert varieties; Grassmann varieties; spherical varieties Grassmannians, Schubert varieties, flag manifolds, Compactifications; symmetric and spherical varieties A classification of spherical Schubert varieties in the Grassmannian | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We introduce a tree-based method to find short double-base (DB) chains. As compared to the classical greedy approach, this new method is not only simpler to implement and faster, experimentally it also returns shorter chains on average. The complexity analysis shows that the average length of a chain returned by this tree-based approach is \(\frac{\log_2 n }{4.6419}\cdot p\). This tends to suggest that the average length of DB-chains generated by the greedy approach is not \(O(\log n/\log \log n)\). We also discuss generalizations of this method, namely to compute Step Multi-Base Representation chains involving more than 2 bases and extended DB-chains having nontrivial coefficients. double-base number system; scalar multiplication; elliptic curve cryptography Doche, C., Habsieger, L.: A tree-based approach for computing double-base chains. In: Mu, Y., Susilo, W., Seberry, J. (eds.) ACISP 2008. LNCS, vol.~5107, pp. 433--446. Springer, Heidelberg (2008) Cryptography, Applications to coding theory and cryptography of arithmetic geometry A tree-based approach for computing double-base chains | 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 Chebyshev curve \(C(a,b,c,\varphi)\) has a parametrization of the form \(x(t)=T_a(t)\); \(y(t)=T_b(t)\); \(z(t)=T_c(t+\varphi)\), where \(a\),\(b\),\(c\) are integers, \(T_n(t)\) is the Chebyshev polynomial of degree \(n\) and \(\varphi\in\mathbb{R}\). When \(C(a,b,c,\varphi)\) is nonsingular, it defines a polynomial knot. We determine all possible knot diagrams when \(\varphi\) varies. When \(a\),\(b\),\(c\) are integers, \((a,b)=1\), we show that one can list all possible knots \(C(a,b,c,\varphi)\) in \(\tilde{\mathcal{O}}(n^2)\) bit operations, with \(n=abc\). We give the parameterizations of minimal degree for all two-bridge knots with 10 crossings and fewer. zero dimensional systems; Chebyshev curves; Lissajous curves; Lissajous knots; polynomial knots; Chebyshev polynomials; minimal polynomial; Chebyshev forms Knots and links in the 3-sphere, Symbolic computation and algebraic computation, Plane and space curves Computing Chebyshev knot diagrams | 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 \(R\) be a discrete valuation ring, \(K\) the field of fractions of \(R\), and \(X_K\) a proper geometrically normal and geometrically connected scheme over \(K\). In this expository article, the author describes the construction of models \(X\) of \(X_K\) whose Picard functor \(\text{Pic}_{X/R}\) satisfies a Néron extension property for étale points. As the author indicates one source of motivation for such models \(X\) is that, when this property is satisfied, the Néron model can be reconstructed from \(\text{Pic}_{X/R}\).
The author also explains how these matters are related to the main results of \textit{C. Pepin} [Algebra Number Theory 6, No. 7, 1315--1348 (2012; Zbl 1321.14038), Math. Ann. 355, No. 1, 147--185 (2013; Zbl 1263.14046)], which are works of the author, as well as to a conjecture of Grothendieck concerning duality theory for Néron models of abelian varieties. Picard variety; Néron model; Grothendieck's pairing Picard schemes, higher Jacobians, Schemes and morphisms, Local ground fields in algebraic geometry, Abelian varieties of dimension \(> 1\) Néron models of Picard varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Consider \(Gr\), the Grassmannian of \(n\)-dimensional subspaces of the space of degree at most \(2n\) polynomials in \(z\). The Wronskian map \(Wr\) maps \(Gr\) to \(P(V)\) where \(V\) is the space of degree \(n(n+1)\) polynomials in \(z\). If \(h\in V\) has only real roots, then there is a well defined bijection between \(Wr^{-1}(h)\) and a certain set of Young tableaux. Young tableaux are combinatorial objects relevant in the combinatorial description of the ring structure of Grassmannians. This bijection is a consequence of the proof of Mukhin-Tarasov-Varchenko of the Shapiro-Shapiro conjecutre.
The paper under review considers a subset \(OGr\) of \(Gr\), the orthogonal Grassmannian. The \(Wr\)-value of points in \(OGr\) are perfect squares. The main theorem of the paper is the following: Suppose \(Wr(x)\) is a perfect square with only real roots. Then \(x\in OGr\) if and only if the tableau corresponding to \(x\) has a certain combinatorial symmetry.
Since the combinatorics of the symmetric tableaux is similar to the combinatorics of the earlier studied ``standard shifted tableaux'', the author can provide an elegant geometric proof of the Littlewood-Richardson rule for the orthogonal Grassmannian. Schubert calculus; Wronski map; orthogonal Grassmannian; symmetric tableaux Purbhoo, K., The Wronski map and shifted tableau theory, Int. math. res. not. IMRN, 24, 5706-5719, (2011) Classical problems, Schubert calculus The Wronski map and shifted tableau theory | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In the paper the sigma model with the target Heisenberg nil-manifold \(X=\Gamma /\;H(\mathbb{R})\) is studied. Here \(H(\mathbb{R})\) is Heisenberg group, which is a vector space \(\mathbb{R}^3\) endowed with a certain multiplication and \(\Gamma\) is a co-compact lattice. Also another sigma model is considered -- the one with target manifold \(\tilde{X}=\mathbb{Z}^3 / \mathbb{R}^3\) endowed with a non-flat \(U(1)\) gerbe. This model is believed to be \(T\)-dual to the first model.
In the paper these models are studied using the double field theory. The authors construct a double torus \(Y\) and identify \(L_2(Y)\) with the space of ground states of both models. From identification one obtains an explicit construction of the hole Hilbert spaces of states of both models. This construction gives an explicit \(T\)-duality isomorphism between two sigma-models. Also this construction endows the Hilbert space with an algebraic structure reminiscent to that of vertex algebra. The authors compute 4-point functions of scalar fields and found out that they contain dilogarithm singularities. It is pointed out how \(n\)-functions naturally reflect the dilogarithm identities. double field theory; sigma model; dilogarithms Aldi, M., Heluani, R.: Dilogarithms, OPE and twisted T-duality. Int. Math. Res. Not. IMRN \textbf{6}, 1528-1575 (2014). arXiv:1105.4280 [math-ph] Model quantum field theories, Spinor and twistor methods applied to problems in quantum theory, Quantum field theory on lattices, PDEs on Heisenberg groups, Lie groups, Carnot groups, etc., Polylogarithms and relations with \(K\)-theory, Applications of vector bundles and moduli spaces in mathematical physics (twistor theory, instantons, quantum field theory) Dilogarithms, OPE, and twisted \(T\)-duality | 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 is a detailed study of the motivic Milnor fibre at infinity of a polynomial \(f\) of two variables in an algebraically closed field \(K\) of characteristic zero, in relation to its Newton polygon at infinity. The motivic Milnor fibre at infinity of \(f\), denoted by \(S_{f,\infty}\), has been defined by Matsui and Takeuchi, and independently by \textit{M. Raibaut} [Bull. Soc. Math. Fr. 140, No. 1, 51--100 (2012; Zbl 1266.14012)], by using the motivic integration introduced by Kontsevich and developed by Denef and Loeser. It follows from their work that \(S_{f,\infty}\) is a motivic incarnation of the topological Milnor fibre at infinity \(F_{\infty}\) of \(f\) endowed with its monodromy action \(T_{\infty}\).
The authors produce, in Theorems 3.8 and 3.23, some detailed formulas for the zeta-functions defining the motivic Milnor fibre at infinity \(S_{f,\infty}\), and the motivic nearby cycles at infinity \(S_{f,a}^{\infty}\), for some value \(a\in K\), in terms of certain motives associated to faces of the Newton polygon at infinity.
\textit{L. Fantini} and \textit{M. Raibaut} [in: Arc schemes and singularities. Hackensack, NJ: World Scientific. 197--220 (2020; Zbl 1440.14072)] proved over \(\mathbb C\) that the Euler characteristic of the motive \(S_{f,a}^{\infty}\) is equal to zero for all \(a\in \mathbb{C}\) except of finitely many values, for which it is equal, modulo a sign, to the Milnor-Lê jump invariant \(\lambda_{a}(f)\). This invariant has been defined and used by several authors before, see e.g. [\textit{M. Tibăr}, Polynomials and vanishing cycles. Cambridge: Cambridge University Press (2007; Zbl 1126.32026)] for more details and references. This leads to the definition of the motivic bifurcation locus \(B_f^{mot}\), in case \(f\) has isolated singularities, by taking into account both the affine singularities and these jumps at infinity.
Classically, the topological bifurcation locus \(B_f^{top}\) is defined as the finite set of values \(a\in \mathbb C\) for which the fibre \(f^{-1}(a)\) is singular, or it is not singular but \(\lambda_{a}(f) \not= 0\).
Here the authors define the Newton bifurcation locus \(B_f^{Newton}\) for any polynomial \(f\) depending effectively on two variables, whatever its Newton polygon might be.
The main results of the paper is the equality (Theorem 3.30) of the bifurcation loci in case of \(f: \mathbb C^{2}\to \mathbb C\) with isolated singularities: \(B_f^{mot} = B_f^{mot} = B_f^{Newton}\). The proof is developed in Section 3 of the paper over several technical lemmas, most of which holding for coefficients in \(K\). motivic Milnor fibre at infinity; Newton polygons; bifurcation values of polynomials; motivic nearby cycles at infinity Arcs and motivic integration, Local ground fields in algebraic geometry, Toric varieties, Newton polyhedra, Okounkov bodies, Singularities in algebraic geometry Newton transformations and motivic invariants at infinity of plane curves | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(F\) be a non-Archimedean local field, \({\breve F}\) be the the completion of the maximal unramified extension of \(F\), and \(G\) be a connected reductive group over \(F\). Further, let \(\sigma\) denote the Frobenius morphism on \(G({\breve F})\) and \({\breve I}\) be a \(\sigma\)-stable Iwahori subgroup of \(G({\breve F})\). Consider the double coset \({\breve I}\dot{w}{\breve I}\) (a Bruhat-like cell for \(G({\breve F})\)) associated to a representative of an element \(w\) in the associated Iwahori-Weyl group. One of the main foci of this survey article is the question of understanding the intersection of \({\breve I}\dot{w}{\breve I}\) with a \(\sigma\)-conjugacy class of \(G({\breve F})\), with particular goals of understanding when such an intersection is nonempty and, if so, determining its dimension. Results are presented for various special cases. For example, non-emptiness is determined for \textit{basic} \(\sigma\)-conjugacy classes, and then, in that setting, a dimension formula is given when \(w\) lies in the \textit{Shrunken Weyl chamber}. Also, some analogous results are given for parahoric subgroups in place of Iwahori subgroups. The various collections of subsets of \(G{(\breve F})\) considered here may be used to provide a group-theoretic model of certain characteristic subsets of Shimura varieties. The author presents a summary of this correspondence.
The first part of the article provides a summary of a number of combinatorial results on conjugacy classes for affine Weyl groups, along with numerous illustrative examples. Topics discussed include straight conjugacy classes; reduction from non-straight to straight conjugacy classes; special partial conjugacy classes; partial orders on the set of straight conjugacy classes and the set of special partial conjugacy classes; and the parameterization of conjugacy classes in terms of standard quadruples.
In the last part of the article, the author considers related affine Hecke algebras and their representation theory. For a finite group, the trace map provides a vector space isomorphism between the cocenter of the group algebra over \({\mathbb C}\) and the dual of the Grothendieck group of finite-dimensional complex representations of the group algebra. The author reviews the extension of this idea to finite Hecke algebras (with examples) and then considers the affine Hecke algebra case, where one must consider the \textit{rigid} cocenter of the Hecke algebra and the \textit{rigid} quotient of the Grothendieck group. In the modular setting (either over a field of positive characteristic or when the defining parameter is a root of unity), the trace map will not be injective. The author makes a conjecture on the kernel and provides some examples in support of this. Again analogous to the group algebra case, for a finite Hecke algebra, one can define a character table and consider its determinant. For an extended affine Hecke algebra, this idea may be replaced with a \textit{rigid} character table, and the author makes a second conjecture about the \textit{rigid} determinant (again with supporting evidence). Coxeter groups; affine Weyl groups; \(p\)-adic groups; conjugacy classes; class polynomials; admissible sets; Shimura varieties; Deligne-Lusztig varieties; Iwahori subgroup; parahoric subgroup; affine Hecke algebras; cocenters; character tables He, X., Hecke algebras and \textit{p}-adic groups, (), 73-135 Reflection and Coxeter groups (group-theoretic aspects), Hecke algebras and their representations, Modular and Shimura varieties, Linear algebraic groups over local fields and their integers, Representations of Lie and linear algebraic groups over local fields, Research exposition (monographs, survey articles) pertaining to group theory Hecke algebras and \(p\)-adic 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 authors develop the Littlewood-Richardson homotopy algorithm. For this purpose, they use numerical continuation to compute solutions to Schubert problems on Grassmannians and it is based on the geometric Littlewood-Richardson rule. The main idea of the algorithm is the new optimal formulation of Schubert problems in local Stiefel coordinates as systems of equations. The implementation presented can solve problem instances with tens of thousands of solutions. Schubert calculus; Grassmannian; Littlewood-Richardson rule; numerical homotopy continuation Classical problems, Schubert calculus, Numerical computation of solutions to systems of equations, Geometric aspects of numerical algebraic geometry Numerical Schubert calculus via the Littlewood-Richardson homotopy algorithm | 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 \(\mathfrak{g}\) be a symmetrizable Kac-Moody Lie algebra. Associated to two dominant integral weights \(\lambda\), \(\mu \in \mathcal{P}^+\) are the integrable, (irreducible) highest weight representations \(V(\lambda)\) and \(V(\mu)\). Then the content of the tensor product decomposition problem is to express the product \(V(\lambda) \otimes V(\mu)\) as a direct sum of irreducible components, i.e., find the decomposition \(V(\lambda) \otimes V(\mu) = \displaystyle{\bigoplus_{\nu \in \mathcal{P}^+}} V(\nu)^{\oplus m_{\lambda, \mu}^\nu}\), where \(m_{\lambda, \mu}^\nu \in \mathbb{Z}_{\geq 0}\) is the multiplicity of \(V(\nu)\) in \(V(\lambda) \otimes V(\mu)\).
Now, let \(\mathfrak{g}\) be an affine Kac-Moody Lie algebra and let \(\lambda, \mu\) be two dominant integral weights for \(\mathfrak{g}\). The authors prove that for any positive root \(\beta\), \(V(\lambda)\otimes V(\mu)\) contains \(V(\lambda+\mu-\beta)\) as a component, where \(V(\lambda)\) denotes the integrable highest weight (irreducible) \(\mathfrak{g}\)-module with highest weight \(\lambda\), extending the case of finite dimensional semisimple Lie algebras to the affine Kac-Moody Lie algebras (Theorem 9.1, page 841). One crucial ingredient in the proof is the action of Virasoro algebra on the tensor product \(V(\lambda)\otimes V(\mu)\). The authors also prove the corresponding geometric results including the higher cohomology vanishing on the \(\mathcal{G}\)-Schubert varieties in the product partial flag variety \(\mathcal{G}/\mathcal{P}\times \mathcal{G}/\mathcal{P}\) with coefficients in certain sheaves coming from the ideal sheaves of \(\mathcal{G}\)-sub-Schubert varieties (Theorem 11.7, page 854), thereby proving the surjectivity of the Gaussian map. affine Kac-Moody Lie algebra; integrable highest weight modules; root components; Virasoro algebra; higher cohomology vanishing; Schubert varieties; partial flag varieties Grassmannians, Schubert varieties, flag manifolds, Homogeneous spaces and generalizations, Kac-Moody groups, Representations of Lie and real algebraic groups: algebraic methods (Verma modules, etc.) Root components for tensor product of affine Kac-Moody Lie algebra modules | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper continues the authors' investigations [see \textit{K. Altmann} and \textit{J. A. Christophersen}, Manuscr. Math. 115, No. 3, 361--378 (2004; Zbl 1071.13008) and \textit{M. Haiman} and \textit{B. Sturmfels}, J. Algebr. Geom. 13, No. 4, 725--769 (2004; Zbl 1072.14007)] on the graph \({\mathcal G}\) of monomial ideals in the polynomial ring \(R=k[x_1,\dots,x_n]\), \(k\) a field. \({\mathcal G}\) is the infinite graph with the monomial ideals in \(R\) as vertex set. Two monomial ideals \(M_1\) and \(M_2\) are connected by an edge if there exists an ideal \(I\) in \(R\) such that the set of all initial monomial ideals of \(I\), with respect to all term orders, is precisely \(\{M_1,M_2\}\). \(I\) is called edge providing in this case. It is well known that \(I,M_1,M_2\) have many invariants in common. Each invariant yields a stratification of \({\mathcal G}\). A first proposition concerns the subgraph \({\mathcal G^r}\) obtained by restriction to artinian ideals of colength \(r\): Each such stratum is a connected component of \({\mathcal G}\). The main result (theorem 8) characterizes edge providing ideals as ``very homogeneous'':
There exists upto multiples a single \(c\in\mathbb Z^n\) such that \(I\) is \(A\)-graded for \(A=\mathbb Z^n/c\mathbb Z\).
This allows to define the Schubert scheme \(\Omega_c(M_1,M_2)=\Omega(M_1,M_2)\) of all \(A\)-homogeneous edge providing ideals connecting \(M_1\) and \(M_2\). A thorough analysis of the settings yields an algorithm that computes, for given \(M_1\) and \(M_2\), the direction \(c\) and \(\Omega(M_1,M_2)\) as affine scheme. More generally, the authors consider \(A\)-homogeneous ideals for arbitrary gradings \(\deg\:\mathbb Z^n\to A\) (including the standard one). Define \(h_I\:A\to \mathbb N\) as the Hilbert function of \(I\), i.e., \(h_I(a),\;a\in A\), is the \(k\)-dimension of the \(a\)-homogeneous part of \(I\). In the above situation, for \(I\in \Omega(M_1,M_2)\) the Hilbert functions of \(I\), \(M_1\) and \(M_2\) coincide. For positive gradings, i.e., \(\mathbb N^n\cap \text{ker}(\deg)=(0)\), this is also the general situation:
\(\Omega_c(M_1,M_2)\not=\emptyset\) implies \(\deg(c)=0\) and hence the Schubert schemes describe an essential part of the multigraded Hilbert scheme \(\text{Hilb}_h\) of all \(A\)-homogeneous ideals with given Hilbert function \(h\). This part is sufficient to detect connectedness: Over \(k=\mathbb R\) or \(k=\mathbb C\), \(\text{Hilb}_h\) is connected if and only if the induced subgraph \({\mathcal G}(\text{Hilb}_h)\) is connected.
Section 4 discusses properties of the Schubert schemes for square-free monomial ideals. The results are more technical and continue the investigations started by \textit{K. Altmann} and \textit{J. A. Christophersen} [loc. cit.]. In particular, it turns out that neighboring square-free ideals are connected by a generalization of the bistellar flip construction [see, e.g., \textit{O. Viro}, Proc. Workshop Differential Geometry Topology, Alghero 1992, World Scientific. 244--264 (1993; Zbl 0884.57015) or \textit{D. Maclagan} and \textit{R. R. Thomas}, Discrete Comput. Geom. 27, No. 2, 249--272 (2002; Zbl 1073.14503)]. The paper ends with a list of open problems about the graph \({\mathcal G}\) and the Schubert schemes. graph of monomial ideals; multigraded Hilbert scheme; Schubert scheme; Gröbner bases; Gröbner degenerations; Stanley-Reisner ideals DOI: 10.1016/j.jpaa.2004.12.030 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Commutative rings defined by monomial ideals; Stanley-Reisner face rings; simplicial complexes, Parametrization (Chow and Hilbert schemes), Grassmannians, Schubert varieties, flag manifolds The graph of monomial ideals | 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 define polar classes associated to a singular holomorphic distribution of tangent subspaces of a projective manifold. We prove that these polar classes can be calculated in terms of the Chern-Mather classes of the tangent sheaf of the distribution and reciprocally. We use their degrees to establish a bound for the degrees of some polar classes associated to an invariant variety. holomorphic foliations; polar classes; Schubert cycles; Chern-Mather classes; invariant varieties Mol (R.S.).- Classes polaires associées aux distributions holomorphes de sous-espaces tangents. Bull. Braz. Math. Soc. (N.S.), 37(1):29-48, (2006). Zbl1120.32019 MR2223486 Singularities of holomorphic vector fields and foliations, Grassmannians, Schubert varieties, flag manifolds Polar classes associated with holomorphic distributions of tangent subspaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This article investigates the additive conjecture of Strassen for bilinear maps, which deals with computational complexity. In this paper this conjecture is considered from an algebraic geometry point of view.
The conjecture is reformulated in terms of symmetric tensors and of Waring rank.
More precisely: the symmetric tensors lead to the consideration of homogeneous polynomials in degree \(d\), also known as forms \(F\) of degree \(d\). The Waring rank of a form \(F\) is defined as the minimum on the set of integers \(r\) such that \(F =\sum_{i=1}^r L_{i}^{d}\), for linear forms \(L_{i}\).
In this new approach the conjecture states that:
For \(F\) and \(G\) non zero forms of degree \(d>1\), defined on disjoint sets of variables one has the following additive property: \(\mathrm{rk}(F+G)=\mathrm{rk}(F)+\mathrm{rk}(G)\).
In this paper it is shown that this conjecture is true in the case of one and two variables and for forms of small Waring rank. This result can be formulated as follows:
Let \(d>1\) and let \(F\in\mathbb{C}[x_{0},\dots,n_{n}], G\in\mathbb{C}[y_{0},\dots,y_{m}]\) be non-zero degree \(d\) forms. If \(n=0\), or \(m=0\) or \(n=1, m=1\) then \(\mathrm{rk}(F+G)=\mathrm{rk}(F)+\mathrm{rk}(G)\).
The main tool to prove these statements is to use Hilbert functions of finite set of points. additive Strassen conjecture; symmetric tensors; computational complexity; polynomials \beginbarticle \bauthor\binitsE. \bsnmCarlini, \bauthor\binitsM. V. \bsnmCatalisano and \bauthor\binitsL. \bsnmChiantini, \batitleProgress on the symmetric Strassen conjecture, \bjtitleJ. Pure Appl. Algebra \bvolume219 (\byear2015), no. \bissue8, page 3149-\blpage3157. \endbarticle \OrigBibText Enrico Carlini, Maria Virginia Catalisano, and Luca Chiantini, Progress on the symmetric Strassen conjecture , J. Pure Appl. Algebra 219 (2015), no. 8, 3149-3157. \endOrigBibText \bptokstructpyb \endbibitem Effectivity, complexity and computational aspects of algebraic geometry Progress on the symmetric Strassen conjecture | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors investigate the cohomology modules \(H^i(X(w),L_\lambda)\), for \(\lambda\) non-dominant on a Schubert variety \(X (w)\) in the generalized flag variety, especially, the vanishing or non-vanishing of \(H^i(X(\omega), L_\lambda)\). For a generic \(\lambda\), the authors give a criterion for vanishing (resp. non-vanishing) of \(H^{l(w)}(X (w), L_\lambda)\) (resp. \(H^0(X(w), L_\lambda)\)). For a general \(j\), the authors give partial results on the vanishing of \(H^j(X(w), L_\lambda)\). cohomology; line bundles; Schubert varieties P. Polo, Variétés de Schubert et excellentes filtrations, Astérisque 173{174 (1989), no. 10-11, 281-311.} Grassmannians, Schubert varieties, flag manifolds, Representation theory for linear algebraic groups Cohomology of line bundles on Schubert varieties. I | 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 quadratic hypersurface over a field \(F\) of characteristic different from two. It is shown here that the torsion subgroup of the Chow group \(\text{CH}^ 3X\) is either trivial or \(\mathbb{Z}/2\mathbb{Z}\). On the other hand, it is known that the groups \(\text{CH}^ pX\), \(p=0,1\), have only trivial torsion, and the torsion part of \(\text{CH}^ 2X\) is either trivial or \(\mathbb{Z}/2\mathbb{Z}\) [the author, Leningr. Math. J. 2, No. 1, 119- 138 (1991); translation from Algebra Anal. 2, No. 1, 141-162 (1990)]. The author and \textit{A. S. Merkur'ev} [Leningr. Math. J. 2, No. 3, 655-671 (1991); translation from Algebra Anal. 2, No. 3, 218-235 (1990)] proved that with suitable choices for \(F\) and \(X\) the groups \(\text{CH}^ pX\), \(p\geq 4\), can have arbitrarily many elements of order two. Grothendieck groups; torsion subgroup of the Chow group Algebraic cycles, Parametrization (Chow and Hilbert schemes), Grothendieck groups (category-theoretic aspects) On cycles of codimension 3 on a projective quadric | 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 formula in the title is for the resultant \(J(d_1,d_2)\) of the class polynomials for the orders of discriminant \(d_1,d_2\) of relatively prime fundamental discriminants of imaginary quadratics, due to \textit{B. H. Gross} and \textit{D. B. Zagier} [J. Reine Angew. Math. 355, 191-220 (1985; Zbl 0545.10015)]. The formula is interesting because it presents a highly composite resultant with factors \(F(m)\) equal to powers of primes of dividing \(m\) as \(m\) runs over positive integers \((d_1d_2-x^2)/4\). The author has conjectural extensions for the cases where \(d_1,d_2\) need not be fundamental and may have \((d_1,d_2)\) a prime power. There is an extended calculation serving as verification. complex multiplication; singular moduli; resultant of the class polynomials; orders; fundamental discriminants of imaginary quadratics Hutchinson, T.: A conjectural extension of the Gross-Zagier formula on singular moduli. Tokyo J. Math. 21, 255--265 (1998) Complex multiplication and moduli of abelian varieties, Modular and automorphic functions, Complex multiplication and abelian varieties A conjectural extension of the Gross-Zagier formula on singular moduli | 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 \(R\) be a regular local ring with field of fractions \(K\). Let \(\mathcal{A} \) be an Azumaya algebra over \(R\). From the paper: ``Our work was motivated by \dots{} a paper by \textit{I.A. Panin} and \textit{M. Ojanguren} [Math. Z. 237, No. 1, 181-198 (2000; Zbl 1042.11024)] on the Grothendieck conjecture for Hermitian spaces. The point was to offer a good axiomatization for the method used in that paper.''
The results of the paper are as follows:
If \(R\) contains a field, then \(R^{\ast }/\)Nrd\((\mathcal{A}^{\ast })\rightarrow K^{\ast }/\)Nrd\((\mathcal{A}_{K}^{\ast })\) is injective; and the canonical map \(H_{\text{ét}}^{1}(R,\text{SL}_{1,\mathcal{A} })\rightarrow H_{\text{ét}}^{1}(K,\text{SL}_{1,\mathcal{A}_{K}})\) is injective. These results were originally proved by \textit{I. Panin} and \textit{A.A. Suslin} [St. Petersbg. Math. J. 9, No. 4, 851-858 (1998); translation from Algebra Anal. 9, No. 4, 215-223 (1997; Zbl 0902.16019)]. Here ``Nrd'' is the reduced norm.
If \(\sigma \) is a unitary involution of \(\mathcal{A}\) and \(R\) contains an infinite field of characteristic not equal to \(2\), then \(U(C)/\text{Nrd}(U( \mathcal{A}))\rightarrow U(C_{K})/ \text{Nrd}(U(\mathcal{A}_{K}))\) is injective and the canonical map
\[
H_{\text{ét}}^{1}(R,\text{SU}_{1,\mathcal{A} })\rightarrow H_{\text{ét}}^{1}(K,SU_{1,\mathcal{A}_{K}})
\]
is trivial. Here \(U(C)\) is the unitary group of the center of \(\mathcal{A}.\)
For \(d\) a natural number, if \(R\) contains a field, then \(R^{\ast }/\)Nrd\(( \mathcal{A}^{\ast })(R^{\ast })^{d}\rightarrow K^{\ast }/\text{Nrd}(\mathcal{A} _{K}^{\ast })(K^{\ast })^{d}\) is injective.
If \(\sigma \) is a unitary involution of \(\mathcal{A}\) and \(R\) contains an infinite field of characteristic not equal to \(2\), then \(U(C)/\text{Nrd}(U( \mathcal{A}))U(C)^{d}\rightarrow U(C_{K})/\text{Nrd}(U(\mathcal{A} _{K}))U(C_{K})^{d}.\)
This paper originally appeared in Russian. The English translation can be difficult to follow. regular local ring; Azumaya algebra; Grothendieck conjecture for Hermitian spaces Homogeneous spaces and generalizations, Regular local rings On Grothendieck's conjecture about principal homogeneous spaces for some classical algebraic 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 In this well-written article, the authors continue the program of removing hypotheses from the Riemann-Roch theorems. The starting point is the Grothendieck-Riemann-Roch theorem [\textit{A. Borel} and \textit{J.-P. Serre}, Bull. Soc. Math. Fr. 86(1958), 97-136 (1959; Zbl 0091.330)] proved for proper morphisms between smooth, quasiprojective varieties. Then \textit{P. Baum}, \textit{W. Fulton} and \textit{R. D. MacPherson} [Publ. Math., Inst. Hautes Études Sci. 45, 101-145 (1975; Zbl 0332.14003)] eliminated the hypothesis that the schemes be smooth varieties, thus generalizing the Grothendieck-Riemann-Roch theorem to the category of quasiprojective schemes (over an arbitrary base field) and proper morphisms; the main point there was to consider homology, \(A_*X_ Q\), and to construct the natural (Riemann-Roch) transformation \(\tau_ X: K_ 0X\to A_*X_ Q\) such that \(\tau_ X({\mathcal O}_ X)=Td(X)\), the Todd class of X. In this article, the authors eliminate the assumption of quasiprojectivity, proving a Riemann-Roch theorem in the category of algebraic schemes (over an arbitrary base field) and proper morphisms. Their method also works to generalize the topological K-theory Riemann- Roch theorem of \textit{P. Baum}, \textit{W. Fulton} and \textit{R. D. MacPherson} [Acta Math. 143, 155-192 (1979; Zbl 0474.14004)] to algebraic \({\mathbb{C}}\)- schemes. The basic ingredients of the proof are the Riemann-Roch theorem for quasiprojective schemes, Chow's lemma allowing one to approximate any algebraic scheme X by a quasiprojective ''envelope'' X'\(\to X\), and an exact sequence from algebraic K-theory (involving only \(K_ 0\) and \(K_ 1)\), which relates \(\tau_{X'}\) to \(\tau_ X\). They also include a proof of an implicit assumption made by \textit{H. Gillet} [Algebraic K- theory, Proc. Conf. Evanston 1980, Lect. Notes Math. 854, 141-167 (1981; Zbl 0478.14011)], concerning the equality of the Riemann-Roch transformations \(\tau^ S_ X\) and \(\tau^ T_ X\) from \(K_ 0X\) to \(A^*X_ Q\) for a given scheme X, regarded as a quasiprojective S- scheme or as a quasiprojective T-scheme. Grothendieck-Riemann-Roch theorem; algebraic schemes; algebraic \({\mathbb{C}}\)-schemes [F-G]W. Fulton, H. Gillet: ``Riemann Roch for general Algebraic Varieties{'', Bull. Soc. Math. France 111 (1983) pp. 287--300.} Riemann-Roch theorems, Applications of methods of algebraic \(K\)-theory in algebraic geometry Riemann-Roch for general algebraic 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 In this paper, we explicitly compute all Littlewood-Richardson coefficients for semisimple and Kac-Moody groups \(G\), that is, the structure constants (also known as the Schubert structure constants) of the cohomology algebra \(H^*(G/P,\mathbb C)\), where \(P\) is a parabolic subgroup of \(G\). These coefficients are of importance in enumerative geometry, algebraic combinatorics and representation theory. Our formula for the Littlewood-Richardson coefficients is purely combinatorial and is given in terms of the Cartan matrix and the Weyl group of \(G\). However, if some off-diagonal entries of the Cartan matrix are \(0\) or \(-1\), the formula may contain negative summands. On the other hand, if the Cartan matrix satisfies \(a_{ij}a_{ji}\geq 4\) for all \(i\), \(j\), then each summand in our formula is nonnegative that implies nonnegativity of all Littlewood-Richardson coefficients. We extend this and other results to the structure coefficients of the \(T\)-equivariant cohomology of flag varieties \(G/P\) and Bott-Samelson varieties \(\Gamma_{\mathbf i}(G)\). Littlewood-Richardson coefficients; flag varieties; Schubert varieties; semisimple groups; Kac-Moody groups; reflection groups; Cartan matrices; Weyl groups Berenstein, A; Richmond, E, Littlewood-Richardson coefficients for reflection groups, Adv. Math., 284, 54-111, (2015) Representation theory for linear algebraic groups, Cohomology theory for linear algebraic groups, Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds, Reflection and Coxeter groups (group-theoretic aspects), Kac-Moody groups Littlewood-Richardson coefficients for reflection 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 We express Hurwitz numbers of polynomials of arbitrary topological type in terms of intersection numbers on the moduli space of curves of genus zero with marked points. Hurwitz numbers of polynomials; moduli space of curves; intersection numbers Shadrin, S.: Polynomial Hurwitz numbers and intersections on M\={}0,k. Funct. anal. Appl. 37, 78-80 (2003) Rational and birational maps, Families, moduli of curves (algebraic), Polynomial rings and ideals; rings of integer-valued polynomials Polynomial Hurwitz numbers and intersections on \(\overline{\mathcal{M}}_{0,k}\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(X\) be a smooth quartic double solid, i.e., \(\pi: X \rightarrow \mathbb{P}^3\) is a double covering branched along a smooth quartic \(Q\subset\mathbb{P}^3\). Hence, \(X\) is a Fano variety of index 2. Recently, moduli of rank two vector bundles with small Chern classes
on these varieties have attracted some interest; \textit{M. Szurek} and \textit{J. A. Wisniewski} [Rev. Roum. Math. Pures Appl. 38, No.7--8, 729--741 (1993; Zbl 0816.14015)] constructed such moduli for \(c_1=-1\), \(c_2=2\) and the reviewer for \(c_1=0\), \(c_2=2\) [in: Proc. Berlin Math. Soc. 1997--2000, 181--200 (2001; Zbl 1082.14045)].
In the paper under review, the author applies the techniques developed in several joint papers
with \textit{D. Markushevich} [J. Algebr. Geom. 10, 37--62 (2001; Zbl 0987.14028); Doc. Math., J. DMV 5, 23--47 (2000; Zbl 0938.14021)]
to study moduli of bundles on certain Fano varieties in terms of an associated Abel-Jacobi map. More precisely,
the main interest here is the moduli space \(M(2;0,3)\) of stable vector bundles of rank two with Chern classes \(c_1=0\), \(c_2=3\), which are related via Serre correspondence to an open set \(H\) of the family of elliptic quintics on \(X\). The author studies the associated Abel-Jabobi map \(\Phi_H: H\rightarrow J(X)\). He shows that \(\Phi_H\) factors through \(M(2;0,3)\) and that this gives the Stein factorization of the Abel-Jacobi map. Furthermore, he proves that the image of \(\Phi_H\) is a translate of the theta divisor on \(J(X)\), using a criterion of
\textit{G. Welters} [``Abel-Jacobi isogenies for certain types of Fano threefolds'', Math. Centre Tracts, 141, Mathematisch Centrum, Amsterdam (1981; Zbl 0474.14028)].
It should be noted that an extended version has recently been published as a joint paper with \textit{D. Markushevich} [Int. Math. Res. Not. 51, 2747--2778 (2003; Zbl 1048.14028)]; containing detailed proofs and further results, showing irreducibility of \(H\) and computing the degree of the Stein factorization as \(84\). quartic double solid; intermediate Jacobian; Abel-Jacobi map; theta divisor A. S. Tikhomirov, New component of the moduli space \?(2;0,3) of stable vector bundles on the double space \?³ of index two, Acta Appl. Math. 75 (2003), no. 1-3, 271 -- 279. Monodromy and differential equations (Moscow, 2001). Picard schemes, higher Jacobians, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, \(3\)-folds, Fano varieties New component of the moduli space \(M\)(2;0,3) of stable vector bundles on the double space \(\mathbb{P}^3\) of index two | 0 |
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