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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 One of the main invariants of a fibration \(f:S\longrightarrow B\) of a surface \(S\) over a smooth complete curve \(B\) of genus \(\gamma\geq 0\) is the slope
\[
s(f)=\frac{(\omega_f\cdot\omega_f)}{\deg f_*\omega_f}
\]
where \(\omega_f=\omega_S\otimes (f^*\omega_B)^{-1}\). It is well-known that, if \(g\geq 2\) is the genus of a fibre, the \textit{slope inequality} holds:
\[
s(f)\geq \frac{4(g-1)}{g}
\]
This bound is sharp and it is reached by some hyperelliptic fibrations.
The two authors show that, for double covers of curves of genus \(\gamma \neq 0\) with \(g\geq 4\gamma+1\), a sharp bound for the slope is given by
\[
s(f)\geq \frac{4(g-1)}{g-\gamma}
\]
They also characterize the fibrations that reach the bound as double covers of locally trival genus-\(\gamma\) fibrations with some mild conditions on the branch locus \(R\). fibration; double covers; slope Cornalba, M.; Stoppino, L., A sharp bound for the slope of double cover fibrations, Michigan math. J., 56, 3, 551-561, (2008) Fibrations, degenerations in algebraic geometry, Surfaces and higher-dimensional varieties A sharp bound for the slope of double cover 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 Die Abhandlung gibt eine systematische Untersuchung der Perioden, welche den Doppelintegralen erster Gattung in einem Körper algebraischer Funktionen zweier Variabeln zugehören, indem sie diese Perioden auf die Perioden analytischer Funktionen von nur einer Veränderlichen reduziert. The periods of double integrals. Surfaces and higher-dimensional varieties On the periods of double integrals in the theory of algebraic functions of two variables. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We define skew Schubert polynomials to be normal form (polynomial) representatives of certain classes in the cohomology of a flag manifold. We show that this definition extends a recent construction of Schubert polynomials due to \textit{N. Bergeron} and \textit{F. Sottile} [Duke Math. J. 95, 373--423 (1998; Zbl 0939.05084)] in terms of certain increasing labeled chains in Bruhat order of the symmetric group. These skew Schubert polynomials expand in the basis of Schubert polynomials with nonnegative integer coefficients that are precisely the structure constants of the cohomology of the complex flag variety with respect to its basis of Schubert classes. We rederive the construction of Bergeron and Sottile in a purely combinatorial way, relating it to the construction of Schubert polynomials in terms of rc-graphs. Schubert polynomial; Bruhat order; Littlewood-Richardson coefficient Lenart, C., Sottile, F.: Skew Schubert polynomials. Proceedings of the American Mathematical Society 131(11), 3319--3328 (2003) Symmetric functions and generalizations, Grassmannians, Schubert varieties, flag manifolds, Combinatorics of partially ordered sets Skew 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 To give a relevant review, we start by recalling some elementary concepts, which is freely referred from [``Unramified division algebras do not always contain Azumaya maximal orders'', Preprint, \url{arXiv.1209.2216}], by \textit{B. Antieau} and \textit{B. Williams}: For a field \(K\) the Artin-Wedderburn Theorem says that every central simple \(K\)-algebra \(A\) is isomorphic to an algebra \(M_n(D)\) of \(n\times n\)- matrices over a finite dimensional
central \(K\)-division algebra \(D\). \(M_n(D)\) and \(M_{n'} (D')\) are Brauer-equivalent if \(D\) and \(D'\) are isomorphic over \(K\). The set of Brauer-equivalence classes is a group under tensor product, \(\operatorname{Br}(K),\) the Brauer group of \(K\). The index of an equivalence class \(a = \operatorname{cl}(M_n(D))\in\operatorname{Br}(K)\) is the degree of \(D\).
Let \(X\) be a connected scheme. A central simple algebra over \(K\) is generalized to the concept of an Azumaya algebra over \(X\), i.e., an Azumaya algebra \(\mathcal A\) is a locally free sheaf of algebras which étale locally takes the form of a matrix algebra. Thus, there is an étale cover \(\pi :U\rightarrow X\) such that \(\pi^\ast\mathcal A=M_n(\mathcal O_U).\)
The degree of A is \(n.\) With this, Brauer equivalence and a contravariant Brauer group functor can be defined. However, over a scheme we don't have Artin-Wedderburn theory and cannot know that a Brauer class \(\alpha\in\operatorname{ Br}(X)\) contains an Azumaya algebra \(\mathcal A\) whose degree divides that of all other Azumaya algebras having class \(\alpha\). The index of \(\alpha\) is defined to be the greatest common divisor of the degrees of Azumaya algebras with Brauer class \(\alpha\). For a regular integral noetherian scheme \(X\) with generic point \(\operatorname{Spec} K\) and a central
simple \(K\)-algebra \(A,\) an order in \(A\) over \(X\) is a torsion-free coherent \(\mathcal O_X\)-algebra \(\mathcal A\) , such that \(\mathcal A\otimes_{\mathcal O_X }K\cong A.\)
A maximal order in \(A\) over \(X\) is an order which is not a proper subalgebra of any other order in \(A\) over \(X.\)
Now we review the present article:
The article consider structure theory of maximal orders over algebraic surfaces. There is a satisfying minimal model program, and the moduli of Azumaya orders in a fixed unramified division algebra and related moduli problems are thoroughly studied. This article extends the moduli theory to orders in a ramified Brauer class, and consider phenomena similar to that which occurs in moduli theory of stable projective surfaces, coming from an analogue of Kollár's condition on the compatibility of the reflexive powers of the dualizing sheaf with base change.
Working over surfaces means that the global dimension of the orders are \(2\), and simpler than in the general theory of Kollár. The authors arrive at a satisfying moduli space with a natural compactification and a virtual fundamental class.
As in the commutative theory, the moduli problem given by fixing the properties of the fibres of a family contains a refined version as a bijective closed substack. The refined moduli problem is described as a moduli problem of Azumaya algebras on stacks rather than orders on varieties, and can also be considered as a moduli theory of parabolic Azumaya algebras. These Azumaya algebras have precise interaction with the ramification divisor coming from the structure of hereditary orders in matrix algebras over discrete valuation rings. This theory was first given by \textit{A. Brumer} [Bull. Am. Math. Soc. 69, 721--724 (1963; Zbl 0113.26002)] who gave the Azumaya algebras under consideration a structure named Brumer log terminal structure, or blt for short.
The paper starts out by relating hereditary algebras over complete discrete valuation rings to Azumaya algebras over root construction stacks. This is then globalized, and leads to a simple approach to families of maximal orders. This gives two resulting moduli problems which are compared, using ideas similar to those of Kollár in his theory of hulls and husks, and a local analysis of reflexive Azumaya algebras on families of rational double points. The comparison leads to several nice and important results.
The final result is the compactification of the Azumaya problem using algebra-objects of the derived category of a stack which can be seen as parabolic generalized Azumaya algebras.
We find room for citing the following definitions, as they are essential for the exposition; verbatim:
Definition 2.2. The hereditary site \(\mathcal F\) of \(\operatorname{Spec} R\) is the site whose underlying category consists of faithfully flat quasi-finite étale \(R\)-schemes \(U\rightarrow\operatorname{Spec} R\) with \(U\) of pure dimension \(1,\) with coverings given by collections of \(R\)-maps \(U_i\rightarrow U\) that are jointly surjective.
Definition 2.3. Given an object \(U\rightarrow\operatorname{Spec}R\) of \(\mathcal F,\) an Azumaya algebra \(\mathcal A\) on \(\mathcal X_U\) is \(n\)-typed if for each closed point \(u\in U\) the restriction of \(\mathcal A\) to \(\mathcal X\otimes_R\mathcal O_{U,u}\) has type \(m\) for some positive integer \(m\) dividing \(n.\)
Definition 2.4. Given an object \(U\rightarrow\operatorname{Spec}R\) of \(\mathcal F\), the stack \(\mathcal A_n\) has as objects over \(U\) the groupoid of \(n\)-typed Azumaya algebras \(\mathcal A\) of degree \(n\) on \(\mathcal X\times_{\operatorname{Spec}R}U.\) The stack \(\mathcal H_n\) has as objects the groupoid of \(n\)-typed hereditary orders on \(U\).
Definition 2.5. Suppose \(n\) is invertible in the residue field \(\kappa\) of \(R\). For any object \(\mathcal A\in\mathcal A_n(U),\) the finite \(\mathcal O_U\)-algebra \(\pi_\ast\mathcal A\) lies in \(\mathcal H_n.\) The resulting map of stacks \(\mathcal A_n\rightarrow\mathcal H_n\) is a \(1\)-isomorphism.
Definition 2.6. An Azumaya algebra \(\mathcal A\) on \(\mathcal X\) is Brumer log terminal (blt) if for every \(i\) the local Azumaya algebra \(\mathcal A_{\eta_i}\) has type \(e_i\).
Under the condition that the reader have the basic knowledge, the article is very well written and easy to follow. The results are important, and comes from comparing different moduli stacks. Brauer group; maximal orders; Azumaya orders; Brauer class; Kollár's condition; compactification; virtual fundamental class; bijective closed substack; parabolic Azumaya algebras; ramification divisor; matrix algebras; Brumer log terminal structure; blt; hulls and husks; reflexive Azumaya algebras; rational double points; compactification; parabolic generalized Azumaya algebras; hereditary site; \(n\)-typed Azumaya algebra; \(1\)-isomorphism; Brumer log terminal Azumaya algebra Brauer groups (algebraic aspects), Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.), Groupoids, semigroupoids, semigroups, groups (viewed as categories), Noncommutative algebraic geometry Blt Azumaya algebras and moduli of maximal orders | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 0653.00009.]
The author describes methods to construct double and triple structures on a given subspace of a complex space and shows how these constructions lead to important applications in complex analysis and algebraic geometry.
In the first paragraph the author studies double structures. In particular he gives a survey about the construction of Ferrand and the method of Serre to construct fibre bundles. From these one gets applications whether a subspace is a set theoretical complete intersection and also to deduce properties of the Maruyama scheme of stable vector bundles of rank 2 on \({\mathbb{P}}^ 3.\)
The second paragraph is concerned with the construction of triple structures which is due to the author and O. Forster. For complete proofs the author refers to his joint paper with \textit{O. Forster} in Algebraic geometry, Proc. Lefschetz Centen. Conf., Mexico City/Mex. 1984, part 1, Contemp. Math. 58, 47-64 (1986; Zbl 0605.14026). There are applications to the existence of holomorphic structures on topological complex vector bundles on \({\mathbb{P}}_ 3\), \({\mathbb{Q}}_ 3\), \({\mathbb{F}}_{1,2}\), \({\mathbb{P}}_ 2\times {\mathbb{P}}_ 1\), \({\mathbb{P}}_ 2\times {\mathbb{P}}_ 1\times {\mathbb{P}}_ 1\) and to the existence of surfaces of low degree in \({\mathbb{P}}_ 3\) with singularities of type \(A_{2k-1}\). double structures; set theoretical complete intersection; Maruyama scheme; triple structures; holomorphic structures on topological complex vector bundles C. BANICA , Structures Doubles et Triples sur un Sous-Espace Complexe (Applications, Banach Center Publications, Vol. 20, 1988 , pp. 15-30). MR 92j:14021 | Zbl 0727.14002 Cycles and subschemes, Complete intersections, Analytic spaces Structures doubles et triples sur un sous-espace complexe. Applications. (Double and triple structures on a complex subspace. Applications.) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Authors' abstract: This is a sequel of previous papers of the same authors on Weierstrass semigroups at ramification points of double coverings of algebraic curves of genus three. In this paper they give a list of possible numerical semigroups when the covering curve is of genus six and show that all of such semigroups are actually of double covering type. This result completes a classification of numerical semigroups of double covering type obtained by ramified double coverings of curves of genus three. Weierstrass semigroups; numerical semigroups; covering curve of genus 6; double coverings , Numerical semigroups of genus six and double coverings of curves of genus three, Semigroup Forum 91 (2015), no. 3, 601--610. Riemann surfaces; Weierstrass points; gap sequences, Commutative semigroups Numerical semigroups of genus six and double coverings of curves of genus three | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 an Enriques surface over an algebraically closed field k, char \(k\neq 2\). The canonical divisor \(K_ X\) which is the only non-zero element of order 2 in Pic X gives rise to an étale double covering \(u: Y\to X\), where Y is a K 3 surface. It is clear that the Enriques surfaces are in a one-to-one correspondence with the K 3 surfaces with a fixed points free involution. A standard example of K 3 surface with a fixed points free involution is given by an intersection of three quadrics in \({\mathbb{P}}^ 5\), where in a suitable system of coordinates the involution is given by the formula \(i(x_ 0:x_ 1:x_ 2:x_ 3:x_ 4:x_ 5)=(- x_ 0:-x_ 1:-x_ 2:-x_ 3:-x_ 4:-x_ 5).\) Roughly speaking, the goal of the paper under review is to show that each K 3 surface which can be represented in the form of étale double covering of an Enriques surface has a projective model of the form described in this example. Enriques surface; double covering; K 3 surface Verra, Alessandro: The étale double covering of an Enriques surface, Rend. semin. Mat. univ. Politec. Torino 41, No. 3, 131-167 (1983) Special surfaces, Coverings in algebraic geometry The étale double covering of an Enriques surface | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author gives an elementary proof of the following result: The quotient of \(\mathbb CP^ 2\) by the relation of complex conjugation is diffeomorphic to \(S^ 4\). This fact is generalized to a number of results on zeros of quadratic forms and more general hyperbolic polynomials. Several interesting conjectures are formulated. complex projective space; complex conjugation on complex projective 2- space; zeros of quadratic forms; hyperbolic polynomials Arnol'd, V. I.: Ramified covering CP2 \(\to S4\), hyperbolicity and projective topology. Siberian math. J. 29, No. 5, 36-47 (1988) Low-dimensional topology of special (e.g., branched) coverings, General binary quadratic forms, Elementary questions in algebraic geometry The ramified covering \(\mathbb CP^ 2\to S^ 4\), hyperbolicity and projective topology | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Double spaces \(X\to {\mathbb{P}}^ 3\) of index two were investigated by \textit{G. E. Welters} [``Abel-Jacobi isogenies for certain types of Fano threefolds'', Math. Centre Tracts 141 (1981; Zbl 0474.14028)], \textit{A. S. Tikhomirov} [Sov. Math., Dokl. 33, 204-206 (1986); translation from Dokl. Akad. Nauk SSSR 286, 821-824 (1986; Zbl 0622.14032)], \textit{C. Voisin} [Duke Math. J. 57, 629-646 (1988)]. In the present paper the author applies methods of A. S. Tikhomirov to the case, when X has a double singular point. Then X is birational to a conic bundle \(X\to {\mathbb{P}}^ 2\) and hence the intermediate Jacobian \(J(\tilde X)\) is a Prym variety. The author gives an effective parametrization of the theta divisor \(\Theta\) \(\subset J(\tilde X)\) and describes in geometrical terms the ramification divisor of the rational Gauss map for \(\Theta\). double spaces of index two; intermediate Jacobian; theta divisor Theta functions and abelian varieties, Picard schemes, higher Jacobians Ramification divisor of the Gauss map for a \(\Theta\)-divisor of the intermediate Jacobian of the double \(P^ 3\) of index 2 with a singularity | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(E\) be an elliptic curve defined over \(K\) given by a Weierstrass equation and let \(P=(x,y)\in E(K)\) be a point. Then for each \(n \geq 1\) we can write the \(x\)- and \(y\)-coordinates of the point \([n]P\) as
\[
[n]P=\biggl(\frac{G_n(P)}{F_n^2(P)},\frac{H_n(P)}{F_n^3(P)}\biggr)
\]
where \(F_n, G_n \), and \(H_n\in K[x,y]\) are division polynomials of \(E\). In this work we give explicit formulas for sequences
\[
(F_n(P))_{n\geq 0}, (G_n(P))_{n\geq 0}, \text{ and } (H_n(P))_{n\geq 0}
\]
associated to an elliptic curve \(E\) defined over \(\mathbb{Q}\) with non-cyclic torsion subgroup. As applications we give similar formulas for elliptic divisibility sequences associated to elliptic curves with non-cyclic torsion subgroup and determine square terms in these sequences. elliptic curves; division polynomials; elliptic divisibility sequences Elliptic curves, Recurrences, Elliptic curves over global fields Sequences associated to elliptic curves with non-cyclic torsion subgroup | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 first observes that the classical Bernstein polynomials
\[
B_{N}(f)(x)=\sum_{j=0}^{N}\binom{N}{j}f\left( \frac{j}{N}\right) x^{j}(1-x)^{N-j}
\]
of a smooth function \(f\in C^{\infty}([0,1])\) and their asymptotic expansion
\[
B_{N}(f)(x)\sim\sum_{\mu=0}^{\infty}L_{\mu}f(x)N^{-\mu}
\]
(where \(L_{\mu}=L_{\mu}\left( x,\frac{\text{d}}{\text{d} x}\right) \) are certain polynomial differential operators), are closely related to the Bergman-Szegö kernels for the Fubini-Study metric on the \(N\)-th powers \(\mathcal{O}(N)\) of the hyperplane line bundle \(\mathcal{O} (1)\rightarrow\mathbb{CP}^{1}\). This holds also for functions \(f\) of \(m\) variables over the cube, with \(\mathbb{CP}^{1}\) replaced by \(\mathbb{CP}^{m}\).
The paper generalizes this picture to a toric projective Kähler variety \(M\) and a Delzant polytope \(P\). Generalized Bernstein polynomial approximations \(B_{h^{N}}\) are defined and studied for \(f\in C(P)\), with respect to a Hermitian metric \(h\) on a line bundle \(L\rightarrow M\). A generalization of the asymptotic expansion of Bernstein polynomials is proved, as well as an asymptotic expansion for Dedekind-Riemann sums. Bernstein polynomials; toric varieties; Delzant polytope; Dedekind-Riemann sums; asymptotic expansions; Bergman-Szegö kernels S. Zelditch. Bernstein polynomials, Bergman kernels, and toric Kähler varieties, J. Symplectic Geom. (to appear); arXiv: 0705.2879. Kähler manifolds, Toric varieties, Newton polyhedra, Okounkov bodies Bernstein polynomials, Bergman kernels and toric Kähler 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 1960s, \textit{J. W. P. Hirschfeld} (see for example [Rend. Mat. Appl., V. Ser. 26, 349--374 (1967; Zbl 0162.52101); ibid. 26, 115--152 (1967; Zbl 0155.29803)]) embarked on a program to classify cubic surfaces with 27 lines over finite fields. This work is a contribution to this problem. We develop an algorithm to classify surfaces with 27 lines over a finite field using the classical theory of double-sixes. This algorithm is used to classify these surfaces over all fields of order \(q\) at most 97. We then construct a family of cubic surfaces over finite fields of odd order. The generic surfaces in this family have six Eckardt points and they are invariant under a symmetric group of degree four. The family turns out to be isomorphic to the example of a family of cubic surface given over the real numbers by Hilbert and Cohn-Vossen. cubic surface; finite field; classification; double-six Combinatorial aspects of finite geometries, Families, moduli, classification: algebraic theory, Other finite nonlinear geometries Cubic surfaces over small 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 Here the authors give explicit equations for any curve (i.e. pure one-dimensional locally Cohen-Macaulay subscheme) on any quadric surface \(Q\). When \(Q\) is a double plane, this was done by \textit{N. Chiarli}, \textit{S. Greco} and \textit{U. Nagel} [J. Pure Appl. Algebra 190, No. 1--3, 45--57 (2004; Zbl 1064.14029)], which generalizes [\textit{R. Harshorne} and \textit{E. Schlesinger}, Commun. Algebra 28, No. 12, 5655--5676 (2000; Zbl 0994.14003)].
This is a very natural problem, because such curves arise quite often (e.g. as the ones with extremal properties) and giving ``equations'' means also giving ``parameter spaces''. As an application they give all Hartshorne-Rao modules of space curves lying on a quadric surface. quadric surface; Hartshorne-Rao module; space curve; double plane; linkage Di Gennaro R., Nagel U.: The equations of space curves on a quadric. Collect. Math. 58(1), 119--130 (2007) Plane and space curves, Local cohomology and commutative rings, Linkage The equations of space curves on a 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 theory of derivators enhances and simplifies the theory of triangulated categories. In this article, a notion of fibered (multi)derivator is developed, which similarly enhances fibrations of (monoidal) triangulated categories. The author presents a theory of cohomological as well as homological descent in this language. The main motivation is a descent theory for Grothendieck's six operations. derivators; fibered derivators; multiderivators; fibered multicategories; Grothendieck's six-functor-formalism; cohomological descent; homological descent; fundamental localizers; well-generated triangulated categories; equivariant derived categories Hörmann, F, Fibered multiderivators and (co)homological descent, Theory Appl. Categ., 32, 1258-1362, (2017) Abstract and axiomatic homotopy theory in algebraic topology, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Monoidal categories (= multiplicative categories) [See also 19D23], Fibered categories, Derived categories, triangulated categories, Homological algebra in category theory, derived categories and functors Fibered multiderivators and (co)homological descent | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This long text (67 pages) surveys some of the important contributions of the author in the last ten years in the field of construction of algorithms in real algebraic geometry and the study of their complexity. It is essentially devoted to present the current solutions to four basic problems in this area. First of all, the author treats the real counting problem. This comes back to Sturm (1835), who gave an exact method to compute the number of real roots of a polynomial in one variable with real coefficients. It is useful to study the problem in a more general setting: Given an ordered domain \(D\) and a real closed field \(R\) containing \(D\), compute the number of roots in \(R\) of a polynomial \(P\in D[X]\). -- There exist several algorithms to solve the problem. The most significant advance with respect to Sturm's work is the so-called Sylvester-Habicht sequence. Using it, just ring operations and exact divisions, performed in polynomial time, are needed. It is also remarkable that it has good specialisation properties. There exist another method, due to Hermite, which essentially consists in the computation of the signature of a quadratic form. In the paper under review, the complexities of these methods are compared and illustrated by means of very well chosen examples. The second problem treated in the paper is to compute the real roots of a polynomial, which is closely related to the evaluation of the sign of a polynomial at a real root of another polynomial. This has nothing to do with approximation theory, but consists in dealing formally with real numbers. The key for that is the use of Thom's lemma, which roughly speaking says that you can distinguish between two distinct real roots of a real polynomial \(P\in R[X]\) by means of the signs of the derivatives of \(P\) at these roots.
The last but most interesting section of the paper concerns multivariate polynomials. The most relevant result provides a bound for the number of connected components of a basic semialgebraic subset \(M\) of \(\mathbb{R}^k\), defined by means of \(s>k\) polynomial equalities or inequalities, as a function of \(s,k\) and the maximum of the degrees of the involved polynomials. Moreover, it is described an algorithm which outputs at least one point in each connected component of \(M\). This section contains a new proof of the weak form of Hilbert's Nullstellensatz, due to Michel Coste. It is a very elegant simplification of the classical proof by Van der Waerden.
The article is very well organized and clearly written. It is interesting for the experts in the field and essential for the beginners. algorithms in real algebraic geometry; complexity; real counting problem; number of real roots; real roots of a polynomial; multivariate polynomials; Hilbert's Nullstellensatz Roy, M. -F.: Basic algorithms in real algebraic geometry and their complexity: from Sturm's theorem to the existential theory of reals, De gruyter expositions in mathematics 23, 1-67 (1996) Semialgebraic sets and related spaces, Analysis of algorithms and problem complexity, Polynomials in real and complex fields: location of zeros (algebraic theorems) Basic algorithms in real algebraic geometry and their complexity: From Sturm's theorem to the existential theory of reals | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(k\) be a field of characteristic \(0\) and let \({\mathbb A}_k\) be the adèle ring of \(k\). A smooth variety \(X\) over \(k\) such that \(X({\mathbb A}_k)\neq \emptyset\) whereas \(X(k)=\emptyset\) is a counterexample to the Hasse principle. The classical way of constructing counterexamples to the Hasse principle is via the Manin obstruction. Let \(\text{Br}(X)=H^2(X,{\mathbb G}_m)\) be the Grothendieck-Brauer group of \(X\). Define \(X({\mathbb A}_k)^{\text{Br}}\) to be the subset of points of \(X({\mathbb A}_k)\) orthogonal to all elements of \(\text{Br}(X)\). The image of \(X(k)\) under the diagonal embedding \(X(k)\hookrightarrow X({\mathbb A}_k)\) is contained in \(X({\mathbb A}_k)^{\text{Br}}\). A counterexample to the Hasse principle is accounted for by the Manin obstruction if already \(X({\mathbb A}_k)^{\text{Br}}=\emptyset\).
In this paper, a counterexample to the Hasse principle not accounted for by the Manin obstruction is constructed. Such a counterexample is given by a smooth proper surface defined over \({\mathbb Q}\) of Kodaira dimension \(\kappa=0\). More specifically, the construction of such a family of smooth proper surfaces over \(k\) is based on two pieces of data:
(1) an elliptic curve \(E\) over \(k\), a \(2\)-isogeny \(\psi: C\to E\) which lifts to a \(4\)-isogeny \(\psi'': C''\to E\) and such that \(C\) has a zero-cycle of degree \(2\) over \(k\),
(2) an unramified double covering \(\phi:D\to D''\) of curves of genus one.
Let \(Y=C\times D\), and let \(X\) be the quotient of \(Y\) by the fixed point free involution \((\sigma, \rho)\), \(f:Y\to X\) where \(\sigma:C\to C\) is the hyperelliptic involution, and \(\rho:D\to D\) is the fixed point free involution interchanging the sheets of the covering \(\phi:D\to D''\). Then \(X\) is a smooth proper geometrically integral surface with \(\kappa = 0\), \(p_g=0\), \(g=1\), \(K_X^2=0\) and \(b_1=b_2=2\). Such a family of surfaces \(X\) may be defined by the affine equations \((x^2+1)y^2=(x^2+2)z^2=3(t^4-54t^2-117t-243)\) taking \(E: y^2=x^3-1221\), \(C: y^2 =3(t^4-54t^2-117t-243)\) and \(D\) defined by \(y^2=x^2+1\), \(z^2=x^2+2\). (In the appendix, \textit{S. Siksek} exhibits \(4\)-descent of the curve \(E\), giving rise to an element of exact order \(4\).)
Theorem: For the family of surfaces \(X\) constructed above, \(X({\mathbb{Q}})=\emptyset\), but \(X({\mathbb{A}}_{\mathbb{Q}})^{\text{Br}}\neq\emptyset\).
A refinement of the Manin obstruction is introduced combining the theory of descent and the Manin obstruction. Then it is shown that this refined Manin obstruction is the only obstruction to the Hasse principle for the family of surfaces. This refinement is defined extending the descent theory of \textit{J.-L. Colliot-Thélène} and \textit{J.-J. Sansuc} [Duke Math. J. 54, 375-492 (1987; Zbl 0659.14028)] to arbitrary torsors under groups of multiplicative type. Brauer-Grothendieck group; Manin obstruction; counterexample to the Hasse principle A. N. Skorobogatov, ''Beyond the Manin obstruction,'' Invent. Math. 135(2), 399--424 (1999). Global ground fields in algebraic geometry, Brauer groups of schemes Beyond the Manin obstruction. -- Appendix A by S. Siksek: 4-descent. -- Appendix B: The Grothendieck spectral sequence and the truncation functor | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials There is much interest nowadays to find lower bounds for polynomials \(f=\sum_\alpha f_\alpha x^\alpha \in \mathbb R[x_1,\dots,x_n]\) on \(\mathbb R^n\). An already classical technique is to bound \(f_*:=\inf \{f(x): x\in \mathbb{R}^n\}\) by means of computing \(f_{\mathrm{sos}}=\sup \{r: f(x)-r \text{ is sum of squares} \}\) via semidefinite programming. A pioneer in this area is Lasserre. \textit{M. Ghasemi} and \textit{M. Marshall} [SIAM J. Optim. 22, No. 2, 460--473 (2012; Zbl 1272.12004)], still using sos-representations have shown how geometric programming can be used to find bounds \(f_{gp}\) so that \(f_{gp}\leq f_{\mathrm{sos}}\leq f_*.\) These bounds are thus usually not as good but much faster to compute. The current paper extends the Ghasemi-Marshall results in two important aspects: i. Rather than searching for sos-representations it searches for sonc-representations (sums of nonnegative circuit polynomials). ii. The constraints for the geometric programs are not anymore formulated in terms of the particular coefficients \(f_{2de_i}\) of monomials \(x_i^{2d}\). Here are some more details.
By a simplex-tail (st) polynomial the authors understand a polynomial \(f\) that can be written in the form \(f=f_0+\sum_{j=1}^n f_{\alpha_j} x^{\alpha_j} + \sum_{\alpha\in \Delta} f_\alpha x^\alpha\), where the essential requirements are these: the set \(V=\{\alpha_0=0, \alpha_1,\dots,\alpha_n\} \subset (2\mathbb{N})^n\) is (usually) the vertex set of the Newton polytope New\((f)\) of \(f;\) the \(f_{\alpha_j}\) are nonnegative; the set \(\Delta \subseteq \text{conv} V\) comprises the \(\alpha\) for which \(f_\alpha x^\alpha\) are not squares. An st-polynomial is called circuit polynomial if \(\Delta\) is a singleton or empty. In their marvelous work [Res. Math. Sci. 3, Paper No. 9, 35 p. (2016; Zbl 1415.11071)] connecting amoeba theory and nonnegative real polynomials, the authors gave necessary and sufficient conditions for a circuit polynomial to be nonnegative which are very much similar to those \textit{C. Fidalgo} and \textit{A. Kovacec} [Math. Z. 269, No. 3--4, 629--645 (2011; Zbl 1271.11045)] gave for an elementary diagonal minus tail form to be nonnegative. An important difference and extension is that nonnegative circuit polynomials need not be sos. The theorem is recorded here as Theorem 2.3. A necessary and sufficient condition in order that a circuit polynomial is sos or a sum of binomial squares (sobs) is that its unique \(\alpha\in \Delta\) is in New\((f)^*,\) the unique maximal New\((f)\)-mediated set in the sense of \textit{B. Reznick} [Math. Ann. 283, No. 3, 431--464 (1989; Zbl 0637.10015)]. Next the authors show an analogue to Ghasemi and Marshall's [loc. cit., Theorem 2.3].
Theorem 3.1. If for every pair \((\alpha,j)\in \Delta \times \{1,\dots,n\}\) there exists a real \(a_{\alpha,j}\geq 0\) such that \quad \(|f_\alpha| \leq \prod_{j=1}^n (a_{\alpha,j}/\lambda_j^\alpha)^{\lambda_j^\alpha};\) and \quad \(f_{\alpha_j} \geq \sum_{\alpha \in \Delta} a_{\alpha,j}\) for all \(j\) \quad hold, then \(f\) is a sonc. Here \((\lambda_1^\alpha,\dots,\lambda_n^\alpha)\) are the barycentric coordinates of \(\alpha\) and undefined factors in the product are understood to be 1.
With this theorem they connect their theory to geometric programming. Theorem 3.4 is formulated for soncs and is roughly similar to Ghasemi and Marshall's [loc. cit., Theorem 3.1] for sobs.
These theorems form the base for section 4 where the geometric programming problems are formulated in detail and the correctness proofs are given. In subsection 4.1 the authors present a number of examples and recount a number of nice features and surprises: Due to the generality of their approach, polynomials where the Ghasemi-Marshall programs cannot be applied, can be treated with their geometric programs geared towards computing \(f_{gp}=\sup\{r: f-r \text{ is sonc }\}\); an example with an acceleration factor of about 10000 over semidefinite programming is given; and finally, while geometric programming in the authors' setting remains fast when compared with semidefinite programming, geometric programming aiming for sonc representations often yields bounds \(f_{gp}>f_{sos},\) that is, they are sometimes better than \(f_{sos}.\) In Section 5 the authors give initial applications to constrained polynomial optimization following the setup \textit{M. Ghasemi} and \textit{M. Marshall} have given in [``Lower bounds for a polynomial on a basic closed semialgebraic set using geometric programming'', Preprint, \url{arXiv:1311.3726}].
The reviewer should inform that a recent paper by \textit{Van Doat Dang} and \textit{Thi Thao Nguyen} [Kodai Math. J. 39, No. 2, 253--275 (2016; Zbl 1398.12004)] similarly presents results ridded of conditions on \(f_{2de_i}\) for sums of squares estimates and geometric programming. That paper has a similar principal bibliography as the present one, and there is some overlap in the results; but it is apparent that the authors didn't know of each other's work. geometric programming; lower bound; nonnegative polynomial; semidefinite programming; simplex; sparsity; sum of nonnegative circuit polynomials; sum of squares S. Iliman and T. De Wolff, \textit{Lower bounds for polynomials with simplex Newton polytopes based on geometric programming}, SIAM J. Optim., 26 (2016), pp. 1128--1146. Fields related with sums of squares (formally real fields, Pythagorean fields, etc.), Real algebraic and real-analytic geometry, Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Convex programming Lower bounds for polynomials with simplex Newton polytopes based on geometric programming | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Numerical algebraic geometry has a close relationship to intersection theory from algebraic geometry. We deepen this relationship, explaining how rational or algebraic equivalence gives a homotopy. We present a general notion of witness set for subvarieties of a smooth complete complex algebraic variety using ideas from intersection theory. Under appropriate assumptions, general witness sets enable numerical algorithms such as sampling and membership. These assumptions hold for products of flag manifolds. We introduce Schubert witness sets, which provide general witness sets for Grassmannians and flag manifolds. intersection theory; numerical algebraic geometry; Schubert variety; witness set Geometric aspects of numerical algebraic geometry, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Grassmannians, Schubert varieties, flag manifolds General witness sets for numerical 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 A 4-semigroup means a numerical semigroup whose minimum positive integer is 4. In [\textit{J. Komeda} and \textit{A. Ohbuchi}, Corrigendum for Weierstrass points with first non-gap four on a double covering of a hyperelliptic curve, Serdica Math. J. 30, No. 1, 43--54 (2004; Zbl 1075.14029); corrigendum ibid. 32, No. 4, 375--378 (2006)], we showed that a 4-semigroup with some conditions is the Weierstrass semigroup of a ramification point on a double covering of a hyperelliptic curve. In this paper we prove that the above statement holds for every 4-semigroup. Weierstrass semigroup of a point; double covering of a hyperelliptic curve; 4-semigroup Komeda, J., Ohbuchi, A.: Weierstrass points with first non-gap four on a double covering of a hyperelliptic curve II. Serdica Math. J. \textbf{34}, 771-782 (2008) Riemann surfaces; Weierstrass points; gap sequences, Coverings of curves, fundamental group, Rational and ruled surfaces Weierstrass points with first non-gap four on a double covering of a hyperelliptic curve. 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 Let \(K\) be a number field, \({\mathbf e}_1,\ldots ,{\mathbf e}_n\) the standard basis of \(K^n\), and \(K^m\) the linear subspace of \(K^n\) generated by \({\mathbf e}_1,\ldots , {\mathbf e}_m\). Further, let \(\alpha =(\alpha_1,\ldots ,\alpha_d)\) be a tuple of integers with \(1\leq\alpha_1<\cdots <\alpha_d\leq n\). For \(B>0\), denote by \(M(\alpha ,B)\) the number of \(d\)-dimensional linear subspaces \(S\) of \(K^n\) of height at most \(B\) satisfying the conditions \(\dim_K(S\cap K^{\alpha_i})=i\) for \(i=1,\ldots , d\). Here the height of \(S\) is some sort of twisted multiplicative height of the exterior product of the vectors from a basis of \(S\). In other words, \(M(\alpha ,B)\) is the number of points of heights at most \(B\) in a Schubert cell. Under the hypothesis \(\alpha_1>1\), the author proves that
\[
M(\alpha ,B)\gg\ll B^{c_1(\alpha )}(\log B)^{c_2(\alpha )-1}\quad\text{for } B\gg 1
\]
where \(c_1(\alpha )=\max\{ \alpha_i-2i+d+1:\, 1\leq i\leq d\}\), \(c_2(\alpha )\) is the number of \(i\) for which the maximum is assumed, and the constants implied by the Vinogradov symbols depend on \(n\) and \(K\). It is conjectured that there is an asymptotic formula of the shape
\[
M(\alpha ,B)= a(\alpha ,K)B^{c_1(\alpha )}(\log B)^{c_2(\alpha )-1}+ O(B^{c_1(\alpha )}(\log B)^{c_2(\alpha )-2})\quad \text{as \(B\to\infty\).}
\]
The author obtains such a formula with slightly larger error term in the special case \(c_1(\alpha )=\alpha_d+1-d\) and \(c_2(\alpha )=1\).
The author derives these results from a counting result of his for rational points of bounde height on flag varieties [Compos. Math. 88, No. 2, 155--186 (1993; Zbl 0806.11030)], using partial summation techniques. heights; Schubert varieties J.L. Thunder, Points of bounded height on Schubert varieties , Inter. J. Number Theory, Heights, Varieties over global fields, Grassmannians, Schubert varieties, flag manifolds Points of bounded height on Schubert varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Dans cet article, l'A. démontre une version K-théorique du théorème de Kashiwara relatif aux images directes de \({\mathcal D}\)- modules. Ce résultat a son origine dans le théorème de Riemann-Roch pour les \({\mathcal D}\)-modules. Soit X une variété analytique compacte et \(V\subset T^*X\) un sous-ensemble analytique fermé homogène. Notons \(K_ V[T^*X]\) le groupe de Grothendieck des gr \({\mathcal D}_ X\)- modules cohérents à support dans V et admettant une bonne filtration. Si M est un \({\mathcal D}_ X\)-module cohérent admettant une bonne filtration, avec car(M)\(\subset V\), on définit l'élément \([M]_ V\subset K_ V[T^*X]\) comme la classe de gr M. La définition de \([M]_ V\) s'étend de la façon habituelle au cas où M est un complexe de \(D^ b({\mathcal D}_ X)_ c\) dont les groupes de cohomologie possèdent les propriétés précédentes. Soit \(f: Y\to X\) un morphisme de variétés analytiques compactes et \(V\subset T^*Y\) un sous-ensemble analytique fermé homogène. Si M est un \({\mathcal D}_ Y\)- module admettant une bonne filtration, avec car(M)\(\subset V\), le théorème de Kashiwara nous dit que les \(H^ if_*M\) sont cohérents et \(car(H^ if_*M)\subset W=\bar fF^{-1}V\), où \(T^*Y\leftarrow^{F}Y\times_ XT^*X\to^{\bar f}T^*X\) sont les morphismes naturels définis par f. Le résultat principal de l'article nous dit que: (1) les \(H^ if_*M\) admettent une bonne filtration; (2) [f\({}_*M]_ W=\bar f_*F^*[M]_ V\), où les morphismes: \(K_ V[T^*Y]\to^{F^*}K_{F^{-1}V}[Y\times_ XT^*X]\to^{\bar f_*}K_ W[T^*X]\) proviennent des morphismes d'espaces annelés suivants: \((T^*Y,gr {\mathcal D}_ Y)\leftarrow^{F}(Y\times_ XT^*X,gr {\mathcal D}_{Y\to X})\to^{f}(T^*X,gr {\mathcal D}_ X).\) Finalement on étudie une version ''graduée'' du résultat précédent par rapport à la filtration introduite par Houzel et Shapira. Grothendieck group; image direct of D-module B. Malgrange, Sur les images directes de \(\scr D\)-modules , Manuscripta Math. 50 (1985), 49-71. Holomorphic bundles and generalizations, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Applications of methods of algebraic \(K\)-theory in algebraic geometry, Sheaves and cohomology of sections of holomorphic vector bundles, general results, Grothendieck groups (category-theoretic aspects) Sur les images directes de \({\mathcal D}\)-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 Let \(k\) be a (topological) field of characteristic \(0\) and denote by \(B_n\) the Artin braid group on \(n\) strands. Due to a construction by \textit{V. Drinfeld} [Leningr. Math. J., 2 829--860 (1991; Zbl 0728.16021)], these data give rise to a certain graded Hopf \(k\)-algebra \(\mathbb{B}_n(k)\) and its completion \(\widehat B_n(k)\) with respect to graduation. Moreover, each \(\lambda\in k\) defines a family \(\text{Ass}_\lambda(k)\) of formal power series in two non-commuting variables, the so-called Drinfeld associators, each of which establishes then a morphism from Drinfeld's \(k\)-pro-unipotent completion \(B_n(k)\) over \(k\) of the Artin braid group \(B_n\) to the group of units in the Hopf algebra \(\widehat{\mathbb{B}}_n(k)\).
In this way, as the author of the present paper has recently observed [On the representation theory of braid groups, 2005, \url{arXiv:math.RT/0502118}], one obtains special functors from the category of (finite-dimensional) representations of the Hopf algebra \(\mathbb{B}_n(k)\) to the category of representations of the Artin braid group \(B_n\) over the field \(K= k((h))\) of Laurent series of the ground field \(k\).
In the paper under review, the author studies these functors more closely. His first motivation is to understand their behavior under changes of the chosen Drinfeld associators, especially with a view toward representations of the Hecke-Iwahori algebra of type A, which classically appear as representations of a quotient of the group algebra of the braid group \(B_n\).
His second motivation concerns the so-called \(k\)-pro-unipotent Grothendieck-Teichmüller group \(GT(k)\) which was also introduced by V. G. Drinfeld in his original paper (1991) cited above. In fact, the Grothendieck-Teichmüller group acts naturally on the set of Drinfeld associators \(\text{Ass}_\lambda(k)\) for a given \(\lambda\in k\) and so does a certain fundamental semi-direct factor. \(GT_1(k)\) of \(GT(k)\). In this context, the author shows how to construct projective representations of these Grothendieck-Teichmüller groups \(GT(k)\) and \(GT_1(k)\), respectively from particular representations of Artin braid groups. The latter are called \(GT\)-rigid representations (of \(\mathbb{B}_n(k)\)), and they may be viewed as natural counterparts to the classical rigid local systems for the ordinary Artin braid groups \(B_n\). After a detailed exposition of his construction of such projective representations of the Grothendieck-Teichmüller groups \(GT(k)\) and \(GT_1(k)\), including the novel framework of \(GT\)-rigid representations of \(\mathbb{B}_n(k)\), their properties are investigated from various points of view. Among the main results are a non-triviality criterion for such projective representations, a description of their decomposition behavior with respect to characters, and the construction of explicit matrix models for the Hecke-Iwahori algebra of type A. Further applications of the author's construction concern a revisited inspection of Drinfeld's associator \(\Phi_{KZ}\in\text{Ass}_1(\mathbb{C})\) obtained from the Knizhnik-Zamolodchikov system of differential equations, mainly with regard to its significance in the study of the Hecke algebra \(H_n(q)\) over \(\mathbb{Q}(q)\) of the symmetric group \(S_n\) à la \textit{J. Gonzalez-Lorca} [C. R. Acad. Sci., Paris, Sér. I, Math. 326, No. 2, 147--152 (1998; Zbl 0914.20013)]. The author's approach produces an explicit unitary matrix model for \(H_n(q)\) associated to a certain Burau representation.
Finally, the various explicit examples exhibited at the end of the paper point to a possibly close connection between the characters of the Grothendieck-Teichmüller group, on the one hand, and the Soulé characters in algebraic K-theory [cf.: \textit{C. Soulé}, in: Journées arithmétiques, Besançon/France 1985, Astérisque 147--148, 225--257 (1987; Zbl 0632.12014) on the other. profinite groups; Grothendieck-Teichmüller group; Hecke algebras; projective representations; braid groups; Hopf algebras; arithmetic algebraic; \(K\)-theory; Young tableaux I. Marin, Caractères de rigidité du groupe de Grothendieck-Teichmüller, Compos. Math. 142 (2006), 657-678. Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory), Representation theory of groups, Hecke algebras and their representations, Braid groups; Artin groups, Projective representations and multipliers, Coalgebras, bialgebras, Hopf algebras [See also 57T05, 16S30--16S40]; rings, modules, etc. on which these act, Polylogarithms and relations with \(K\)-theory Rigidity characters of the Grothendieck-Teichmüller group | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 \((G:P)\) be the varieties of Chevalley groups \(G\) by their maximal parabolic subgroups \(P\). The author discusses interpretations of varieties \((G:P)\) and small Schubert cells in terms of linear algebra, generalizations of varieties to the case of Kac-Moody algebras and superalgebras, and applications to extremal graph theory. varieties of Chevalley groups; maximal parabolic subgroups; small Schubert cells; Kac-Moody algebras; superalgebras [25] Ustimenko V.\ A., ''On the Varieties of Parabolic Subgroups, their Generalizations and Combinatorial Applications'', Acta Applicandae Mathematicae, 52 (1998), 223--238 Representation theory for linear algebraic groups, Linear algebraic groups over arbitrary fields, Groups with a \(BN\)-pair; buildings, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Buildings and the geometry of diagrams, Group varieties, Group actions on varieties or schemes (quotients) On the varieties of parabolic subgroups, their generalizations and combinatorial applications | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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-Serre conjecture for principal bundles asserts that for any regular local ring \(R\) with \(K = \mathrm{Frac}(R)\) and a reductive \(R\)-group scheme \(G\), the map \(H^1_{\mathrm{\'{e}t}}(R, G) \to H^1_{\mathrm{\'{e}t}}(K, G)\) induced by \(R \hookrightarrow K\) has trivial kernel. The main theorem of this paper under review proves such an injectivity result for \(G\) isotropic, simple and simply connected and for certain rings \(R\). Here isotropy means that \(G\) contains a \(\mathbb{G}_{m,R}\).
Specifically, it is proved that that \(H^1_{\mathrm{\'{e}t}}(\mathcal{O} \otimes_k A, G) \to H^1_{\mathrm{\'{e}t}}(K \otimes_k A, G)\) has trivial kernel in the following situation: \(k\) is an infinite field, \(\mathcal{O}\) is the semi-local ring of finitely many closed points on a smooth irreducible affine \(k\)-variety, \(K = \mathrm{Frac}(\mathcal{O})\), \(G\) is an isotropic and simply connected \(\mathcal{O}\)-group scheme and \(A\) is any Noetherian \(k\)-algebra.
As a corollary, one proves for any regular domain \(R \supset \mathbb{Q}\) and any isotropic simple simply connected \(R\)-group scheme \(G\), the map \(H^1_{\mathrm{\'{e}t}}(R[t_1, \ldots, t_n], G) \to H^1_{\mathrm{\'{e}t}}(R, G)\) induced by evaluation at \(t_1 = \cdots = t_n = 0\) has trivial kernel. This result is false if the isotropy condition is dropped. reductive group schemes; principal bundles; Grothendieck-Serre conjecture [80] Panin I., Stavrova A., Vavilov N., On Grothendieck--Serre's conjecture concerning principal \(G\)-bundles over reductive group schemes. I, 2009, 28 pp., arXiv: Group schemes, Exceptional groups, Linear algebraic groups over adèles and other rings and schemes, Linear algebraic groups and related topics On Grothendieck-Serre's conjecture concerning principal \(G\)-bundles over reductive group schemes: 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 We give the first examples of smooth Fano and Calabi-Yau varieties violating the (narrow) canonical strip hypothesis, which concerns the location of the roots of Hilbert polynomials of polarized varieties. They are given by moduli spaces of rank 2 bundles with fixed odd-degree determinant on curves of sufficiently high genus, hence our Fano examples have Picard rank 1, index 2, are rational, and have moduli. The hypotheses also fail for several other closely related varieties. geography of Fano manifolds; geography of Calabi-Yau manifolds; roots of Hilbert polynomials; Verlinde formula; moduli spaces of vector bundles on algebraic curves; toric varieties Fano varieties, Calabi-Yau manifolds (algebro-geometric aspects), Algebraic moduli problems, moduli of vector bundles, Vector bundles on curves and their moduli, Toric varieties, Newton polyhedra, Okounkov bodies Examples violating Golyshev's canonical strip hypotheses | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 probabilistic analysis of condition numbers is of central importance to turn condition-based complexity analyses of numerical algorithsm into probabilistic complexity analyses. While the former explain how the algorithm behaves at a particular input, the former allows us to see how the algorithm behaves in general.
Unfortunately, passing from a condition-based complexity analysis to a probabilistic complexity analysis requires to choose a probabilistic distribution of the input. With the notable exception of numerical linear algebra and the theory of random matrices, probabilistic analyses of condition numbers in numerical algebraic geometry rely exclusively on some form of Gaussian assumption of the input.
In their previous paper [\textit{A. A. Ergür} et al., Found. Comput. Math. 19, No. 1, 131--157 (2019; Zbl 1409.65033)], they performed an average complexity analysis in which they consider the condition number, as introduced in [\textit{F. Cucker} et al., J. Complexity 24, No. 5--6, 582--605 (2008; Zbl 1166.65021)], of random real polynomial systems under robust assumptions. In this paper, they extend their results to the smoothed analysis framework of \textit{D. Spielman} and \textit{S.-H. Teng} [in: Proceedings of the thirty-third annual ACM symposium on theory of computing, STOC 2001. Hersonissos, Crete, Greece, July 6--8, 2001. New York, NY: ACM Press. 296--305 (2001; Zbl 1323.68636)]. In these framework, we don't just consider a random input, but an arbitrary input perturbed by random noise. As numerical algorithms work with inputs submitted to errors, this is a more realistic framwork.
Moreover, the authors do not only provide an smoothed complexity analysis, but they also do so for a class of estructured random polynomial systems. The structure is chosen by taking the random polynomials out of subspaces of the space of polynomials, for which one can guarantee nice evaluation bounds. This is the first probabilistic analysis for random structured polynomials.
Finally, let us note that to achive this, Ergür, Paouris and Rojas rely on results coming from geometric functional analysis for subgaussian and anti-concentrated random variables such as those in [\textit{R. Vershynin}, High-dimensional probability. An introduction with applications in data science. Cambridge: Cambridge University Press (2018; Zbl 1430.60005); \textit{M. Rudelson} and \textit{R. Vershynin}, Int. Math. Res. Not. 2015, No. 19, 9594--9617 (2015; Zbl 1330.60029)]. condition number; random polynomials; grid method; smoothed analysis; structured polynomials Geometric aspects of numerical algebraic geometry, Numerical computation of matrix norms, conditioning, scaling, Real algebraic and real-analytic geometry, Complexity and performance of numerical algorithms Smoothed analysis for the condition number of structured real polynomial systems | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper provides an interesting approach to the topological study of Schubert varieties, as well as a self-contained exposition on Schubert geometry.
Let \(G_ k(\mathbb{R}^ n)\) denote the Grassmannian of \(k\)-planes in \(\mathbb{R}^ n\). For a sequence \(\underline n=(n_ 0,n_ 1,\dots,n_ s)\) of natural numbers with \(n_ 0=1<n_ 1<\cdots<n_ s=n\), we define Schubert strata \(S(d)\), \(d=(d_ 1,\dots,d_ s)\), by the sets of \(V\in G_ k(\mathbb{R}^ n)\) satisfying \(\dim V\cap\mathbb{R}^{n_ i}=d_ i\), \(i=1,\dots,s\), with respect to the flag \(\mathbb{R}^{n_ 1}\subset\cdots\subset\mathbb{R}^{n_ s}\). Let \(\sigma=(\sigma_ 1,\dots,\sigma_ k)\) be a Schubert symbol and \(\overline{e(\sigma)}\subset G_ k(\mathbb{R}^ n)\) the corresponding Schubert variety. -- Then the main theorem 2.12 in this paper says that there exists an \(\underline n\), such that the stratification of the Schubert variety \(\overline{e(\sigma)}\) given by \(\{S(d)\}\) is coarsest among all topological stratifications of \(\overline{e(\sigma)}\). topological study of Schubert varieties; Schubert geometry; Grassmannian; Schubert symbol; stratification Buoncristiano, S.; Veit, A. B.: The intrinsic stratification of a Schubert variety. Adv. math. 91, No. 1, 1-26 (1992) Grassmannians, Schubert varieties, flag manifolds, Stratifications in topological manifolds, Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations) The intrinsic stratification of a Schubert variety | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Double affine Hecke algebras for reduced root systems were introduced by \textit{I. Cherednik} [Int. Math. Res. Not. 1992, No. 9, 171--180 (1992; Zbl 0770.17004)] in order to prove the Macdonald conjectures. Those of type \(C^\vee C_n\) were introduced in the works of Noumi, Sahi and Stokman as a generalization of Cherednik algebras of types \(B_n\) and \(C_n\), in order to prove the Macdonald conjectures for Koornwinder polynomials. This article defines and studies new algebras \(H(t,q)\) which are generalizations of double affine Hecke algebras of type \(C^\vee C_n\) in the case \(n=1\): Fix a star-shaped simply laced affine Dynkin diagram \(\widehat{D}\) (\(\tilde{D}_4\),\(\tilde{E}_6\),\(\tilde{E}_7\), or \(\tilde{E}_8\)). Let \(m\) be the number of legs of \(\widehat{D}\) and \(d_j-1\), \(j=1,\dots ,m,\) be the length of the \(j\)th leg. Then the authors define a family of algebras \(H(t,q)\) depending on parameters \(q\in\mathbb{C}^\ast\) and \(t=(t_{ij})\), \(t_{kj}\in\mathbb{C}^\ast\), \(k=1,\dots,m,\) \(j=1,\dots,d_k\), by generators \(T_k\), \(k=1,\dots,m\), with explicitly defined relations
\[
\prod_{j=1}^{d_k}(T_k-e^{2\pi ij/d_k}t_{kj})=0,\text{ }k=1,\dots,m;\text{ }\prod_{k=1}^m T_k=q.
\]
For \(\widehat{D}=\tilde{D}_4\) the result is exactly the double affine Hecke algebra of type \(C^\vee C_1.\) In the cases \(\widehat{D}=\tilde{E}_{6,7,8}\) the results are new algebras which are the main subject of this article. If \(t_{kj}=1\) and \(q=1\), the algebra \(H(t,q)=H(1,1)\) is the group algebra for some group \(G\) known to be isomorphic to a 2-dimensional crystallographic group \(\mathbb{Z}_l\ltimes\mathbb{Z}^2,\) \(l=2,3,4,6\) in the cases \(\widehat{D}=\tilde{D}_4,\tilde{E}_6,\tilde{E}_7,\tilde{E}_8,\) respectively. Moreover, \(H(1,q)\) is a twisted group algebra of \(G\) so that \(H(t,q)\) is a deformation of the twisted group algebra of \(G\). The authors prove that if they regard \(\log(t_{kj})\) as formal parameters then this deformation is flat, and \(H(t,qe^{\varepsilon})\) is the universal deformation of \(H(1,q)\) if \(q\) is not a root of unity (\(\varepsilon\) is a new formal parameter).
The authors also prove a more delicate algebraic PBW-theorem, claiming that some filtrations on \(H(t,q)\) have certain explicit Poincaré series, independent of \(t\) and \(q\). It is known that for \(\widehat{D}=\tilde{D}_4\) and \(q=1\) the algebra \(H(t,q)\) is finite over its center \(Z(t,q)\), and the spectrum of \(Z(t,q)\) is an affine cubic surface, obtained from a projective one by removing three lines forming a triangle. The authors prove that this result is valid also for \(q\) being a root of unity, and they generalize it to the cases \(\widehat{D}=\tilde{E}_{6,7,8}\). In these cases the spectrum of \(Z(t,q)\), \(q\) a root of unity, turns out to be an affine surface \(S(t,q)\) obtained from a projective del Pezzo surface \(S(t,q)\) of degrees \(3,2,1\) respectively by removing a nodal \(\mathbb{P}^1.\) This means that for \(q\neq 1\), the spherical subalgebra \(eH(t,q)e\) in \(H(t,q)\) should be viewed as an algebraic quantization of the surface \(S(t,1)\).
Moreover, the algebraic PBW theorem for \(H(t,q)\) implies that the Rees algebra of \(eH(t,q)e\) with respect to an appropriate filtration provides a (noncommutative) quantization of the Poisson surface \(S(t,q)\). The authors recall the basics of crystallographic groups in the plane. They consider the twisted group algebra \(B(q)=H(1,q)\) of a planar crystallographic group \(G\), and deform it into an algebra \(\mathbf{H}(q)\), which is a version of \(H(t,q)\) in which \(\log t_{kj}\) are formal parameters. They prove the formal PBW theorem for \(\widehat{\mathbf{H}}(q)\), and give results on the cohomology of \(B(q)\) and on its universal deformation.
The authors then define an increasing filtration on \(H(t,q)\), the length filtration. They prove that the Poincaré series for this filtration is independent of \(t,q\). They then use this result to establish general properties of \(H(t,q)\). Using the Riemann-Hilbert correspondence they define a homomorphism from a formal version of the generalized double affine Hecke algebra to the completion of the deformed preprojective algebra of the quiver associated to the graph \(\widehat{D}\). Thus a holomorphic map from the universal deformation of the Kleinian singularity \(\mathbb{C}^2/\Gamma\) to the family of surfaces \(S(t,1)\) can be defined. This is a local isomorphism of analytic varieties near \(0\in\mathbb{C}^2/\Gamma\), which in the case \(\widehat{D}=\tilde{D}_4\) encodes generic solution to the Painlevé VI equation.
The authors study closely the surfaces \(S(t,q)\): Let \(\mathbf{G}\) be the simple Lie group corresponding to the diagram \(D\). They then prove that the algebra \(H(t,q)\) depends only on the projection of \(t\) to the maximal torus \(\mathbf{T}\subset\mathbf{G}\), and that the map \(t\mapsto S(t,1)\) from \(\mathbf{T}\) to the moduli space of affine del Pezzo surfaces is Galois and has Galois group isomorphic to the Weyl group \(W\) of \(G\). This also implies that the coefficients of the equation of \(S(t,1)\), as functions of \(t\), are polynomials of characters of irreducible representations of \(\mathbf{G}\), and the authors compute these polynomials explicitly. The authors give the necessary definitions and results of deformation theory, and they organize the article such that the results depending on computer calculations are left to the end of the article.
This article covers a lot of results depending heavily on nontrivial theories, both analytic and algebraic. It is a very nice article covering nearly everything possible on the algebras \(H(t,q)\) and their deformation theory. It contains high quality results, and it takes some time to fully understand them and their proofs. generalized double affine Hecke algebras of rank 1; quantized del Pezzo surfaces Etingof, P.; Oblomkov, A.; Rains, E., \textit{generalized double affine Hecke algebras of rank 1 and quantized del Pezzo surfaces}, Adv. Math., 212, 749-796, (2007) Noncommutative algebraic geometry, Quantum groups (quantized enveloping algebras) and related deformations, Hecke algebras and their representations Generalized double affine Hecke algebras of rank 1 and quantized del Pezzo 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 [For part I, see the preceding review Zbl 0890.12005.]
Part II is concerned with applications of classical invariant theory to statistical physics and to theta functions. The main theorem in chapter 2 is stated as follows: For a partition function \(\xi(s)= \sum^\infty_{l=1} \gamma_l s^{dl}\) satisfying \(\gamma_t\geq 0\) \((l\geq 1)\) and \(\alpha>0\), the \(2n\)-apolar of \(\xi (s)\)
\[
A_{2n} \bigl(\xi(s), \xi(s)\bigr) =s^{2n} \sum^{2n}_{l=0} (-1)^k {2n \choose k} \left({d \over ds} \right)^{2n-k} \xi(s) \left({d \over ds} \right)^k \xi(s)
\]
has the expansion
\[
A_{2n} \bigl(\xi(s), \xi(s)\bigr) =\sum^\infty_{l=2} \beta_{n,l} s^{-\alpha l}
\]
such that \(\beta_{n,l} \geq 0\) \((l\geq 2)\). This means, for a given partition function \(\xi(s)\) with nonnegative relative probabilities, we construct a sequence of partition functions \(A_{2n} (\xi(s), \xi(s))_{n\geq 1}\) with the same properties, which may be considered a sequence of symbolic higher derivatives of \(\xi(s)\).
The main theorem in chapter 3 is stated as follows: For given theta functions \(\varphi_1(z)\) and \(\varphi_2(z)\) of level \(n_1\) and \(n_2\), respectively, in \(g\) variables \(z=(z_1, z_2, \dots, z_g)\), the \(r=(r_1,r_2, \dots, r_g)\)-apolar
\[
A_r \bigl(\varphi_1(z), \varphi_2(z)\bigr) =\sum_{0\leq j\leq r-j} {(-1)^{|j|} \over n_1^{|j|} n_2^{|r-j|}} {r\choose j} \left({\partial \over \partial z} \right)^j \varphi_1(z) \left({\partial \over\partial z} \right)^{r-j} \varphi_2(z)
\]
is a theta function of level \(n_1+n_2\), and
\[
\begin{multlined} \left({\partial\over \partial z} \right)^h \varphi_1(z) \left({\partial \over\partial z} \right)^k \varphi_2(z)= \sum_{0\leq j\leq h+k} \sum_l(-1)^{|l|} n_1^{|l|} n_2^{|j-l|} {h\choose l} {k\choose j-l} \\ {(-1)^{|h|} (n_1 n_2)^{|h+k-j |} \over (n_1+n_2)^{|h+k|}} \left({\partial \over\partial z} \right)^j A_{h+k-j} \bigl(\varphi_1(z), \varphi_2(z)\bigr). \end{multlined}
\]
{}. differential algebra; decomposition of differential polynomials; apolars; invariant theory; theta functions; statistical partition function Morikawa, H, On differential polynomials II, Nagoya Math. J, 148, 73-112, (1997) Differential algebra, Theta functions and abelian varieties, Exact enumeration problems, generating functions, Equilibrium statistical mechanics On differential polynomials. 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 Let \(A\subset\mathbb{Z}^{n-1}\) be a finite subset, \(\mathbb{C}^ A\) the linear \(\mathbb{C}\)-space of Laurent polynomials \(f=\sum_{\omega\in A}a_ \omega X^ \omega\), \(a_ \omega\in\mathbb{C}\) in some indeterminates \(X\) and \(\nabla_ 0\subset\mathbb{C}^ A\) the set of those \(f\) for which there is \(\kappa\in(\mathbb{C}^*)^{n-1}\) such that \(f(\kappa)=(\partial f/\partial X_ i)(\kappa)=0\) for all \(i\). The closure \(\nabla_ A\) of \(\nabla_ 0\) is an irreducible variety defined in fact on \(\mathbb{Z}\). When \(\nabla_ A\) has codimension 1 then an irreducible polynomial \(\Delta_ A\in\mathbb{Z}[a_ \omega;\omega\in A]\), which is zero on \(\nabla_ A\), is unique up to the sign and it is called the \(A\)-discriminant. If \(\text{codim}(\nabla_ A)>1\) then put \(\Delta_ A=1\). The \(A\)- discriminant is homogeneous and satisfies the following quasi homogeneous \((n-1)\)-conditions: ``\(\sum_{\omega\in A}m(\omega)\cdot\omega\in\mathbb{Z}^{n-1}\) is constant for all monomials \(\prod_{\omega\in A}a_ \omega^{m(\omega)}\) which enter in \(\Delta_ A\)''. This notion extends the classical notions of discriminant and resultant. --- Let \(A=\{\omega_ 1,\ldots,\omega_ N\}\) and \(Y_ A\) be the closure of the set \(\{(\kappa^{\omega_ 1},\ldots,\kappa^{\omega_ N}\mid\kappa\in\mathbb{C}^{*n-1}\}\) in \(\mathbb{P}^{N-1}\). Then \(\nabla_ A\) and \(Y_ A\) are dual projective varieties and the description of \(\Delta_ A\) follows if we can describe the equations of the dual projective variety of a given projective one \(Y\subset\mathbb{P}^{N-1}\) [see the authors' previous paper in Sov. Math., Dokl. 39, No. 2, 385-389 (1989); translation from Dokl. Akad. Nauk SSSR 305, No. 6, 1294-1298 (1989; Zbl 0715.14042)].
Let \(G\) be a free abelian group of rank \(n\), \(G_ \mathbb{C}:=\mathbb{C}\otimes_ \mathbb{Z} G\), \(\lambda:G\to(\mathbb{Q},+)\) a nonzero group morphism, \(S\subset G\) a finitely generated semigroup such that \(o\in S\) and \(\lambda(s)\geq 1\) for all \(s\in S\), \(S_ e=\{t\in S\mid\lambda(t)=e\}\) for \(e\in\mathbb{Q}\) and \(A\subset S_ 1\) a finite subset generating in \(G_ \mathbb{R}=\mathbb{R}\otimes_ \mathbb{Z} G\) the same convex cone as \(S\). For \(k\in\mathbb{Z}_ +\), \(e\in\mathbb{Q}\), \(\omega\in A\) let \(\bigwedge^ k(e)\) be the space of all maps \(S_{k+e}\to\bigwedge^ kG_ \mathbb{C}\), \(\partial_ \omega:\bigwedge^ k(e)\to\bigwedge^{k+1}(e)\) the map given by \(\partial_ \omega(\gamma)(u)=\omega\wedge\gamma(u-\omega)\), if \(u-\omega\in S_{k+e}\), otherwise \(\partial_ \omega(\gamma)(u)=0\) and \(\partial_ f=\sum_{\omega\in A}a_ \omega\partial_ \omega\) if \(f=\sum a_ \omega X^ \omega\in\mathbb{C}^ A\). The complex \((\overset{.}\bigwedge (e),\partial_ f)\) is called the Cayley-Koszul complex. --- Choose a basis \(u\) in terms of \(\overset{.}\bigwedge(e)\) and let \(E_ e(f)\) be the determinant of the complex \((\overset{.}\bigwedge (e),\partial_ f)\) with respect to \(u\) [see \textit{F. Fischer}, Math. Z. 26, 497-550 (1927) or \textit{J.-K. Bismut} and \textit{D. S. Freed}, Commun. Math. Phys. 106, 159-176 (1986; Zbl 0657.58037)]. For \(e\) sufficiently high \(E_ A(f):=E_ e(f)\) is a polynomial of \((a_ \omega)\), \(f=\sum a_ \omega X^ \omega\) which depends on \(e\) only by a constant multiple. --- Let \(Q_ A\) be the convex closure of \(A\) in \(G_ \mathbb{R}\). If \(f=\sum a_ \omega X^ \omega\) we can express \(E_ A(f)=\sum_ \varphi c_ \varphi\prod_{\omega\in A}a_ \omega^{\varphi(\omega)}\), where \(\varphi\) runs in the set \(\mathbb{Z}^ A_ +\) of the maps \(A\to\mathbb{Z}_ +\). Let \(M(E_ A)\subset\mathbb{R}^ A\) be the convex closure of those \(\varphi\in\mathbb{Z}^ A_ +\) for which \(c_ \varphi\neq 0\). Then there exists a nice correspondence between the vertices of \(M(E_ A)\) and some special triangulations of \(Q_ A\).
The theory is applied to the following examples: the discriminant of a polynomial in two indeterminates, the resultant of two quadratric polynomials, the elliptic curve in Tate normal form\dots Laurent polynomials; Cayley-Koszul complex; determinant; discriminant of a polynomial in two indeterminates; elliptic curve Gel'fand, I.; Zelevinskiǐand, A.; Kapranov, M., Discriminants of polynomials in several variables and triangulations of Newton polyhedra, Algebra i Analiz, 2, 1, (1990) Toric varieties, Newton polyhedra, Okounkov bodies, Polynomial rings and ideals; rings of integer-valued polynomials, Complexes, Determinantal varieties Discriminants of polynomials in several variables and triangulations of Newton polyhedra | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 that Lusztig's Frobenius map (for quantum groups at roots of unity) can be, after dualizing, viewed as a characteristic zero lift of the geometric Frobenius splitting of \(G/B\) (in char \(p>0\)) introduced by \textit{V. B. Mehta} and \textit{A. Ramanathan} [Ann. Math. (2) 122, 27-40 (1985; Zbl 0601.14043)]. Frobenius map; Schubert variety; quantum group; geometric Frobenius splitting Kumar, S., Littelmann, P.: Frobenius splitting in characteristic zero and the quantum Frobenius map. J. Pure Appl. Algebra 152, 201--216 (2000) Quantum groups (quantized enveloping algebras) and related deformations, Grassmannians, Schubert varieties, flag manifolds, Quantum groups (quantized function algebras) and their representations, Cohomology theory for linear algebraic groups Frobenius splitting in characteristic zero and the quantum Frobenius map | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 compare the role of set theory in formulating notions of `monster model' or `universe' in the work of Shelah and Grothendieck monster model; Grothendieck universe; simple theory; bounded orbit; saturation and omission Classification theory, stability, and related concepts in model theory, Foundations of algebraic geometry How big should the monster model be? | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 hyperelliptic curves arising from Chebyshev polynomials. The aim of this paper is to characterize the pairs \((q, d)\) such that the hyperelliptic curve \(\mathcal{C}\) over a finite field \(\mathbb{F}_{q^2}\) given by \(y^2 = \varphi_d(x)\) is maximal over the finite field \(\mathbb{F}_{q^2}\) of cardinality \(q^2\). Here \(\varphi_d(x)\) denotes the Chebyshev polynomial of degree \(d\). The same question is studied for the curves given by \(y^2 = (x \pm 2) \varphi_d(x)\), and also for \(y^2 = (x^2 - 4) \varphi_d(x)\). Our results generalize some of the statements in [\textit{T. Kodama} et al., Finite Fields Appl. 15, No. 3, 392--403 (2009; Zbl 1184.11020)]. finite field; maximal curves; hyperelliptic curves; Chebyshev polynomials; Dickson polynomials Curves over finite and local fields, Zeta and \(L\)-functions in characteristic \(p\), Finite ground fields in algebraic geometry, Arithmetic ground fields for curves On certain maximal hyperelliptic curves related to Chebyshev 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 aim of this short note is to give a simple proof of the non-rationality of the double cover of the three-dimensional projective space branched over a sufficiently general quartic. double solid; intermediate Jacobian; Torelli theorem Rationality questions in algebraic geometry, \(3\)-folds, Fano varieties, Automorphisms of surfaces and higher-dimensional varieties A simple proof of the non-rationality of a general quartic double solid | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials It is well-known that every non-negative univariate real polynomial can be written as the sum of two polynomial squares with real coefficients. When one allows a (non-negatively) \textit{weighted} sum of \textit{finitely many} squares instead of a sum of two squares, then one can choose all coefficients in the representation to lie in the field generated by the coefficients of the polynomial. In particular, this allows for an effective treatment of polynomials with rational coefficients.
In this article, we describe, analyze and compare, from both the theoretical and practical points of view, two algorithms computing such a weighted sum of squares decomposition for univariate polynomials with rational coefficients.
The first algorithm, due to the third author, relies on real root isolation, quadratic approximations of positive polynomials and square-free decomposition, but its complexity was not analyzed. We provide bit complexity estimates, both on the runtime and the output size of this algorithm. They are exponential in the degree of the input univariate polynomial and linear in the maximum bitsize of its complexity. This analysis is obtained using quantifier elimination and root isolation bounds.
The second algorithm, due to Chevillard, Harrison, Joldes and Lauter, relies on complex root isolation and square-free decomposition, and was introduced for certifying positiveness of polynomials in the context of computer arithmetic. Again, its complexity was not analyzed. We provide bit complexity estimates, both on the runtime and the output size of this algorithm, which are polynomial in the degree of the input polynomial and linear in the maximum bitsize of its complexity. This analysis is obtained using Vieta's formula and root isolation bounds.
Finally, we report on our implementations of both algorithms and compare them in practice on several application benchmarks. While the second algorithm is, as expected from the complexity result, more efficient on most of examples, we exhibit families of non-negative polynomials for which the first algorithm is better. non-negative univariate polynomials; Nichtnegativstellensätze; sum of squares decomposition; root isolation; real algebraic geometry Real algebra, Semialgebraic sets and related spaces, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Analysis of algorithms, Symbolic computation and algebraic computation Algorithms for weighted sum of squares decomposition of non-negative univariate 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 \(w\) be a permutation of \(\{1, 2, \dots, n\}\), and let \(D(w)\) be the Rothe diagram of \(w\). The Schubert polynomial \({\mathfrak{S}_w}\left(x \right)\) can be realized as the dual character of the flagged Weyl module associated with \(D(w)\). This implies the following coefficient-wise inequality:
\[
\mathrm{Min}_w\left(x \right) \leq{\mathfrak{S}_w}\left(x \right) \le\mathrm{Max}_w\left(x \right),
\] where both \(\mathrm{Min}_w(x)\) and \(\mathrm{Max}_w(x)\) are polynomials determined by \(D(w)\). \textit{A. Fink} et al. [Adv. Math. 332, 465--475 (2018; Zbl 1443.05179)] found that \({\mathfrak{S}_w}\left(x \right)\) equals the lower bound \(\mathrm{Min}_w(x)\) if and only if \(w\) avoids twelve permutation patterns. In this paper, we show that \({\mathfrak{S}_w}\left(x \right)\) reaches the upper bound \(\mathrm{Max}_w(x)\) if and only if \(w\) avoids two permutation patterns 1432 and 1423. Similarly, for any given composition \(\alpha \in \mathbb{Z}_{ \succcurlyeq 0}^n\), one can define a lower bound \(\mathrm{Min}_\alpha (x)\) and an upper bound \(\mathrm{Max}_\alpha (x)\) for the key polynomial \(\kappa_\alpha (x)\). \textit{R. Hodges} and \textit{A. Yong} [``Multiplicity-free key polynomials'', Preprint, \url{arXiv:2007.09229}] established that \(\kappa_\alpha (x)\) equals \(\mathrm{Min}_\alpha (x)\) if and only if \(\alpha\) avoids five composition patterns. We show that \(\kappa_\alpha (x)\) equals \(\mathrm{Max}_\alpha (x)\) if and only if \(\alpha\) avoids a single composition pattern (0, 2). As an application, we obtain that when \(\alpha\) avoids (0, 2), the key polynomial \(\kappa_\alpha (x)\) is Lorentzian, partially verifying a conjecture of \textit{J. Huh} et al. [``Logarithmic concavity of Schur and related polynomials'', Preprint, \url{arXiv:1906.09633}]. Schubert polynomial; key polynomial; flagged Weyl module; upper bound; Lorentzian polynomial Symmetric functions and generalizations, Classical problems, Schubert calculus Upper bounds of 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 The concept of sums of nonnegative circuit (SONC) polynomials was recently introduced as a new certificate of nonnegativity especially for sparse polynomials. In this paper, we explore the relationship between nonnegative polynomials and SONCs. As a first result, we provide sufficient conditions for nonnegative polynomials with general Newton polytopes to be a SONC, which generalizes the previous result on nonnegative polynomials with simplex Newton polytopes. Second, we prove that every SONC admits a SONC decomposition without cancellation. In other words, SONC decompositions preserve sparsity of nonnegative polynomials, which is dramatically different from the classical sum of squares decompositions and is a key property to design efficient algorithms for sparse polynomial optimization based on SONC decompositions. nonnegative polynomial; sum of nonnegative circuit polynomials; SONC; certificate of nonnegativity; sum of squares Semialgebraic sets and related spaces, Polynomials in real and complex fields: location of zeros (algebraic theorems), Fields related with sums of squares (formally real fields, Pythagorean fields, etc.), Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.), Polynomial optimization, Nonconvex programming, global optimization Nonnegative polynomials and circuit 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 present paper investigates the higher-order Sawada-Kotera-type equation and the higher-order Lax-type equation in fluids. The Bell polynomials approach is employed to directly bilinearize the two equations. For the Lax-type equation, bilinear Bäcklund transformation, Lax pair, Darboux covariant Lax pair and infinitely many conservation laws are obtained by means of binary Bell polynomials. Moreover, based on its bilinear form, \(N\)-soliton solutions are also obtained. For the Sawada-Kotera-type equation, with the help of the Riemann theta function and Hirota bilinear method, its one periodic wave solution is obtained. A limiting procedure is presented to analyze in detail the relations between the one periodic wave solution and one soliton solution. Bell polynomials; Hirota bilinear method; Sawada-Kotera-type equation; Lax-type equation Wang, Y. H; Chen, Y., Bell polynomials approach for two higher-order KdV-type equations in fluids, Nonlinear Anal. Real World Applications, 31, 533-551, (2016) KdV equations (Korteweg-de Vries equations), Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems, Soliton solutions, Theta functions and abelian varieties Bell polynomials approach for two higher-order KdV-type equations in fluids | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 reduced separated scheme of finite type over a complete discrete valuation field \(K\) of characteristic 0 with perfect residue field of positive characteristic \(p\). Fontaine conjectured the existence of a natural isomorphism
\[
H_{\text{dR}}^*(X)\otimes_{\bar{K}}B_{\text{dR}} \;\longrightarrow\; H_{\text{ét}}^*(X_{\bar{K}},\mathbb{Q}_p)\otimes_{\mathbb{Q}_p}B_{\text{dR}}
\]
(where \(\bar{K}\) is an algebraic closure of \(K\)) relating de Rham cohomology and \(p\)-adic étale cohomology, where \(B_{\text{dR}}\) is Fontaine's periods field. This important result has been proved by \textit{G. Faltings} [Astérisque. 279, 185--270 (2002; Zbl 1027.14011)], \textit{W. Nizioł} [Duke Math. J. 141, No. 1, 151--178 (2008; Zbl 1157.14009)], \textit{T. Tsuji} [Invent. Math. 137, No. 2, 233--411 (1999; Zbl 0945.14008)] and more recently by \textit{A. Beilinson} [J. Am. Math. Soc. 25, No. 3, 715--738 (2012; Zbl 1247.14018)]. It was also proved by \textit{P. Scholze} [Forum Math. Pi 1, Article ID e1, 77 p. (2013; Zbl 1297.14023)].
This paper is based on several talks given by the author on that Beilinson's paper.
The ideas of Beilinson have been further exploited in two recent papers: [\textit{A. Beilinson}, Camb. J. Math. 1, No. 1, 1--51 (2013; Zbl 1351.14011)] and [\textit{B. Bhatt}, ``\(p\)-adic derived de Rham cohomology'', \url{arXiv:1204.6560}]. de Rham cohomology; \(p\)-adic étale cohomology; Fontaine's rings; cotangent complex; log scheme; alteration; semi-stable morphism; Grothendieck topology \(p\)-adic cohomology, crystalline cohomology, de Rham cohomology and algebraic geometry, Étale and other Grothendieck topologies and (co)homologies, Varieties over finite and local fields, Picard schemes, higher Jacobians, Simplicial sets, simplicial objects (in a category) [See also 55U10], Nonabelian homotopical algebra Around the Poincaré lemma, after Beilinson | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 propose an algebraic geometry framework for the Kakeya problem. We conjecture that for any polynomials \(f, g \in \mathbb{F}_{q_0} [x, y]\) and any \(\mathbb{F}_q / \mathbb{F}_{q_0}\), the image of the map \(\mathbb{F}_q^3 \to \mathbb{F}_q^3\) given by \((s, x, y) \mapsto(s, s x + f(x, y), s y + g(x, y))\) has size at least \(\frac{q^3}{4} - O(q^{5 / 2})\) and prove the special case when \(f = f(x), g = g(y)\). We also prove it in the case \(f = f(y), g = g(x)\) under the additional assumption \(f^\prime(0) g^\prime(0) \neq 0\) when \(f, g\) are both affine polynomials. Our approach is based on a combination of Cauchy-Schwarz and Lang-Weil. The algebraic geometry inputs in the proof are various results concerning irreducibility of certain classes of multivariate polynomials. Kakeya problem; image set on \(F_q\)-points; Lang-Weil bound; reducibility of polynomials in several variables; number of irreducible components of a variety; indecomposable polynomials; affine polynomials; permutation polynomials K. Slavov, An algebraic geometry version of the Kakeya problem, in preparation. Configurations and arrangements of linear subspaces, Finite ground fields in algebraic geometry, Finite geometry and special incidence structures An algebraic geometry version of the Kakeya problem | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 continues the recent progress regarding an enumerative problem in algebraic geometry first studied in the 19th century. The basic problem is to count the number of plane curves of a given degree \(d\) with precisely \(r\) nodal singularities, and no other singularities, passing through the appropriate number of points in general position. A generalization of this problem is to allow the ambient surface (\(\mathbb{P}^2\) in the above situation) to be an arbitrary smooth projective surface, and to consider nodal curves lying in a given linear system (\(|\mathcal{O}_{\mathbb{P}^2}(d)|\) above). Limited progress towards the basic problem was made by mathematicians such as Cayley, but it wasn't until the methods of Gromov-Witten theory began to permeate modern algebraic geometry in the 1990s that progress really took off. Tremendous progress was made in the past couple decades in papers by Göttsche, Bryan-Leung, Caporaso-Harris, Kleiman-Piene, Kool-Shende-Thomas, and Tzeng, using modern enumerative geometric methods as well as elegant combinatorics---and from the tropical geometry school in works of Fomin-Mikhalkin and Block, where the problem is translated by Mikhalkin's powerful Correspondence Theorem into an intriguing problem in graph theory. The precise results and history leading up to the present paper are nicely summarized and explained in its introduction.
The main result in this paper is a proof of a conjecture of \textit{L. Göttsche} [Commun. Math. Phys. 196, No. 3, 523--533 (1998; Zbl 0934.14038)], refining an earlier conjecture of Di Francesco-Itzykson, concerning the polynomials counting the nodal curves in the basic setup of the above enumerative problem. The author derives this result by first showing that in the general setup (i.e., arbitrary surface and linear system) the generating series for these polynomials admits a close connection to the so-called Bell polynomials from combinatorics. In fact, the techniques in this paper are entirely combinatorial: all the geometry in this enumerative problem is absorbed into the papers of the above-cited authors, on which this paper relies heavily. This is a clever excursion into formal power series manipulation based on some nice combinatorics and the explicit formulae/properties of enumerating polynomials and their generating functions studied in earlier works by the aforementioned authors. Göttsche conjecture; nodal curves; plane curves; Bell polynomials; generating function N. Qviller, The Di Francesco-Itzykson-Göttsche conjectures for node polynomials of \mathbb{P}^{2}, Internat. J. Math. 23 (2012), No. 1250049. Enumerative problems (combinatorial problems) in algebraic geometry, Plane and space curves, Factorials, binomial coefficients, combinatorial functions The di Francesco-Itzykson-Göttsche conjectures for node polynomials 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 This paper originates in part from the Habilitation mémoire (Université de Bourgogne, Dijon, 2016) of the author and in part from an introductory talk he gave at the RAQIS'16 conference held at Geneva, Switzerland, in August 2016. It deals with a novel construction that associates an integrable, tau-symmetric hierarchy and its quantization to a cohomological field theory on the moduli space of stable curves, without the semi-simplicity assumption which is needed for the Dubrovin-Zhang hierarchy. It is inspired by Eliashberg, Givental and Hofer's symplectic field theory [\textit{Y. Eliashberg} et al., in: GAFA 2000. Visions in mathematics---Towards 2000. Proceedings of a meeting, Tel Aviv, Israel, August 25--September 3, 1999. Part II. Basel: Birkhäuser. 560--673 (2000; Zbl 0989.81114)] and is the fruit of a joint project of the author with A. Buryak, B. Dubrovin and J. Guéré (see e.g. [\textit{A. Buryak}, et al. ``Tau-structure for the double ramification hierarchies'', \url{arXiv:1602.05423}; ``Integrable systems of double ramification type'', \url{arXiv:1609.04059}]).
After a self contained introduction to the language of integrable systems in the formal loop space and the needed notions from the geometry of the moduli space of stable curves the author explains the double ramification hierarchy construction and presents its main features, with an accent on the quantization procedure, concluding with a list of examples worked out in detail. This paper does not contain new results and most of this material originates from an ongoing collaboration with A. Buryak, B. Dubrovin and J. Guéré. It is however a complete reorganization and, in part, a rephrasing of those results with the aim of showcasing the power of these methods and making them more accessible to the mathematical physics community. moduli space of stable curves; integrable systems; cohomological field theories; double ramification cycle; double ramification hierarchy Rossi, P., Integrability, quantization and moduli spaces of curves, SIGMA, 13, (2017) Families, moduli of curves (algebraic), Relationships between algebraic curves and integrable systems, Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) Integrability, quantization and 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 Given a symmetric group and a maximal parabolic subgroup, minimal length coset representatives can be indexed by Young diagrams fitting inside a rectangle. It is shown that parabolic Kazhdan-Lusztig basis elements in the corresponding Hecke algebra can be written as a product of factors which are differences between a standard generator and a rational function in \(v\). The factors depend in a combinatorial way on the Young diagram corresponding to the index of the basis element. A factorization for the dual Kazhdan-Lusztig basis is also obtained. Kazhdan-Lusztig polynomials; Young diagrams; Hecke algebras; Kazdan-Lusztig bases Kirillov, A., Jr., Lascoux, A.: Factorization of Kazhdan-Lusztig elements for Grassmanians. In: Koike, K. et al. (eds.) Combinatorial Methods in Representation Theory, pp. 143--154. Kinokuniya, Tokyo (2000) Hecke algebras and their representations, Grassmannians, Schubert varieties, flag manifolds, Representations of finite symmetric groups, Combinatorial aspects of representation theory Factorization of Kazhdan-Lusztig elements for Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This is the second paper on Hall polynomials for symplectic groups. The definition is analogous to that of Hall polynomials for general linear groups. In both papers we compute the number of all totally isotropic subspaces \(W\) of type \(\mu\) in a vector space with symplectic geometry \(V\) of type \(\lambda\) denoted \(g^\lambda_\mu\) (see Section 0.1 for definitions). Let the dimensions of \(V\) and \(W\) be \(2m\) and \(m\), respectively. We represent the basis of \(V\) of type \(\lambda= r_1^{d_1} r_2^{d_2} \dots r_s^{d_s}\) with a diagram consisting of \(s\) blocks. The entries of this diagram represent elements of a basis of \(V\) consisting of \(m\) hyperbolic pairs (see Definition 0.1.1). Because of the existence of \(S\)-\(S\) basis (see 0.2 for definition) we always have an obvious totally isotropic subspace. Our method is to ``deform'' this initial subspace into all others by adding and discarding basis elements from the current basis. We must keep the track of the type of subspace we thus obtain and the geometric structure (that is, the positions of hyperbolic pairs in the diagram) of the entire space, ensuring isotropy at each step. In our first paper [ibid. 174, No. 1, 53-76 (1995; Zbl 0832.20068)] we obtain the result for the case when \(\lambda = r^d\), corresponding to a diagram with one block. In this case we can choose our basis so that each type of \(W\) corresponds to one geometric structure of \(V\) only. This makes our task and notation simpler and we obtain a closed formula for \(g^\lambda_\mu\) (see Theorem 0.5.5). Unfortunately it is not so in the case of \(\lambda = r^{d_1}_1 r^{d_2}_2 \cdots r^{d_s}_s\), \(s \neq 1\), where two subspaces may have the same type but induce different geometric structures in \(V\) (see the example in Section 1). This case is studied below. The Hall polynomials for general linear groups have been computed by T. Klein (later, I. G. Macdonald and F. M. Maley). Our method differs from that of either author substantially. Many of the results of part I are used in this paper. A quick review of those results is given in the first section of this paper. diagrams; Hall polynomials; symplectic groups; totally isotropic subspaces; symplectic geometry; hyperbolic pairs; closed formula; general linear groups Representation theory for linear algebraic groups, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Classical groups (algebro-geometric aspects), Symmetric functions and generalizations Hall polynomials for symplectic groups. 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 A theorem of the first author states that the cotangent bundle of the type \(\mathbf{A}\) Grassmannian variety can be canonically embedded as an open subset of a smooth Schubert variety in a two-step affine partial flag variety (see [\textit{V. Lakshmibai}, Transform. Groups 21, No. 2, 519--530 (2016; Zbl 1390.14148)]).
That theorem was inspired by works of Lusztig and Strickland. In particular, \textit{G. Lusztig} [J. Am. Math. Soc. 3, No. 2, 447--498 (1990; Zbl 0703.17008)] related certain orbit closures arising from the type \(\mathbf{A}\) cyclic quiver with \(h\) vertices to affine Schubert varieties. In the case \(h = 2\), Strickland [\textit{E. Strickland}, J. Algebra 75, 523--537 (1982; Zbl 0493.14030)] relates such orbit closures to conormal varieties of determinantal varieties; furthermore, any determinantal variety can be canonically realized as an open subset of a Schubert variety in the Grassmannian (see [\textit{V. Lakshmibai} and \textit{C. S. Seshadri}, Proc. Indian Acad. Sci., Sect. A 87, No. 2, 1--54 (1978; Zbl 0447.14011)]).
In this paper, the authors extend the result of [\textit{V. Lakshmibai}, Transform. Groups 21, No. 2, 519--530 (2016; Zbl 1390.14148)] to cominuscule generalized Grassmannians of arbitrary finite type (such Grassmannians occur in types \(\mathbf{A}-\mathbf{E}\)). Schubert varieties; Grassmannian; affine flag varieties Grassmannians, Schubert varieties, flag manifolds The cotangent bundle of a cominuscule Grassmanian | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 review of results on the structure of homogeneous ind-varieties \(G/P\) of the ind-groups \(G = \operatorname{GL}_\infty (\mathbb{C})\), \(\operatorname{SL}_\infty (\mathbb{C})\), \(\operatorname{SO}_\infty (\mathbb{C} )\), and \(\operatorname{Sp}_\infty (\mathbb{C} )\), subject to the condition that \(G/P\) is the inductive limit of compact homogeneous spaces \(G_n /P_n \). In this case, the subgroup \(P \subset G\) is a splitting parabolic subgroup of \(G\) and the ind-variety \(G/P\) admits a ``flag realization''. Instead of ordinary flags, one considers generalized flags that are, in general, infinite chains \(\mathcal{C}\) of subspaces in the natural representation \(V\) of \(G\) satisfying a certain condition; roughly speaking, for each nonzero vector \(\upsilon\) of \(V\), there exist the largest space in \(\mathcal{C} \), which does not contain \(\upsilon \), and the smallest space in \(\mathcal{C} \), which contains \(v\). We start with a review of the construction of ind-varieties of generalized flags and then show that these ind-varieties are homogeneous ind-spaces of the form \(G/P\) for splitting parabolic ind-subgroups \(P \subset G\). Also, we briefly review the characterization of more general, i.e., nonsplitting, parabolic ind-subgroups in terms of generalized flags. In the special case of the ind-grassmannian \(X\), we give a purely algebraic-geometric construction of \(X\). Further topics discussed are the Bott-Borel-Weil theorem for ind-varieties of generalized flags, finite-rank vector bundles on ind-varieties of generalized flags, the theory of Schubert decomposition of \(G/P\) for arbitrary splitting parabolic ind-subgroups \(P \subset G\), as well as the orbits of real forms on \(G/P\) for \(G = \operatorname{SL}_\infty (\mathbb{C} )\). ind-variety; ind-group; generalized flag; Schubert decomposition; real form Infinite-dimensional Lie groups and their Lie algebras: general properties, Infinite-dimensional Lie (super)algebras, Grassmannians, Schubert varieties, flag manifolds Ind-varieties of generalized flags: a survey | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 Vénéreau polynomials are a sequence of polynomials
\[
b_{m}=y+x^{m}(xz+y(yu+z^{2}))\in\mathbb{C}[x][y,z,u],
\]
\(m\geq1\), which were proposed by S. Vénéreau as potential counterexamples to important conjectures in affine geometry: the Abhyankar-Sathaye embedding conjecture which asserts that every closed embedding of an affine space into another is equivalent to an embedding as a linear subspace, and the Dolgachev-Weisfeiler conjecture which asks whether every flat fibration from an affine space to another with all fibers also isomorphic to affine spaces is a trivial affine bundle.
It is known that the level hypersurfaces of Vénéreau polynomials are isomorphic to affine spaces of dimension \(3\) and that for every \(m\geq1\), the fibration \((b_{m},x):\mathbb{A}^{4}\rightarrow\mathbb{A}^{2}\) is flat with all fibers isomorphic to affine spaces \(\mathbb{A}^{2}\). So \(b_{m}\) provides a counterexample to the Abhyankar-Sathaye embedding conjecture unless it is a \textit{variable} of the polynomial ring \(\mathbb{C}[x,y,z,u]\), i. e., there exists polynomials \(f_{1},f_{2},f_{3}\) such that \(\mathbb{C}[x,y,z,u]=\mathbb{C}[b_{m},f_{1},f_{2},f_{3}]\), and a counterexample to the Dolgachev-Weisfeiler conjecture unless it has the stronger property to be a \(\mathbb{C}[x]\)-\textit{variable} of \(\mathbb{C}[x,y,z,u]\) in the sense that there exists polynomials \(g_{1},g_{2}\) such that \(\mathbb{C}[x,y,z,u]=\mathbb{C}[x,b_{m},g_{1},g_{2}]\). It was established by S. Vénéreau that \(b_{m}\) is indeed a \(\mathbb{C}[x]\)-coordinate for every \(m\geq3\) but the question for \(m=1,2\) remained open.
In the article under review, the author introduces a more general class of \textit{Vénéreau -type polynomials} of the form \(f_{Q}=y+xQ\) with \(Q\in\mathbb{C}[x][v,w]\) and he proves that \(b_{m}\) is a coordinate if and only if so is \(f_{Q}\) for \(Q=x^{2m-1}w\). This is applied to recover the fact that \(b_{m}\) is a \(\mathbb{C}[x]\)-coordinate for \(m\geq3\) and to prove the new result that \(b_{2}\) is a \(\mathbb{C}[x]\)-coordinate too. Other properties of polynomials \(f_{Q}\) in relation with the aforementioned conjectures are established. polynomial rings; Vénéreau polynomials; coordinates; Dolgachev-Weisfeiler conjecture Lewis, D., Vénéreau-type polynomials as potential counterexamples, J. Pure Appl. Algebra, 217, 5, 946-957, (2013) Polynomials over commutative rings, Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem) Vénéreau-type polynomials as potential counterexamples | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 version of the author's report intended for the 25th Arbeitstagung Bonn 1984, Proc. Meet. Max-Placnk-Inst. Math., Lect. Notes 1111, 59-101 (1985). Let \(x_ i\) be commutative and \(\xi_ k\) skew- commutative variables. The scheme Spec \({\mathbb{Z}}[x_ 1,...,x_ m;\xi_ 1,...,\xi_ n]\) may be considered as a geometric object of dimension (1,m\(| n)\) where 1, m and n correspond to the ''arithmetic'' direction \({\mathbb{Z}}\), ''even'' directions \(x_ 1,...,x_ m\) and ''odd'' directions \(\xi_ 1,...,\xi_ n\), respectively. According to the author, this paper aims at giving meaning to the following metaphysical principle: ''all three types of dimensions have equal rights'' - thus developing A. Weil's ideas. The first part of the paper deals with the analogy between the ''arithmetic'' and the ''even'' dimensions. Here, an attemps is made to lay the foundations of ''arithmetic geometry'' (''A-geometry'') which would contain analogues of the main notions and theorems of algebraic and analytic geometries. In the second part of the paper, the relation between ''even'' and ''odd'' dimensions is discussed on the basis of the principle: ''even geometry is the collective effect of infinite- dimensional odd geometry''. The technical part of the paper (the material involved) is clear from the contents: 1. A-manifolds and A-divisors; 2. Riemann-Roch theorems; 3. Problems and perspectives of A-geometry; 4. superspaces; 5. Schubert supercells, 6. Geometry of supergravity.
Throughout the text, the interplay of mathematical and physical notions is stressed, the latter pertaining mostly to elementary particles, field theory and supergravity. supermanifold; grand unification; arithmetic geometry; A-geometry; even geometry; odd geometry; Riemann-Roch theorems; Schubert supercells; supergravity DOI: 10.1070/RM1984v039n06ABEH003181 Generalizations (algebraic spaces, stacks), Supergravity, Grassmannians, Schubert varieties, flag manifolds, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Riemann-Roch theorems New directions in 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 Let \(B\) be a surface of degree 8 in \(\mathbb{P}^3\). Assume that \(B\) is the union of two smooth surfaces \(B_1\) and \(B_2\) of degree \(d\) and \(e\) respectively intersecting transversally along a smooth curve \(C\). The double covering of \(\mathbb{P}^3\) branched along \(B\) has a non-singular model which is a Calabi-Yau manifold. In this paper, the author computes the Hodge numbers \(h^{11}\) and \(h^{12}\) of this manifold. Denote by \(\sigma:\widetilde \mathbb{P}^3\to \mathbb{P}^3\) the blow-up of \(\mathbb{P}^3\) with center \(C\) and \(\pi:X\to \widetilde \mathbb{P}^3\) the double covering of \(\widetilde\mathbb{P}^3\) branched along the strict transform \(\widetilde B\) of \(B\).
Main theorem:
\[
h^{11}(X)=2,\quad h^{12} (X)= \begin{cases} 122 & \text{ if } d=1,\;e=7\\ 102 & \text{ if } d=2,\;e=6\\ 90 & \text{ if } d=3,\;e=5\\ 86 & \text{ if } d=4,\;e=4\end{cases}.
\]
double covering; singularity; Calabi-Yau manifold; Hodge numbers S. Cynk \({ref.surNamesEn}, Hodge numbers of double octic with non-isolated singularities,, \)Ann. Pol. Math.\(, 73, 221, (2000)\) Singularities of surfaces or higher-dimensional varieties, Transcendental methods, Hodge theory (algebro-geometric aspects), Calabi-Yau manifolds (algebro-geometric aspects) Hodge numbers of a double octic with non-isolated singularities | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The survey is devoted to the concept of ``standard monomials'', introduced in the 1940's by Hodge to study the Schubert varieties in the Grassmannian/flag varieties, and developed in the 1970's into a ``theory'' by Seshadri, in collaboration with Musili, Lakshmibai, Littelmann, etc., to study the Schubert varieties in the generalized flag varieties \(G/Q\), where \(G\) is a semisimple group and \(Q\) is a parabolic subgroup of \(G\).
The author begins with the classical results on the Grassmannians and the Schubert varieties, such as the Plücker coordinates, the quadratic relations and the standard monomials in the Plücker coordinates that form a basis in the algebra of regular functions on the affine cones over the Grassmannians. This nice basis, parametrized by the standard Young tableaux, allows to prove many important geometric properties of the Grassmannians and the Schubert varieties -- the projective normality, the projective factoriality (for the Grassmannians), the projective Cohen-Macaulay property.
In the next sections possible generalizations to the cases of the flag variaties corresponding to irreducible \(G\)-modules with 1) a minuscule highest weight; 2) a quasi-minuscule highest weight; 3) a classical type highest weight are given. Some applications of the theory to the study of determinantal varieties, ladder determinantal varieties, varieties of idempotents, varieties of complexes, quiver varieties and singular loci of the Schubert varieties are discussed. For the later developments of the standard monomial theory see the preceding review [\textit{V.Lakshmibai}, in: A tribute to C. S. Seshadri. Birkhäuser, Trends in Mathematics, 283--309 (2003; Zbl 1056.14065)], which may be found in the same volume.
This report may be recommended both to beginners who are looking for main definitions, problems and results in the area, and to specialists who would like to have a systematic historical review of the subject. Grassmannians; semisimple algebraic groups; flag varieties; Schubert varieties; minuscule weights Musili, C.: The development of standard monomial theory. I. In: A tribute to C. S. Seshadri, Chennai, 2002. Trends Math., pp. 385--420. Birkhäuser, Basel (2003) Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Synthetic treatment of fundamental manifolds in projective geometries (Grassmannians, Veronesians and their generalizations), History of mathematics in the 20th century The development of standard monomial theory. 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 Considérons l'espace des modules \(\mathcal{M}_{g,n}\) des courbes algébriques de genre \(g\) avec \(n\) points marqués. Pour tout \(k\geq0\) et \(n\)-uplet \(\mathbf{m}=(m_{1},\dots,m_{n})\) dont les somme des éléments est \(k(2g-2)\) on peut définir les sous-ensembles \[\mathcal{H}_{g}^{k}(\mathbf{m}) = \left\{ (C;p_{1},\dots,p_{n}) : \omega_{C}^{\otimes k} = \mathcal{O}_{C}(\sum_{i=1}^{n}m_{i}p_{i}) \right\}.\] Un problème important consiste à étendre ces ensembles et leur classes fondamentales au bord de l'espace de Deligne-Mumford et à calculer les cycles correspondants. Il existe jusqu'à aujourd'hui trois façon de résoudre ce problème pour \(k\geq1\).
La première repose sur le fait que \(\mathcal{H}_{g}^{k}(\mathbf{m})\) est le tiré-en-arrière de la section nulle du fibré jacobien universel via l'application d'Abel-Jacobi. De nombreuses extensions de cette section ont été proposées. On pourra en particulier cité [\textit{D. Holmes}, J. Inst. Math. Jussieu 20, No. 1, 331--359 (2021; Zbl 1462.14031)] dont la construction est un ingrédient essentiel de la preuve du résultat principal de ce papier. Elles produisent la même classe dans la compactification de Deligne-Mumford, qui est dénotée \(\overline{\textrm{DRC}}\).
La seconde, proposée par Pixton (dans une prépublication que le rapporteur n'a pas réussi à trouvé), consiste à définir une classe \(P_{g}^{g,k}(\tilde{\mathbf {m}})\) dans l'anneau tautologique. Une introduction à cette formule est donnée dans [\textit{F. Janda} et al., Publ. Math., Inst. Hautes Étud. Sci. 125, 221--266 (2017; Zbl 1370.14029)].
Enfin la troisième, proposée dans [\textit{J. Schmitt}, Doc. Math. 23, 871--894 (2018; Zbl 1395.14021)] consiste à définir un cycle \(H_{g,\mathbf{m}}^{k}\) comme une somme pondérée de fermés explicites de l'espace de Deligne-Mumford.
Il a été montré dans une récente prépublication [\textit{Y. Bae} et al., Pixton's formula and Abel-Jacobi theory on the Picard stack, arXiv:2004.08676] que \(2^{-g}P_{g}^{g,k}(\tilde{\mathbf {m}}) = \overline{\textrm{DRC}}\). Le résultat principal de cet article (Theorem 1.1) est l'égalité \(H_{g,\mathbf{m}}^{k} = \overline{\textrm{DRC}}\) montrant l'égalité des trois classes.
Il serait trop long de parler de la preuve, mais il convient de noter que celle-ci est excellemment présentée dans la section 1.4 de la longue et très intéressante introduction de ce papier. double ramification cycle; deformation theory; Abel-Jacobi; differentials Families, moduli of curves (algebraic), Jacobians, Prym varieties, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Infinitesimal structure of the pluricanonical double ramification locus | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 is devoted to some aspects of the algebraic theory of cyclotomic \({\mathbb{Z}}_ p\)-fields \(K=k(\mu_{p^{\infty}})\). The author uses Iwasawa's theory of sheaves for algebraic number fields [\textit{K. Iwasawa}, Ann. Math., II. Ser. 69, 408-413 (1959; Zbl 0090.029)] in order to improve some classical results, and to poke at Greenberg's conjecture from various new perspectives. The main originality of this work is to deal with all the absolute values (i.e., including the Archimedean ones) in the definition of the Pics.
By this way, the author is able to generalize \textit{R. Greenberg}'s conjecture [Am. J. Math. 98, 263-284 (1976; Zbl 0334.12013)], some of B. Gross' results [\textit{L. J. Federer} and \textit{B. H. Gross}, Invent. Math. 62, 443-457 (1980; Zbl 0468.12005)], and also the \textit{L. V. Kuz'min}'s duality pairing [Math. USSR, Izv. 14, 441-498 (1980); translation from Izv. Akad. Nauk SSSR, Ser. Mat. 43, 483-546 (1979; Zbl 0434.12006)] from the class of C.M. fields to arbitrary ones. For instance, the generalized Greenberg conjecture is the assertion that the canonical map
\[
(*)\quad Pic_ p(K)\quad \to \quad C\ell_ p(K)
\]
is an isomorphism from the p- part of the Picard group of K onto the p-part of the ideal class group. The author proves that (*) is (almost always) injective, and observes that it is also surjective if one replaces the naive topology in the definition of Pics by the flat quasi-finite Grothendieck topology. Iwasawa theory; cyclotomic \({bbfZ}_ p\)-fields; sheaves for algebraic number fields; duality pairing; Greenberg conjecture; Picard group; ideal class group; Pics; Grothendieck topology Cyclotomic extensions, Iwasawa theory, Étale and other Grothendieck topologies and (co)homologies On the role of the points at infinity in Iwasawa 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 this text, we study factorizations of polynomials over the tropical hyperfield and the sign hyperfield, which we call \textit{tropical polynomials} and \textit{sign polynomials}, respectively. We classify all irreducible polynomials in either case. We show that tropical polynomials factor uniquely into irreducible factors, but that unique factorization fails for sign polynomials. We describe division algorithms for tropical and sign polynomials by linear terms that correspond to roots of the polynomials. hyperfields; factorization into irreducible elements; division algorithm for polynomials Foundations of tropical geometry and relations with algebra, Combinatorial aspects of matroids and geometric lattices Factorizations of tropical and sign polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper we calculate Reidemeister torsion of flag manifold \(K/T\) of compact semi-simple Lie group \(K = SU_{n+1}\) using Reidemeister torsion formula and Schubert calculus, where \(T\) is maximal torus of \(K\). We find that this number is 1. Also we explicitly calculate ring structure of integral cohomology algebra of flag manifold \(K/T\) of compact semi-simple Lie group \(K = SU_{n+1 }\) using root data, and Groebner basis techniques. Reidemeister torsion; flag manifolds; Weyl groups; Schubert calculus; Groebner-Shirshov bases; graded inverse lexicographic order Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc., Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Gröbner-Shirshov bases, Semisimple Lie groups and their representations, Loop groups and related constructions, group-theoretic treatment On Reidemeister torsion of flag manifolds of compact semisimple Lie 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 Soient \(R\) un anneau et \(I=(x_1, \dots, x_m)\) un idéal de type fini de \(R\). On note \(x=(x_1, \dots, x_m)\in R^m\). Soient \(f\in R[X_1, \dots, X_m]\), \(f=\sum_{a\in\mathbb{N}^m} r_aX^a\) et \(S\subset\mathbb{N}^m\) le support de \(f\). On dit que \(f\) est un polynôme strict sur \((R,I)\) si \(r_a\in R\setminus I\) et pour \(a,a'\in S\) on a \(a\not \preceq a'\), ou \(a\preceq a'\) est la relation \(a_i\leq a_i'\), \(a=(a_1, \dots, a_m)\), \(a'=(a_1', \dots, a_m')\) sur \(\mathbb{N}^m\). On dit que \(x \in R\) est quasiséparable s'il existe un \(j\in\mathbb{N}\) tel que pour tout sous-ensemble fini \(V\) de \(\mathbb{N}^m\) et \(G=\{x^b,b\in V\}\), on a \(\bigcap^\infty_{d=0} (I^jG+I^d) =I^jG\). L'A. demontre que pour \(x\in R\) quasiséparable il existe un polynôme strict \(f\in R[X]\) sur \((R,I)\) tel que \(u=f(x)\) et l'image \(\overline f\) de \(f\) dans \(R/I[X]\) est appelée une polyforme de \(u\) relative a \((R,x)\). On demontre aussi que, si de plus, la suite \(x\) est quasirégulière la polyforme de \(u\) relative à \((R,x)\) est unique. L'A. utilise le concept de polyforme pour donner une démonstration simplifiée pour la désingularisation d'une courbe [voir l'exposé de l'A.: \textit{S. S. Abhyankar} in: Singularities, Summer Inst., Arcata 1981, Proc. Sumpos. Pure Math. 40, Part 1, 1-45 (1983; Zbl 0521.14005)]. resolution of singularities; polynomials Abhyankar, S.S.: Polynomial Expansion. Proceedings of the American Mathematical Society, vol. 126, pp. 1583--1596 (1998) Global theory and resolution of singularities (algebro-geometric aspects), Polynomials over commutative rings Polynomial expansion | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 rotation minimizing frame (RMF) of a space curve is an adapted frame whose angular velocity about the curve tangent vanishes. It is used in motion design and for the generation of swept surfaces. For general curves, RMFs cannot be computed in closed form. An important exception in this respect are Pythagorean hodograph curves (PH curves) whose transcendental RMFs are obtained by integrating a rational function.
The article under review presents two new criteria for the existence of \textit{rational} rotation minimizing frames on quintic PH curves. One of the criteria pertains to the Hopf map description of PH curves, the other one to the description via quaternions. In contrast to the earlier criteria of \textit{R.~T.~Farouki, C.~Giannelli, C.~Manni,} and \textit{A.~Sestini} [Comput. Aided Geom. Design 26, No.~5 580--592 (2009; Zbl 1205.65079)] they are of low algebraic degree and exhibit the expected symmetry in the coefficients of the defining polynomials. The quaternion characterization is especially nice and concise. rotation-minimizing frames; Pythagorean-hodograph curves; angular velocity; Hopf map; complex polynomials; quaternions Farouki, R.T.: Quaternion and Hopf map characterizations for the existence of rational rotation-minimizing frames on quintic space curves. Adv. Comput. Math. 33, 331--348 (2010) Curves in Euclidean and related spaces, Computational aspects of field theory and polynomials, Special algebraic curves and curves of low genus, Plane and space curves, Computational aspects of algebraic curves, Computer graphics; computational geometry (digital and algorithmic aspects), Computer science aspects of computer-aided design Quaternion and Hopf map characterizations for the existence of rational rotation-minimizing frames on quintic space 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 This paper develops some of the theory of strongly algebraic vector bundles over real algebraic varieties. A continuous vector bundle on a real algebraic variety \(X\) is isomorphic to a strongly algebraic vector if some stabilization is. Furthermore if two strongly algebraic bundles are continuously isomorphic, they are algebraically isomorphic. Hence one is reduced to studying a certain subgroup \(\tilde K_{F-\text{alg}}\) of the reduced Grothendieck group of \(F\) vector bundles, where \(F\) is the reals, complexes or the quaternions.
This paper shows that this subgroup tends to be small. For example, if the complexification of \(X\) is a nonsingular complete intersection this paper shows that for \(X\) odd dimensional this subgroup is finite and for \(X\) even dimensional it is finite \(\mathbb{Z}\oplus\)finite group. In case \(X\) has no torsion in even dimensional homology, the finite groups above are \((\mathbb{Z}/2)^ k\). algebraic vector bundles over algebraic varieties; Grothendieck group J. Bochnak, M. Buchner, and W. Kucharz, Vector bundles over real algebraic varieties, \?-Theory 3 (1989), no. 3, 271 -- 298. Topology of real algebraic varieties, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], \(K\)-theory of schemes, Grothendieck topologies and Grothendieck topoi Vector bundles over real 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 This note describes moduli spaces of complexes in the derived category of a Veronese double cone \(Y\). Focusing on objects with the same class \(\kappa_1\) as ideal sheaves of lines, we describe the moduli space of Gieseker stable sheaves and show that it has two components. Then, we study the moduli space of stable complexes in the Kuznetsov component of \(Y\) of the same class, which also has two components. One parametrizes ideal sheaves of lines and it appears in both moduli spaces. The other components are not directly related by a wall-crossing: we show this by describing an intermediate moduli space of complexes as a space of stable pairs in the sense of \textit{R. Pandharipande} and \textit{R. P. Thomas} [Invent. Math. 178, No. 2, 407--447 (2009; Zbl 1204.14026)]. derived categories; Bridgeland stability conditions; Fano threefolds; moduli spaces; Veronese double cone Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Fano varieties, Algebraic moduli problems, moduli of vector bundles A note on the Kuznetsov component of the Veronese double cone | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let E and F be vector bundles of ranks n and m on an scheme X and \(\phi: E\to F\) a morphism of vector bundles. The degeneracy locus of rank r of \(\phi\), \(D_ r(\phi)\), is locally defined, as a subscheme of X, by the annulation of all \((r+1)\)-order minors of \(\phi\). The author gives an explicit description of the ideal \(P_ r\) of the polynomials in the Chern classes of E and F that give classes supported in \(D_ r(\phi)\) in a universal way: to be more explicit, if \(i_ r: D_ r(\phi)\to X\) denotes the inclusion morphism, \(P_ r\) is the ideal of the polynomials \(P\in {\mathbb{Z}}[X_ 1,...,X_ n,Y_ 1,...,Y_ m]\) such that \(P(c.(E),c.(F))\cap \alpha \in Im(i_ r)_*\) for any morphism \(\phi\) of vector bundles as above on an arbitrary scheme X and any \(\alpha\) in the Chow ring of X. Such an ideal clearly includes the Thom-Porteous polynomials giving the fundamental classes of the \(D_ i(\phi)\) for \(i\leq r\), but it is not generated by such polynomials.
A rule for computing the Chern numbers of kernel and cokernel bundles and an algorithm for the Chern numbers of smooth degeneracy locus are also obtained. degeneracy locus; Chern classes; Thom-Porteous polynomials; Chern numbers Pragacz, Piotr, Enumerative geometry of degeneracy loci, Ann. Sci. École Norm. Sup. (4), 21, 3, 413-454, (1988) Enumerative problems (combinatorial problems) in algebraic geometry, Determinantal varieties, Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Singularities in algebraic geometry, Characteristic classes and numbers in differential topology Enumerative geometry of degeneracy loci | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials For a connected reductive group \(G\) and a Borel subgroup \(B\), we study the closures of double classes \(BgB\) in a \((G\times G)\)-equivariant ``regular'' compactification of \(G\). We show that these closures \(\overline{BgB}\) intersect properly all \((G\times G)\)-orbits, with multiplicity one, and we describe the intersections. Moreover, we show that almost all \(\overline {BgB}\) are singular in codimension two exactly. We deduce this from more general results on \(B\)-orbits in a spherical homogeneous space \(G/H\); they lead to formulas for homology classes of \(H\)-orbit closures in \(G/B\), in terms of Schubert cycles. Bruhat decomposition; equivariant compactification; regular embedding; spherical homogeneous space; Schubert cycles M. Brion. ''The behaviour at infinity of the Bruhat decomposition''. Comment. Math. Helv. 73(1998), pp. 137--174.DOI. Group actions on varieties or schemes (quotients), Homogeneous spaces and generalizations, Grassmannians, Schubert varieties, flag manifolds The behavior at infinity of the Bruhat decomposition | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 article is the complement of the preprint ``Sextic double solids'' by \textit{I. Cheltsov} and \textit{J. Park} [\url{arXiv:math/0404452}]. The authors state the following conjecture (Conjecture 1.4).
Let \(S\subset \mathbb P^3\) be a nodal surface of degree \(2r\). Suppose that the surface \(S\) has at most \(r(2r-1)+1\) singular points. Then the double cover of \(\mathbb P^3\) ramified along \(S\) is not \(\mathbb Q\)-factorial (that is, there exists a Weil divisor such that any multiple of it is not Cartier divisor) if and only if the surface \(S\) is defined by the equation of the form
\[
f_r(x,y,z,w)^2+h_1(x,y,z,w)g_{2r-1}(x,y,z.w)=0,
\]
where \(f_r\), \(g_{2r-1}\), \(h_1\) are homogeneous polynomials of degrees \(r\),\(2r-1\), and \(1\), respectively.
The authors prove this conjecture for \(r=2\) (Theorem 4.3) and for \(r=3\) (Theorem 5.3). The main instrument of the proofs is the method of H. Clemens, which reduces the computation of the rank of 4-th integral cohomology of double solids to a combinatorial problem. double solid; nodal singularity; threefold; \(\mathbb Q\)-factoriality Hong, Kyusik; Park, Jihun, On factorial double solids with simple double points, J. Pure Appl. Algebra, 208, 1, 361-369, (2007) Divisors, linear systems, invertible sheaves, Singularities of surfaces or higher-dimensional varieties, \(3\)-folds On factorial double solids with simple double points | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The main result of the article is: Let \(X\) be a Schubert variety in the Grassmann manifold \(G(m,n)\) of \(m\)-dimensional subspaces of an \(n\)- dimensional vector space over the real or complex numbers. Then there exists a Riccati flow on \(G(m,n)\), and a stable manifold \(W\) of this flow, such that \(W\) is exactly the smooth locus of \(X\). The approach to the study of the singularities of Schubert varieties via the Riccati flow gives a new point of view on the topic, with interesting consequences. It is, for example, possible to prove, without using representation theory, that Schubert varieties over the complex numbers are smooth if and only if their singular cohomology satisfies Poincaré duality. stable manifold of Riccati flow; singularities of Schubert varieties Wolper, J.S.: The Riccati flow and singularities of Schubert varieties. Proceedings of the AMS 123 (1995) 703--709 Grassmannians, Schubert varieties, flag manifolds, Singularities in algebraic geometry, Dynamics induced by flows and semiflows, Topology of real algebraic varieties The Riccati flow and singularities 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 Let \(C\) be a smooth curve over an algebraically closed field \(\mathbf{k}\), and let \(E\) be a locally free sheaf of rank \(r\). We compute, for every \(d > 0\), the generating function of the motives \([ \operatorname{Quot}_C(E, \boldsymbol{n})] \in K_0( \operatorname{Var}_{\mathbf{k}})\), varying \(\boldsymbol{n} = (0 \leq n_1 \leq \cdots \leq n_d)\), where \(\operatorname{Quot}_C(E, \boldsymbol{n})\) is the \textit{nested Quot scheme of points}, parametrising 0-dimensional subsequent quotients \(E \twoheadrightarrow T_d \twoheadrightarrow \cdots \twoheadrightarrow T_1\) of fixed length \(n_i = \chi( T_i)\). The resulting series, obtained by exploiting the Białynicki-Birula decomposition, factors into a product of shifted motivic zeta functions of \(C\). In particular, it is a rational function. Quot scheme; zeta function Parametrization (Chow and Hilbert schemes), Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), (Equivariant) Chow groups and rings; motives, Applications of methods of algebraic \(K\)-theory in algebraic geometry On the motive of the nested Quot scheme of points on 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 In this paper, the authors try to generalize their own results [Rev. Mat. Complut. 30, No. 3, 589--620 (2017; Zbl 1390.13074)] concerning the algebraicity of Puiseux series in one variable to the case of several variables.
From the abstract: ``We show that the algebraicity of such a series for given bounded degrees is determined by a finite number of explicit universal polynomial formulas. Conversely, given a vanishing polynomial, there is a closed-form formula for the coefficients of the series in terms of the coefficients of the polynomial and of a bounded initial part of the series.'' multivariate polynomials; algebraic power series; implicitization; closed form of coefficients Power series rings, Formal power series rings, Surfaces and higher-dimensional varieties About algebraic Puiseux series in several variables | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We give a embedding of the Lagrangian Grassmannian \(\mathrm{LG}(n)\) inside an ordinary Grassmannian that is well-behaved with respect to the Wronski map. As a consequence, we obtain an analogue of the Mukhin-Tarasov-Varchenko theorem for \(\mathrm{LG}(n)\). The restriction of the Wronski map to \(\mathrm{LG}(n)\) has degree equal to the number of shifted or unshifted tableaux of staircase shape. For special fibres one can define bijections, which, in turn, gives a bijection between these two classes of tableaux. The properties of these bijections lead to a geometric proof of a branching rule for the cohomological map \(H^\ast(\mathrm{Gr}(n, 2 n)) \otimes H^\ast(\mathrm{LG}(n)) \rightarrow H^\ast(\mathrm{LG}(n))\), induced by the diagonal inclusion \(\mathrm{LG}(n) \hookrightarrow \mathrm{LG}(n) \times \mathrm{Gr}(n, 2 n)\). We also discuss applications to the orbit structure of jeu de taquin promotion on staircase tableaux. Schubert calculus; Lagrangian Grassmannian; Wronski map; Young tableaux Combinatorial aspects of representation theory, Classical problems, Schubert calculus A marvellous embedding of the Lagrangian Grassmannian | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We discuss four methods of proving modularity of Calabi--Yau threefolds with \(h^{12}=1\): existence of elliptic ruled surfaces inside (Hulek-Verrill), correspondence with a product of an elliptic curve and a \(K3\) surface (Livné-Yui), correspondence with a (modular) rigid Calabi-Yau threefold, and existence of an involution splitting the fourdimensional representation into twodimensional subrepresentations.
We apply these methods to prove modularity of 17 out of 18 double octic Calabi-Yau threefolds for which ``numerical evidence of modularity'' was found in the second author's thesis.
We observe that modularity holds for those elements in a pencil having some additional geometric properties. In the proofs we use representations of the considered Calabi-Yau threefolds as a Kummer fibration associated to a fiber product of rational elliptic fibrations. Calabi-Yau; double coverings; modular forms Cynk, S.; Meyer, C., Modularity of some non-rigid double octic Calabi-Yau threefolds, Rocky Mt. J. Math., 38, 1937-1958, (2008) Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Relations with algebraic geometry and topology, Calabi-Yau manifolds (algebro-geometric aspects) Modularity of some non-rigid double octic Calabi-Yau threefolds | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials \textit{A. Björner} and \textit{T. Ekedahl} [Ann. Math. (2) 170, No. 2, 799--817 (2009; Zbl 1226.05268)] prove that general intervals \([e, w]\) in Bruhat order are ``top-heavy'', with at least as many elements in the \(i\)-th corank as the \(i\)-th rank. Well-known results of \textit{J. B. Carrell} [Proc. Symp. Pure Math. 56, 53--61 (1994; Zbl 0818.14020)] and of \textit{V. Lakshmibai} and \textit{B. Sandhya} [Proc. Indian Acad. Sci., Math. Sci. 100, No. 1, 45--52 (1990; Zbl 0714.14033)] give the equality case: \([e, w]\) is rank-symmetric if and only if the permutation \(w\) avoids the patterns 3412 and 4231 and these are exactly those \(w\) such that the Schubert variety \(X_w\) is smooth. In this paper we study the finer structure of rank-symmetric intervals \([e, w]\), beyond their rank functions. In particular, we show that these intervals are still ``top-heavy'' if one counts cover relations between different ranks. The equality case in this setting occurs when \([e, w]\) is self-dual as a poset; we characterize these \(w\) by pattern avoidance and in several other ways. Weyl group; Bruhat order; Schubert variety; intersection cohomology; Kazhdan-Lusztig polynomial Combinatorial aspects of algebraic geometry, Grassmannians, Schubert varieties, flag manifolds, Other algebro-geometric (co)homologies (e.g., intersection, equivariant, Lawson, Deligne (co)homologies), Classical problems, Schubert calculus Self-dual intervals in the Bruhat 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 In the paper under review, the author generalizes a theorem of Ogg on supersingular \(j\)-invariants to supersingular elliptic curves with level. Ogg observed that the level one case yields a characterization of the primes dividing the order of the monster. He shows that the corresponding analyses for higher levels give analogous characterizations of the primes dividing the orders of other sporadic simple groups (e.g., baby monster, Fischer's largest group). This situates Ogg's theorem in a broader setting. More generally, he characterizes, in terms of supersingular elliptic curves with level, the primes arising as orders of Fricke elements in centralizer subgroups of the monster. He also presents a connection between supersingular elliptic curves and umbral moonshine. Finally, he presents a procedure for explicitly computing invariants of supersingular elliptic curves with level structure. moonshine; modular curves; supersingular elliptic curves; supersingular polynomials Elliptic curves, Structure of modular groups and generalizations; arithmetic groups, Holomorphic modular forms of integral weight, Relationship to Lie algebras and finite simple groups, Forms of half-integer weight; nonholomorphic modular forms, Simple groups: sporadic groups Supersingular elliptic curves and moonshine | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 of the paper under review deduce string-theoretically the conjecture of \textit{T. Hausel} et al. [Duke Math. J. 160, No. 2, 323--400 (2011; Zbl 1246.14063)] on the cohomology of character varieties with marked points (\textit{HLRV conjecture} for short). In doing so, they provide strong evidence for this conjecture.
We start by briefly reviewing the HLRV conjecture. In broadest terms, the HLRV conjecture expresses a weighted sum of mixed Poincaré polynomials of character varieties as the \(\log\) of a generating function made up of modified MacDonald polynomials. As this may appear unmotivated, the present paper provides strong physical reasons why it should be so.
We review the definition of character varieties. Denote by \(C\) a smooth genus \(g\geq 0\) projective curve over \(\mathbb{C}\) and let \(D = p_1 + \cdots + p_k\) be a divisor of marked points, which are assumed to be distinct and reduced. Consider the fundamental group \(\pi_1(C\backslash D)\). Each marked point \(p_i\) determines a generator \(\gamma_i\) of \(\pi_1(C\backslash D)\). Let \(r\geq 1\). For any non-empty partition \(\mu=(\mu^1,\dots,\mu^l)\) of \(r\), we choose a semisimple conjugacy class \(C_\mu\) in \(GL(r,\mathbb{C})\) with the property that the eigenvalues of any matrix in \(C_\mu\) have multiplicities given by \(\left\{\mu^1,\dots,\mu^l\right\}\). Consider a collection \(\boldsymbol{\mu}=(\mu_1,\dots,\mu_k)\) of partitions of \(r\). The moduli problem consists in parametrizing conjugacy classes of representations
\[
f : \pi_1(C\backslash D) \to GL(r,\mathbb{C}),
\]
with the property that \(f(\gamma_i)\in C_{\mu_i}\). The moduli space of such representations yields the \textit{character variety} \(\mathcal{C}(C,D;\boldsymbol{\mu})\). The dependence on the choice of eigenvalues is suppressed in the notation because the topological invariants below do not depend on these choices.
Via a weight filtration, the compactly supported cohomology of \(\mathcal{C}(C,D;\boldsymbol{\mu})\) is packaged into a mixed Poincaré polynomial \(P_c(\mathcal{C}(C,D;\boldsymbol{\mu});u,t)\). To each marked point \(p_i\) is associated an infinite collection of formal variables \(\boldsymbol{x}_i=(\boldsymbol{x}_{i,1},\boldsymbol{x}_{i,2},\dots)\). Summing over \(n\geq 1\) and over all all partitions of \(n\) of an appropriate weighting of \(P_c(\mathcal{C}(C,D;\boldsymbol{\mu});u,t)\) in the variables \(\boldsymbol{x}_i\) yields a generating function \(F_{HLRV}(z,w,\boldsymbol{x})\). The authors consider another generating function \(Z_{HLRV}(z,w,\boldsymbol{x})\) made up of modified MacDonald polynomials. Then the HLRV conjecture claims that
\[
(1) \; \; Z_{HLRV}(z,w,\boldsymbol{x}) = \exp(F_{HLRV}(z,w,\boldsymbol{x})).
\]
The strategy of the authors to derive (1) is as follows. For each curve \(C\), they construct a local Calabi-Yau orbifold curve \(\widetilde{Y}\) and consider its generating function \(Z^{ref}_{\widetilde{Y}}(q,\boldsymbol{x},y)\) of refined stable pair invariants. They conjecture that it agrees with \(Z_{HLRV}\) after a change of variables, namely that:
\[
(2) \; \; Z^{ref}_{\widetilde{Y}}(z^{-1}w,\boldsymbol{x},z^{-1}w^{-1})=Z_{HLRV}(z,w,\boldsymbol{x}).
\]
The authors make use of the \textit{parabolic \(P=W\) conjecture}, which is as follows. Motivated by the fact that the character variety \(\mathcal{C}(C,D;\boldsymbol{\mu})\) is diffeomorphic to a moduli space of strongly parabolic Higgs bundles on \(C\), it conjectures that the mixed Poincaré polynomial \(P_c(\mathcal{C}(C,D;\boldsymbol{\mu});u,t)\) is identified with a certain mixed Poincaré polynomial for Higgs bundles.
Assuming the parabolic \(P=W\) conjecture and the conjecture of (2), the authors prove using geometric engineering that the same change of variables as in (2) turns (1) into the \textit{refined} stable pair/\textit{refined} Gopakumar-Vafa expansion. The latter is an enumerative correspondence widely expected to hold true. Its unrefined version (taking the Euler characteristic) was proven for toric Calabi-Yau 3-folds in genus 0.
Throughout the paper, the authors give ample evidence for the various involved conjectures to hold true. They review the various objects that are involved in great detail, contributing to the ease of reading. HLRV conjecture; refined stable pair invariants; refined Gopakumar-Vafa invariants; refined BPS expansion; parabolic P=W conjecture; MacDonald polynomials; geometric engineering; parabolic Higgs bundles; local orbifold curves Chuang, W-y; Diaconescu, D-E; Donagi, R.; Pantev, T., Parabolic refined invariants and Macdonald polynomials, Commun. Math. Phys., 335, 1323, (2015) 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, Symmetric functions and generalizations, Combinatorial aspects of representation theory, Relationships between surfaces, higher-dimensional varieties, and physics, Calabi-Yau manifolds (algebro-geometric aspects), Combinatorial aspects of partitions of integers, Mirror symmetry (algebro-geometric aspects) Parabolic refined invariants and Macdonald 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 \noindent The author studies lambda operations on mixed motives. This structure is induced from the natural tensor structure on the category \(D:=DM_{\text{gm}}(\mathrm{Spec}(F),{\mathbb Q}).\) Let
\[
{\zeta}(X,t)={\sum}_{n\geq 0}{\text{cl(Sym}}^{n}(X))t^{n}
\]
be the power series with coefficient in \(K_{0}(D),\) where the class of a motive \(X\) in \(K_{0}\) is is denoted as \({\text{cl}}(X)\). \(K_{0}(D)\) is defined to be the Grothendieck ring of the additive category \(D\) modulo the subgroup generated by \({\text{cl}}(Z)-{\text{cl}}(X))-{\text{cl}}(Y)\) for distinguished triangles \(X\rightarrow Z\rightarrow Y\rightarrow X[1]\). \textit{M. Kapranov} [``The elliptic curve in the S-duality theory and Eisenstein series for Kac-Moody groups'', \url{arXiv:math/0001005}] considered the zeta function of a motive \(X\). Denote \({\lambda}(X):={\zeta}(X,-t)^{-1}.\) The assignment \({\text{cl}}(X)\rightarrow {\lambda}(X)\) defines a \({\lambda}\)-structure on \(K_{0}(D).\) The author considers the Adams operations on this \({\lambda}\)-ring in the equivariant context i.e. there is an action of a finite group \(G\) on a motive \(X.\) lambda ring; Grothendieck ring: motivic zeta function (Equivariant) Chow groups and rings; motives, \(J\)-homomorphism, Adams operations On lambda operations on mixed motives. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \({\mathbb C}_p\) be an algebraically closed field of finite characteristic \(p\), complete with respect a non-arhimedean absolute value \(|\cdot |\). Consider \(\Lambda \subseteq {\mathbb C}_p\) a discrete \({\mathbb F}_p\)-submodule such that there exists an \({\mathbb F}_p\)--basis \(\{\lambda_0,\lambda_1,\dots\}\) of \(\Lambda\) such that \(0<|\lambda_0|<|\lambda_1|< \cdots \) and \(|\lambda_i|\to\infty\). For a natural number \(k\) define the meromorphic function
\[
C_{k,\Lambda}(z):=\sum_{\lambda\in \Lambda}\frac{1} {(z-\lambda)^k}
\]
on \({\mathbb C}_p\). A weak form of the main result of the paper establishes that all zeroes of \(C_{k,\Lambda}\) lie on critical spheres \(S_{i,\Lambda}=\partial B(0,|\lambda_i|)=\{ y\in {\mathbb C}_p\mid |y|=|\lambda_i|\}\) of \(\Lambda\). The strong form of the result gives an accurate account of the number of zeroes of \(C_{k,\Lambda}\) on \(S_{i,\Lambda}\), which depends only on \(i\) and \(k\), but not on \(\Lambda\).
The paper is organized as follows. In Section 2 the authors collect some facts about \(p\)-adic expansions. In Section 3 power sums on elements of finite sublattices are discussed. Goss polynomials attached to \(\Lambda\) are introduced in Section 4. Section 5 is devoted to the study of the fundamental domain \({\mathfrak F}_{\Lambda}\). The proof of the main result, Theorem 6.1, is given in Section 6. The proof is based on properties of power sums and non-archimedean contour integration. In the last section the authors give examples and applications. rigid-analytic period functions; Goss polynomials; distribution of zeros of \(p\)-adic functions Arithmetic theory of algebraic function fields, Modular forms associated to Drinfel'd modules, Rigid analytic geometry On the zeroes of certain periodic functions over valued fields of positive characteristic | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 \textit{standard} cardioid is the set of points in the complex plane formed by reflecting the point 1 in every tangent to the unit circle. These points constitute a simple closed curve that is the boundary of two open disjoint regions, a bounded \textit{inner} region that is heart-shaped, and an \textit{outer} region. We verify that the outer region consists of all the points from which three tangents can be drawn to the cardioid, a statement that is part of the folklore of the theory of the cardioid -- and deemed by many to be geometrically obvious! It was the key observation that led Frank Morley to his celebrated trisector theorem. cardioid; cubic polynomials Polynomials and rational functions of one complex variable, Special algebraic curves and curves of low genus, History of mathematics in the 20th century Cardioids and self-inversive cubic polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A Brauer factor set (Bfs) \(c\) is said to be normalized [cf. \textit{L. H. Rowen}, Trans. Am. Math. Soc. 282, 765-772 (1984; Zbl 0539.16016)]\ if the Galois group acts on it by the sign representation: \(c^\sigma=c^{\text{sgn }\sigma}\). Rowen (l.c.) showed that for any central simple \(F\)-algebra \(R\), \(R\otimes R\) is similar to an algebra with normalized Bfs, so any algebra of odd index has a normalized Bfs. Here the authors show that any \(R\) of even degree with normalized Bfs cannot be a division algebra and in fact is often a square in \(\text{Br}(K/F)\), where \(K\) is a separable splitting field for the algebra.
The authors begin by finding a cohomological description of normalized Bfs's, as a certain subgroup of the kernel of the restriction map \(H^2(G,M)\to H^2(H,M)\) where \(G\) is the Galois group (corresponding to a Galois extension \(E\) of \(F\) containing \(K\)), \(H\) is the subgroup corresponding to \(K\) and \(M\) is a \(G\)-module which is \(H^1\)-trivial. The normalized Bfs's can then be described as elements restricting to 0 on certain subgroups. As a consequence they find that a central simple algebra whose degree and index are divisible by the same (positive) power of 2 cannot have a normalised Bfs. -- Next the authors consider the space \([K,D]\) spanned by \(xy-yx\), where \(x\in K\), \(y\in D\). They show that (for \(K=F[a]\)) the following are equivalent: (a) \(KvK=[K,D]\) for all \(v\neq 0\) in \([K,D]\); (b) the Galois group \(G\) (of the normal closure \(E\) of \(K\)) is doubly transitive on the conjugates of \(a\), (c) if the minimal polynomial of \(a\) over \(F\), \(f(\lambda)\) factorizes as \(f=(\lambda-a)g\) in \(K[\lambda]\), then \(g\) is irreducible over \(K\). Other similar but somewhat technical conditions on \(D\) are examined, which could have an influence on the structure of \(D\). Brauer factor sets; Galois groups; separable splitting fields; central simple algebras; degree; index; minimal polynomials DOI: 10.1006/jabr.1997.7148 Finite-dimensional division rings, Brauer groups of schemes, Skew fields, division rings, Quaternion and other division algebras: arithmetic, zeta functions Normalized Brauer factor sets | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The equations of motion of integrable systems involving hyperelliptic Riemann surfaces of genus 2 and one relevant degree of freedom are integrated in the framework of the Jacobi inversion problem, using a reduction to the \(\theta\)-divisor on the Jacobi variety, i.e., to the set of zeros of the \(\theta\)-function. Explicit solutions are given in terms of Kleinian \(\sigma\)-functions and their derivatives. The procedure is applied to the planar double pendulum without gravity, but it is worked out for any Abelian integral of first or second kind. equations of motion; integrable systems; hyperelliptic Riemann surfaces; Jacobi inversion problem; reduction; \(\theta\)-function; Kleinian \(\sigma\)-functions; planar double pendulum Enolski, V. Z.; Pronine, M.; Richter, P. H., Double pendulum and \textit{ {\(\theta\)}}-divisor, J Nonlinear Sci, 13, 2, 157-174, (2003) Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.), Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions, KdV equations (Korteweg-de Vries equations), Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics, Analytic theory of abelian varieties; abelian integrals and differentials, Relationships between algebraic curves and integrable systems, Other completely integrable equations [See also 58F07] Double pendulum and \(\theta\)-divisor | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials From the abstract: Given a homogeneous ideal \(I \) in a polynomial ring over a field, one may record, for each degree \(d\) and for each polynomial \(f\in I_d\), the set of monomials in \(f\) with nonzero coefficients. These data collectively form the \(tropicalization\) of \(I\) . Tropicalizing ideals induces a ``matroid stratification'' on any (multigraded) Hilbert scheme. Very little is known about the structure of these stratifications. In this paper, we explore many examples of matroid strata, including some with interesting combinatorial structure, and give a convenient way of visualizing them. We show that the matroid stratification in the Hilbert scheme of points (\(\mathbb{P}^1)^{[k]}\) is generated by all Schur polynomials in \(k\) variables. We end with an application to the \(T\) -graph problem of (\(\mathbb{A}^2)^{[n]}\); classifying this graph is a longstanding open problem, and we establish the existence of an infinite class of edges. Schur polynomials; tropical ideals; multigraded Hilbert schemes Parametrization (Chow and Hilbert schemes), Algebraic combinatorics The matroid stratification of the Hilbert scheme of points on \(\mathbb{P}^1\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(\mathbb{F}_q\) denote a finite field having \(q\) elements, where \(q\) is a prime power. For a vector space \(E\) of finite dimension \(m\) over \(\mathbb{F}_q\) and \(\ell\le m\), let \(G(\ell, m)\) denote the Grassmannian variety of vector subspaces of dimension \(\ell\) of \(E\). A projective variety \(X\) is called a linear section of the of the Grassmannian \(G(\ell,m)\) if \(X=G(\ell, m)\cap Z(g_1,\ldots, g_N)\), where \(g_1, g_2, \ldots, g_N\) are linearly independent functionals in the ideal that they generate and \(X(\mathbb{F}_q)=\{P_1, \ldots, P_M\}\) is a non-empty set of \(\mathbb{F}_q\)-rational points of \(X\). In this paper, authors study parity-check codes by showing that for every linear section of a Grassmannian, there exists a parity check code with good properties depending on the linear sections. For the Lagrangian-Grassmannian variety, they reveal that these parity-check codes are the low density parity check (LDPC) codes. They also obtained some properties of parity check codes associated to linear sections of Grassmannians. algebraic geometry codes; Grassmann codes; Lagrangian-Grassmannian codes; Schubert codes; parity check codes; LDPC codes Grassmannians, Schubert varieties, flag manifolds, Applications to coding theory and cryptography of arithmetic geometry, Geometric methods (including applications of algebraic geometry) applied to coding theory, Linear codes (general theory) Codes on linear sections of 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 By the classical theory of symmetric functions, there are four special bases \(\{s_\lambda\}_\lambda\), \(\{Q^\vee_\lambda\}_\lambda\), \(\{Q_\lambda\}_\lambda\), and \(\{S_\lambda\}_\lambda\) of the space of symmetric functions, called the Schur, Hall-Littlewood \(P\)-, Hall-Littlewood \(Q\)-, and big Schur functions, respectively. They satisfy nice properties such as orthogonality, triangularity, Pieri and Littlewood-Richardson rules on the multiplication, which are extensively used in various area. The paper under review gives a unified interpretation of these bases in terms of graded modules over the ring \(A := \mathbb{C}[\mathfrak{S}_n] \ltimes \mathbb{C}[x_1,\dotsc,x_n]\) along the theory of Kosoka system introduced by the author in [Ann. Sci. Éc. Norm. Supér. (4) 48, No. 5, 1035--1074 (2015; Zbl 1367.20038)]. The simple graded \(A\)-modules are parametrized by partitions \(\lambda\) of \(n\), and denoted by \(L_\lambda\). Then one has a sequence of \(A\)-module surjections \(P_\lambda \to \widetilde{K}_\lambda \to K_\lambda \to L_\lambda\) with \(P_\lambda\) a chosen projective cover of \(L_\lambda\). The crucial point of the argument is the usage of the Garsia-Procesi basis [\textit{A. M. Garsia} and \textit{C. Procesi}, Adv. Math. 94, No. 1, 82--138 (1992; Zbl 0797.20012)], which enables one to analyze \(K_\lambda^*\) in great detail. Then one obtains homological results on the graded \(A\)-modules \(P_\lambda\), \(\widetilde{K}_\lambda\), \(K_\lambda\) and \(L_\lambda\) (Theorems A,C). The four bases mentioned in the beginning are recovered as the twisted Frobenius characteristic of these graded modules, and several properties of the symmetric functions can also be re-derived (Corollaries 2.39,2.40). symmetric functions; Springer representations; Kostka polynomials Group actions on varieties or schemes (quotients), Representation theory for linear algebraic groups, Hecke algebras and their representations, Representations of finite symmetric groups, Symmetric functions and generalizations, Derived categories of sheaves, dg categories, and related constructions in algebraic geometry, Representations of Lie and linear algebraic groups over local fields Symmetric functions and Springer representations | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This Bourbaki seminar primarily describes the work of \textit{J.-B. Bost} [Publ. Math., Inst. Hautes Étud. Sci. 93, 161--221 (2001; Zbl 1034.14010)], who showed the following. Let \(K\) be a number field embedded in \(\mathbb C\), let \(X\) be a smooth algebraic variety over \(K\) (i.e., an integral separated scheme of finite type over \(K\)), and let \(F\) be an algebraic subbundle of the tangent bundle \(T_X\). We assume that \(F\) is involutive; i.e., closed under the Lie bracket. Then \(F\) defines a holomorphic foliation of the complex manifold \(X(\mathbb C)\). Bost showed that the leaf \(\mathcal F\) through a rational point \(P\in X(K)\) is algebraic if the following local conditions are satisfied: (i) for almost all prime ideals \(\mathfrak p\) of the ring of integers \(\mathcal O_K\) of \(K\), the \(p\)-curvature of the reduction modulo \(\mathfrak p\) of the subbundle \(F\subseteq T_X\) vanishes at \(P\) (here \(p\) is the prime of \(\mathbb Z\) lying below \(\mathfrak p\)); and (ii) the manifold \(\mathcal F\) satisfies the Liouville property: every plurisubharmonic function on \(\mathcal F\) bounded from above is constant. For example, \(\mathcal F\) satisfies the Liouville property if it is a holomorphic image of a complex algebraic variety minus a closed analytic subset.
The article also shows how Bost's theorem is related to a conjecture of Grothendieck, predicting when a linear system \((d/dz)Y=A(z)Y\), \(A(z)\in M_d(\mathbb Q(z))\), has a basis of algebraic solutions. In addition, the article shows how Bost's theorem implies a theorem of Y.~André giving a local criterion for when a differential form on a smooth variety is exact; a local criterion for two elliptic curves to be isogenous (special case of a theorem of Faltings); and even a theorem of Kronecker stating that if \(\alpha\) is an element of a number field \(K\) such that \(\alpha\) is congruent to an element of \(\mathbb F_p\) for all but finitely many primes \(\mathfrak p\) of \(K\), then \(\alpha\in\mathbb Q\). algebraicity; foliation; Hermitian vector bundle; \(p\)-curvature; slope; Grothendieck conjecture; Arakelov geometry; Liouville property Chambert-Loir, A., Théorèmes d'algébricite en géométrie diophantienne (d'après J.-B. bost, Y. André, D. and G. chudnovsky), Asterisque, 282, 175-209, (2002) Varieties over global fields, Transcendence (general theory), Arithmetic varieties and schemes; Arakelov theory; heights Algebraicity theorems in Diophantine geometry following J.-B. Bost, Y. André, D. \& G. Chudnovsky. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 Schur polynomials \(s_{\lambda }\) are essential in understanding the representation theory of the general linear group. They also describe the cohomology ring of the Grassmannians. For \(\rho = (n, n-1, \dots , 1)\) a staircase shape and \(\mu \subseteq \rho\) a subpartition, the Stembridge equality states that \(s_{\rho /\mu } = s_{\rho /\mu^T}\). This equality provides information about the symmetry of the cohomology ring. The stable Grothendieck polynomials \(G_{\lambda }\), and the dual stable Grothendieck polynomials \(g_{\lambda }\), developed by \textit{A. S. Buch} [Acta Math. 189, No. 1, 37--78 (2002; Zbl 1090.14015)], \textit{T. Lam} and \textit{P. Pylyavskyy} [Int. Math. Res. Not. 2007, No. 24, Article ID rnm125, 48 p. (2007; Zbl 1134.16017)], are variants of the Schur polynomials and describe the \(K\)-theory of the Grassmannians. Using the Hopf algebra structure of the ring of symmetric functions and a generalized Littlewood-Richardson rule, we prove that \(G_{\rho /\mu } = G_{\rho /\mu^T}\) and \(g_{\rho /\mu } = g_{\rho /\mu^T}\), the analogues of the Stembridge equality for the skew stable and skew dual stable Grothendieck polynomials. Stembridge equality; Grothendieck polynomial; Young tableau; Hopf algebra Symmetric functions and generalizations, Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Connections of Hopf algebras with combinatorics The Stembridge equality for skew stable Grothendieck polynomials and skew dual stable Grothendieck 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 Schubert calculus on the space of \(d\)-dimensional linear subspaces of a smooth \(n\)-dimensional quadric lying in the complex projective space is the object of study in this article. Following Hodge and Pedoe the author develops the intersection theory of this space in a purely combinatorial manner. It is proved in particular that if a triple intersection of Schubert cells on this space is nonempty then a certain combinatorial relation holds among the Schubert symbols involved, similar to the classical one. The sufficiency of these necessary conditions is also studied. Several examples are examined to illustrate the necessity and sufficiency of these conditions. subspaces of a quadric; Schubert calculus; intersection theory; intersection of Schubert cells Sertöz, S.: A triple intersection theorem for the varieties \(SO(n)/pd\), Fund. math. 142, No. 3, 201-220 (1993) Grassmannians, Schubert varieties, flag manifolds, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry A triple intersection theorem for the varieties \(SO(n)/P_ d\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 investigates maps between local Picard groups. The main goal is to understand the kernel in two cases: pull-back to the normalization and restriction to a hyperplane section. We give sufficient (and almost necessary) conditions that guarantee that the kernel is finite or of finite type. local Picard group; Grothendieck-Lefschetz hyperplane theorem; normalization Kollár, János, Maps between local Picard groups, (2014) Singularities of surfaces or higher-dimensional varieties, Picard groups, Topological properties in algebraic geometry, Deformations of complex singularities; vanishing cycles, Topological aspects of complex singularities: Lefschetz theorems, topological classification, invariants, Local deformation theory, Artin approximation, etc. Maps between local Picard groups | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(X \subset \mathbb P^r_k\) be an integral non-degenerate variety of dimension \(n\) over an algebraically closed field \(k\) of characteristic zero. The \(a\)th secant variety \(\sigma_a (X)\) is the closure of the union of all linear subspaces of \(\mathbb P^r\) spanned by \(a\) points of \(X\). The variety \(X\) is \textit{not defective} if \(\sigma_a (X)\) has the expected dimension \(\min\{r,a(n+1)-1\}\) for all \(a > 0\). The \textit{tangential variety} \(\tau (X) \subset X\) is the closure of the union of all Zariski tangent spaces \(T_q X \subset \mathbb P^r\) taken over \(q \in X_{\text{reg}}\) and is a variety of dimension at most \(2n\). \textit{Abo and Vannieuvenhoven} described all integers \(\dim \sigma_a (\tau (X))\) for the image \(X \subset \mathbb P^r\) of the \(d\)-Veronese embedding of \(\mathbb P^n\) [\textit{H. Abo} and \textit{N. Vannieuwenhoven}, Trans. Am. Math. Soc. 370, No. 1, 393--420 (2018; Zbl 1387.14136)], proving a conjecture of \textit{A. Bernardi} et al. [J. Algebra 321, No. 3, 982--1004 (2009; Zbl 1226.14065)].
In the paper under review, the author uses results of \textit{J. Alexander} and \textit{A. Hirschowitz} [J. Algebr. Geom. 4, No. 2, 201--222 (1995; Zbl 0829.14002); Invent. Math. 140, 303--325 (2000; Zbl 0973.14026)] and methods from Abo and Vannieuvenhoven [loc. cit.] to compute the dimension of the join \(\sigma_{a,b} (X)\) of \(a\) copies of \(X\) and \(b\) copies of \(\tau (X)\) when \(X \subset \mathbb P^r\) is the degree \(d\) Veronese embedding of \(\mathbb P^n\). After proving some conditions under which a product of a variety and a curve under a Segre embedding are not defective, he proves that the tangential variety of \((\mathbb P^1)^n\) embedding by the linear system \(|{\mathcal O}_{(\mathbb P^1)^n} (d_1, \dots, d_n)|\) is not defective if \(d_i \geq 3\) and \((d_1+1)/(d_2+1) \geq 38\) and a similar statement for \(\mathbb P^n \times \mathbb P^1\). He also proves some asymptotic results giving generic non-defective behavior. tangential variety; secant variety; additive decompositions of homogeneous polynomials; defectivity; Segre-Veronese variety Projective techniques in algebraic geometry, Secant varieties, tensor rank, varieties of sums of powers On the secant varieties of tangential 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 set of conjugacy classes appearing in a product of conjugacy classes in a compact, \(1\)-connected Lie group \(K\) can be identified with a convex polytope in the Weyl alcove. In this paper we identify linear inequalities defining this polytope. Each inequality corresponds to a non-vanishing Gromov-Witten invariant for a generalized flag variety \(G/P\), where \(G\) is the complexification of \(K\) and \(P\) is a maximal parabolic subgroup. This generalizes the results for \(SU(n)\) of \textit{S. Agnihotri} and \textit{C. Woodward} [Math. Res. Lett. 5, No. 6, 817--836 (1998; Zbl 1004.14013)] and \textit{P. Belkale} [Compos. Math. 129, No. 1, 67--86 (2001; Zbl 1042.14031)] on the eigenvalues of a product of unitary matrices and quantum cohomology of Grassmannians. conjugacy classes; parabolic bundles; quantum cohomology; generalized flag variety; Grassmannians; Schubert calculus C. Teleman and C. Woodward, ''Parabolic bundles, products of conjugacy classes and Gromov-Witten invariants,'' Ann. Inst. Fourier \((\)Grenoble\()\), vol. 53, iss. 3, pp. 713-748, 2003. Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics, Group actions on varieties or schemes (quotients) Parabolic bundles, products of conjugacy classes, and Gromov-Witten invariants. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 too technical to describe the results in detail. Roughly speaking, the author examines surjective holomorphic maps \(\varphi : X \to Y\) of complex projective manifolds which are locally biholomorphic outside \(\varphi^{-1} (S)\), where \(S \subset Y\) is a hypersurface with only simple normal crossings, and which satisfy additional conditions of a monomial type (in terms of local coordinates). The main theorem gives formulas for \(\varphi^* ch(Y) - ch(X)\) in \(A(X) \otimes Q\). As applications, the author examines the case when \(\varphi \) is (a) a finite ramified covering and (b) a blow up, then he calculates \(c_i (X) - \varphi^*c_i (Y)\) for \(i = 1,2\) and \(i = 1,2,3\) respectively. Finally he investigates \(\varphi\) when it is a finite ramified covering of \(Y = \mathbb{P}^n\) and \(X\) is minimal of general type. Then he gets \(K^n_X < 2c_2 (X) K_X^{n - 2}\). When \(X\) is singular, he examines Kawai covers. If such a cover is minimal of general type, he obtains the last formula also for it. exponential Chern character; Grothendieck-Riemann-Roch theorem; Grothendieck group; minimal manifold of general type; divisor with simple normal crossings Yamamoto, S.: The behaviour of Chern characters of projective manifolds under certain holomorphic maps. Comment. math. Univ. st. Paul 44, 175-199 (1995) Proper holomorphic mappings, finiteness theorems, Ramification problems in algebraic geometry, Minimal model program (Mori theory, extremal rays), Analytic subsets and submanifolds, Compact complex \(n\)-folds The behaviour of Chern characters of projective manifolds under certain holomorphic maps | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The authors study the lifting of the Schubert stratification of the homogeneous space of complete real flags of \(\mathbb R^{n+1}\) to its universal covering group \(\mathrm{Spin}_{n+1}\). They call the lifted strata the Bruhat cells of \(\mathrm{Spin}_{n+1}\), in keeping with the homonymous classical decomposition of reductive algebraic groups. They present explicit parameterizations for these Bruhat cells in terms of minimal-length expressions \(\sigma = a_{i_1} \dots a_{i_k}\) for permutations \(\sigma \in S_{n+1}\) in terms of the n generators \(a_i = (i, i + 1)\). These parameterizations are compatible with the Bruhat orders in the Coxeter-Weyl group \(S{n+1}\). This stratification is an important tool in the study of locally convex curves; they present a few such applications. Schubert stratifications; signed Bruhat cells; Bruhat stratifications; locally convex curves; Coxeter group; symmetry group Grassmannians, Schubert varieties, flag manifolds, Stratifications in topological manifolds, Covering spaces and low-dimensional topology Locally convex curves and the Bruhat stratification of the spin group | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 deals with the problem of deciding whether an sparse polynomial vanishes on a given algebraic set. They motivate the problem via some applications and comment the achievements, as well as the techniques used as tropical geometry matrix theory or computational number theory, by other authors on the topic. In this paper, they deal with the computation of all polynomials, of a fix degree having at most a fixed upper bound of terms, that vanish on an algebraic set that is represented by a witness set. They present algorithms and illustrate their method through with examples from kinematics, chemical reaction and networks. binomial ideals; sparse polynomials; computational algebraic geometry Geometric aspects of numerical algebraic geometry, Numerical algebraic geometry, Symbolic computation and algebraic computation, Foundations of tropical geometry and relations with algebra Binomiality testing and computing sparse polynomials via witness sets | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials As a result of the solution of Horn's conjecture and the saturation conjecture by \textit{A. Klyachko} [Sel. Math., New Ser. 4, No. 3, 419--445 (1998; Zbl 0915.14010)], and \textit{A. Knutson} and \textit{T. Tao} [J. Am. Math. Soc. 12, No. 4, 1055--1090 (1999; Zbl 0944.05097)], one can decide when an intersection of Schubert classes in a Grassmannian is non-zero by solving a series of inequalities coming from the answer to similar questions for smaller Grassmannians. This indicates that there should be geometric proofs of the results of Klyachko and Knutson/Tao [see \textit{W. Fulton}, Astérisque 252, 255--269, Exp. No. 845 (1998; Zbl 0929.15006)]. In this article the author provides such proofs and obtains results that generalize the original conjecture of Horn, and also give a stronger form of the saturation conjecture.
For relations to the characterization of eigenvalues of a sum of hermitian matrices in terms of the summands, and the connections to representation theory, combinatorics and geometric invariant theory see the beautiful article [\textit{W. Fulton}, Bull. Am. Math. Soc., New Ser. 37, No. 3, 209--249 (2000; Zbl 0994.15021)]. Schubert classes; Grassmannians; hermitian matrices; eigenvalues; transversality Geometric proofs of Horn and saturation conjectures. \textit{Journal of Alge-} \textit{braic Geometry }15(2006), 133--173.arXiv:math/0208107.Zbl 1090.14014 MR 2177198 Grassmannians, Schubert varieties, flag manifolds, Determinantal varieties, Representation theory for linear algebraic groups, Classical problems, Schubert calculus Geometric proofs of Horn and saturation conjectures | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Aufstellung der Gleichung für den Fall von 3 Doppelpunkten und 2 Spitzen. double points; fifth order curves Euclidean analytic geometry, Questions of classical algebraic geometry, Plane and space curves Equations of certain fifth order 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 The main result of the present paper is the computation of a certain natural divisor class on the universal family over a partial compactification of \(\mathcal{A}_g\), the moduli space of principally polarized abelian varieties of dimension \(g\). Namely, denote by \(\mathcal{A}_g'\) Mumford's partial compactification of \(\mathcal{A}_g\) obtained by adding semi-abelic varieties of torus rank one, i.e., compactifications of \(\mathbb{C}^*\)-extensions of abelian varieties of dimension \((g-1)\). The boundary \(\mathcal{A}_g' \setminus \mathcal{A}_g\) is isomorphic to \(\mathcal{X}_{g-1}\), the universal family over \(\mathcal{A}_{g-1}\) (the isomorphism is explicitly described in the paper). Furthermore it has the property that it is contained in every toroidal compactification of \(\mathcal{A}_g\). It is thus a natural starting point for the study of the Chow ring and cohomology groups of any compactification of \(\mathcal{A}_g\).
The universal family \(\mathcal{X}_g \to \mathcal{A}_g\) admits a zero section \(z_g : \mathcal{A}_g \to \mathcal{X}_g\), which set-theoretically assigns to any abelian variety \(A\) with origin \(0 \in A\) the moduli point \((A, 0)\) in \(\mathcal{X}_g\). This setup can be extended to a universal family \(\mathcal{X}_g'\) and a zero section \(z_g': \mathcal{A}_g' \to \mathcal{X}_g'\) over the partial compactification. The main result is then the computation of the class of the image of this section, expressed as a polynomial in certain geometrically defined classes of codimension \(1\) and \(2\).
This result is then used to compute the class of the closure of the \textit{double ramification cycle} on a partial compactification of \(\mathcal{M}_{g,n}\), the moduli space of \(n\)-pointed curves of genus \(g\). Given an \(n\)-tuple \(\underline{d} = (d_1, \dots, d_n) \in \mathbb{Z}^n\) of integers summing to zero, the latter is defined as the locus of pointed curves \((C; p_1, \dots, p_n) \in \mathcal{M}_{g,n}\) such that \(\mathcal{O}_C(\sum d_i p_i) = 0 \in \mathrm{Jac}(C)\). This locus can be expressed as the pull-back of the zero section discussed above under the Abel-Jacobi map \(s_{\underline{d}}: \mathcal{M}_{g,n} \to \mathcal{X}_g\) defined set-theoretically by \(s_{\underline{d}}(C; p_1, \dots, p_n) = (\mathrm{Jac}(C), \mathcal{O}_C(\sum d_i p_i))\). It can also be interpreted as the Hurwitz locus of curves admitting a map to \(\mathbb{P}^1\) with prescribed preimages and ramification over two points (hence the name).
This construction of the double ramification cycle is not restricted to the case of smooth curves: It can be extended to curves of \textit{compact type}, i.e., stable curves with no non-separating nodes. The class of the closure of the double ramification cycle on \(\mathcal{M}_{g,n}^{\mathrm{ct}}\) was computed by \textit{R. M. Hain} [Math. Sci. Res. Inst. Publ. 28, 97--143 (1995; Zbl 0868.14006)]. Here the authors take this result one step further by extending the computation to \(\mathcal{M}_{g,n}^o\), the moduli spaces of stable curves with at most one non-separating node. Moreover they show that the Abel-Jacobi map does not extend to the locus of curves having more than one non-separating node. principally polarized abelian varieties; moduli space; toroidal compactification, semi-abelic varieties, double ramification cycle S. Grushevsky and D. Zakharov, The zero section of the universal semiabelian variety, and the double ramification cycle, preprint, arXiv:1206.3534. Algebraic moduli of abelian varieties, classification, Families, moduli of curves (algebraic) The zero section of the universal semiabelian variety and the double ramification cycle | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 give combinatorial formulas for the Hilbert coefficients, \(h\)-polynomial and the Cohen-Macaulay type of Schubert varieties in Grassmannians in terms of the posets associated with them. As a consequence, necessary conditions for a Schubert variety to be a complete intersection and combinatorial criteria are given for a Schubert variety to be Gorenstein and almost Gorenstein, respectively. Grassmannian; Schubert variety; Hilbert coefficients; Gorenstein ring; almost Gorenstein ring Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.), Dimension theory, depth, related commutative rings (catenary, etc.), Grassmannians, Schubert varieties, flag manifolds Hilbert coefficients of Schubert varieties in Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials For an elliptic curve \(E\) over a finite field \(\mathbb {F}_q\), where \(q\) is a prime power, we propose new algorithms for testing the supersingularity of \(E\). Our algorithms are based on the polynomial identity testing problem for the \(p\)-th division polynomial of \(E\). In particular, an efficient algorithm using points of high order on \(E\) is given. division polynomials; polynomial identity testing; elliptic curves Curves over finite and local fields, Number-theoretic algorithms; complexity, Applications to coding theory and cryptography of arithmetic geometry, Elliptic curves, Cryptography On division polynomial PIT and supersingularity | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This book presents the current state of the art and ongoing work on \textit{affine Schubert calculus} with an accent on the combinatorics of a family of polynomials called \(k\)-Schur functions. Several generalizations of \(k\)-Schur functions are also discussed.
In [Duke Math. J. 116, No. 1, 103--146 (2003; Zbl 1020.05069)], \textit{L. Lapointe} et al. found computational evidence of a conjectural property for a family of new bases for a filtration on the symmetric function space: the property is that Macdonald polynomials expand positively in terms of it (see Section 4.11). This gave rise to \(k\)-Schur functions, which then were proven to be connected to a vast set of subjects, see the introduction of the book.
Chapter 2 (which occupies 2/3 of the book) presents basics on \(k\)-Schur functions, emphasizing combinatoric aspects in the symmetric function setting. In particular, \(k\)-Pieri rule for the product of \(k\)-Schur functions is discussed. Also, \(k\)-Schur functions (resp. their duals) generate so called \textit{strong} (resp., \textit{weak}) tableaux, which is explained by means of an affine insertion algorithm. A lot of example in \textit{Sage} is given, the authors hope that this will encourage the reader to generate new data and new conjectures.
Chapter 3 explains the combinatorial connections between Stanley symmetric functions (appeared when Stanley was enumerating reduced words in the symmetric group) and \(k\)-Schur functions, using root systems, nilCoxeter and nilHecke rings. There are exercises in this chapter. Several geometric interpretations of the material are listed at the end of this chapter.
\textit{T. Lam} showed [Am. J. Math. 128, No. 6, 1553--1586 (2006; Zbl 1107.05095)] that the dual \(k\)-Schur functions are a special case of affine analogs of Stanley symmetric functions. Then, the way how Stanley symmetric functions are related to nilCoxeter algebra [\textit{S. Fomin} and \textit{R. P. Stanley}, Adv. Math. 103, No. 2, 196--207 (1994; Zbl 0809.05091)] can be reproduced in the affine setting.
Chapter 4 presents the nilHecke ring in the general Kac-Moody setting, and then this ideology is applied for affine Grassmannians. The nilHecke ring was introduced to study the torus equivariant cohomology of Kac-Moody partial flag varieties, and so this chapter presents this geometric aspect of the story. The algebraic part of correspondence between polynomial representatives for the Schubert classes of the affine Grassmannian, and \(k\)-Schur functions in homology and the dual \(k\)-Schur functions in cohomology is presented. Schur functions; affine Schubert calculus; \(k\)-Schur functions; Macdonald positivity; Pieri rule; nilCoxeter ring; nilHecke ring; Kac-Moody variety Lam, Thomas; Lapointe, Luc; Morse, Jennifer; Schilling, Anne; Shimozono, Mark; Zabrocki, Mike, \(k\)-Schur functions and affine Schubert calculus, Fields Institute Monographs 33, viii+219 pp., (2014), Springer, New York; Fields Institute for Research in Mathematical Sciences, Toronto, ON Research exposition (monographs, survey articles) pertaining to algebraic geometry, Classical problems, Schubert calculus, Homogeneous spaces and generalizations, Symmetric functions and generalizations, Combinatorial aspects of representation theory, Kac-Moody groups, Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.) \(k\)-Schur functions and affine Schubert calculus | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The concept of exotic (ordered) configuration spaces of points in a space was proposed by Baryshnikov. He obtained formulas for the (exponential) generating series of their Euler characteristics. We explore unordered analogs of the spaces. Considering a complex quasiprojective variety, we give a formula for the generating series of the classes of these configuration spaces in the Grothendieck ring of complex quasiprojective varieties. We state the result in terms of the natural power structure over this ring. This yields formulas of generating series of additive invariants of configuration spaces like the Hodge-Deligne polynomial and the Euler characteristic. configuration space; generating series; Grothendieck ring of complex quasiprojective varieties Generating series of the classes of exotic unordered configuration spaces | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials By extending the notion of grid classes to include infinite grids, we establish a structural characterisation of the simple permutations in Av(4231, 35142, 42513, 351624), a pattern class which has three different connections with algebraic geometry, including the specification of indices of Schubert varieties defined by inclusions. This characterisation leads to the enumeration of the class. Schubert varieties; pattern classes; enumeration Michael H. Albert and Robert Brignall, Enumerating indices of Schubert varieties defined by inclusions. J. Combin. Theory Ser. A 123 (2014), 154--168. Permutations, words, matrices, Exact enumeration problems, generating functions, Classical problems, Schubert calculus Enumerating indices of Schubert varieties defined by inclusions | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 following is a classic open problem in the Schubert calculus of the flag variety:
Given any reasonably nice subvariety \(Y \subset \mathrm{Flags}(\mathbb C^n)\), express the homology class of \(Y\) as an integral linear combination of Schubert classes.
In this paper, the authors consider the case where \(Y\) is the Peterson variety. By analyzing the cellular structure of the Peterson variety and the group action of a one-dimensional torus on this variety, they reduce the computations in the intersection theory of the flag variety to a systematic combinatorial analysis of the elements of the symmetric group.
In the process, they give a partial solution to the first problem introduced above.
their proof counts the points of intersection between certain Schubert varieties in the full flag variety and the Peterson variety, and shows that these intersections are proper and transverse. Schubert calculus; intersection theory; Peterson variety Insko, Erik, Schubert calculus and the homology of the Peterson variety, Electron. J. Combin., 22, 2, (2015) Grassmannians, Schubert varieties, flag manifolds, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Classical problems, Schubert calculus, Classical real and complex (co)homology in algebraic geometry Schubert calculus and the homology of the Peterson variety | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(X\) be a Fano manifold of Picard number one defined over the field of complex numbers. For a general point \(x\) in \(X\), consider the normalized space \(\mathcal{K}_x\) of rational curves of minimal degree with respect to \(-K_X\) through \(x\). It is well known that \(\mathcal{K}_x\) is a smooth projective variety and the map \(\tau_x : \mathcal{K}_x \to \mathbb{P}T_x(X)\) sending a member of \(\mathcal{K}_x\) to its tangent direction at \(x\) is the normalization morphism of the image \(\mathcal{C}_x := \tau_x(\mathcal{K}_x)\), which is called the variety of minimal rational tangents at \(x\). Since \(\tau_x\) is an embedding in many examples, it has been asked whether \(\tau_x\) is always an immersion, or even an embedding. Note that \(\tau_x\) is an immersion at \([C] \in \mathcal{K}_x\) if and only if \(C\) is a standard rational curve, i.e., under the normalization \(f : \mathbb{P}^1 \to C \subset X\), the vector bundle \(f^*T(X)\) decomposes as \(\mathcal{O}_{\mathbb{P}^1}(2) \oplus \mathcal{O}_{\mathbb{P}^1}(1)^p \oplus \mathcal{O}_{\mathbb{P}^1}^q\) for some nonnegative integers \(p\) and \(q\) with \(p + q = \dim X - 1\). In fact, a general rational curve of minimal degree through \(x\) is standard, and the above question is asking whether all members of \(\mathcal{K}_x\) are standard. If we extend the setting to an arbitrary uniruled projective manifold and a locally unsplit dominating family of rational curves, then there is a counterexample to the above question given by \textit{C. Casagrande} and \textit{S. Druel} [Int. Math. Res. Not. 2015, No. 21, 10756--10800 (2015; Zbl 1342.14088)]. However, this example is a Fano manifold of Picard number three, and the locally unsplit dominating family is neither minimal nor unsplit.
In the paper under review, the authors show a negative answer to the above question by explicitly studying \(\tau_x\) for Veronese double cones, which are a Fano manifold of Picard number one. More precisely, let \(X^f\) be a Veronese double cone of dimension \(n\) associated with a general weighted homogeneous polynomial of degree \(2d\) with \(n\) weight one variables and \(1\) weight two variable such that \(d \geq 3\) is an odd integer and \(n \geq d\) is an integer. Then \(X^f\) is a Fano manifold of Picard number one, and there exists a minimal dominating family consisting of minimal rational curves on \(X^f\). Furthermore, if \(2d \leq n\), then \(\tau_x\) at a general point \(x \in X^f\) is not an immersion. This example is an optimal counterexample in some senses. Veronese double cone; Fano manifolds; varieties of minimal rational tangents Hwang, J. M.; Kim, H., Varieties of minimal rational tangents on Veronese double cones, Algebraic Geom., 2, 176-192, (2015) Fano varieties, Rationally connected varieties Varieties of minimal rational tangents on Veronese double cones | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 2005, Kayal suggested that Schoof's algorithm for counting points on elliptic curves over finite fields might yield an approach to factor polynomials over finite fields in deterministic polynomial time. We present an exposition of his idea and then explain details of a generalization involving Pila's algorithm for abelian varieties. factoring polynomials; zeta function; abelian variety; finite fields Number-theoretic algorithms; complexity, Polynomials over finite fields, Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture), Finite ground fields in algebraic geometry, Arithmetic ground fields for abelian varieties Using zeta functions to factor polynomials 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 From the preface to the whole volume: In this extended lecture notes, the author introduces us to the theory of algebraic cycles on abelian varieties from yet an other viewpoint than his predecessor [see \textit{S. Müller-Stach}, ibid. 285-305 (2000; see the preceding review Zbl 0991.14005)] over algebraically closed fields. Both results are proved by using the theory of Fourier transforms, in particular the author presents the so-called main theorem of Fourier theory on abelian varieties. This, in turn, can be proved by using the Grothendieck Riemann-Roch theorem. Altogether, this contribution provides deep insights into the structure of Chow groups of abelian varieties and serves as an ideal model for arbitrary varieties. Fourier theory on abelian varieties; Grothendieck Riemann-Roch theorem; Chow groups of abelian varieties Murre, J.: Algebraic cycles on abelian varieties: application of abstract Fourier theory. The arithmetic and geometry of algebraic cycles, 307-320 (1998) Algebraic cycles, Algebraic theory of abelian varieties, Motivic cohomology; motivic homotopy theory, Riemann-Roch theorems, (Equivariant) Chow groups and rings; motives Algebraic cycles on abelian varieties: Application of abstract Fourier 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 The author constructs an interesting Zariski pair, consisting of two plane curves of degree 12 with 30 cusps each, having the same Alexander polynomial, but being distinguished by a height pairing. For doing this, the Alexander polynomials are related to the Mordell-Weil rank of an isotrivial family of Jacobians. ideals of quasi-adjunction; Alexander polynomials; Mordell-Weil rank; Zariski pairs Plane and space curves, Singularities of curves, local rings, Coverings of curves, fundamental group, Jacobians, Prym varieties, Elliptic surfaces, elliptic or Calabi-Yau fibrations, \(3\)-folds, Hypersurfaces and algebraic geometry Mordell-Weil lattices and toric decompositions 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 In the paper under review the author studies algebraic varieties defined by minors of a fixed size of a matrix such that all minors are restricted to lie in a ladder shaped region. The main result are explicit formulas for the Hilbert functions and Hilbert series related to these varieties. In particular, the author shows that, although the formulas in the general case are complicated, they can be used to derive fairly simple estimates for some useful geometric invariants of the varieties such as the degree of the Hilbert polynomial. determinantal ideals; Hilbert functions; determinantal varieties; Schubert varieties; ladder determinantal varieties; Hilbert series DOI: 10.1016/S0012-365X(01)00256-4 Determinantal varieties, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Exact enumeration problems, generating functions, Combinatorial identities, bijective combinatorics, Combinatorial aspects of representation theory, Grassmannians, Schubert varieties, flag manifolds, Linkage, complete intersections and determinantal ideals Hilbert functions of ladder determinantal varieties | 0 |
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