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In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The problem of computing products of Schubert classes in the cohomology ring can be formulated as the problem of expanding skew Schur polynomial into the basis of ordinary Schur polynomials. We reformulate the problem of computing the structure constants of the Grothendieck ring of a Grassmannian variety with respect to its basis of Schubert structure sheaves in a similar way; we address the problem of expanding the generating functions for skew reverse-plane partitions into the basis of polynomials which are Hall-dual to stable Grothendieck polynomials. From this point of view, we produce a chain of bijections leading to Buch's \(K\)-theoretic Littlewood-Richardson rule. Grothendieck polynomial; Littlewood-Richardson rule; tabloid; sign-reversing involution; Yamanouchi word Combinatorial aspects of representation theory, Classical problems, Schubert calculus A dual approach to structure constants for \(K\)-theory of Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We find the algebras of unipotent invariants of Cox rings for all double flag varieties of complexity 0 and 1, the algebra in question is either a free algebra or a hypersurface. Knowing the structure of this algebra permits one to effectively decompose tensor products of irreducible representations into direct sums of irreducible representations. Double flag variety; Cox ring; complexity; linear representation; tensor product of representations; branching problem Classical groups (algebro-geometric aspects), Homogeneous spaces and generalizations Invariants of the Cox rings of low-complexity double flag varieties for classical 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 As confirmed by the title, this is a book about G-functions. Moreover, as far as I know, it is the first book on this subject. So it is worth while to give an elaborate discussion. Before we consider its contents further, let us give a brief description of G-functions.
Consider a Taylor series of the form \(f(z)=\sum^{\infty}_{n=0}a_ nz^ n\), where the numbers \(a_ n\) belong to the same algebraic number field K \(([K:{\mathbb{Q}}]<\infty)\). Suppose it satisfies the following conditions, (i) f satisfies a linear differential equation with polynomial coefficients. (ii) \(| a_ n| =O(c^ n_ 1)\) for all \(n>0\) and a fixed \(c_ 1>0\). (iii) (common denominator of \(a_ 0,...,a_ n)=O(c^ n_ 2)\) for all \(n>0\) and a fixed \(c_ 2>0.\)
Roughly speaking, they can be considered as (very interesting) variations on the geometric series. They are not the same as the Meyer G-functions. The most common examples are -log(1-z), arctg(z) and the ordinary hypergeometric functions with rational parameters. They were defined by C. L. Siegel in 1929, along with their relatives, the E-functions, which can be considered as variations on \(e^ z\). Although Siegel states some irrationality results for values of G-functions at algebraic points, he never published the details of his computations. This turned out to be understandable when subsequent work of Galochkin and others showed that there are many more obstacles to get arithmetic results for values of G- functions then for E-functions. Significant progress was achieved in the 1980's, notably by E. Bombieri and G. V. Chudnovsky. Much of this progress was related to the properties of G-functions themselves and to the question of what G-functions really are.
If one takes any linear differential equation with coefficients in \({\mathbb{Q}}(z)\), its power series solutions will usually not be G-functions. Having a G-function solution poses very large constraints on the arithmetic of a linear differential equation. There exist several conjectures in this direction, the most important being the Bombieri- Dwork conjecture that the differential equation should come from algebraic geometry in a suitable sense. All known G-functions actually arise in this way. The converse statement is known to be true. So statements on the arithmetic nature of values of G-functions can also have consequences for problems in algebraic geometry. This explains the `Geometry' part of the title of the book. Despite all progress the amount of arithmetic results on values of G-functions at algebraic points is still very meager indeed.
The book under review is in the first place an account of recent developments in connection with G-function, which are otherwise scattered over the literature. Secondly, it is a very interesting attempt to point out directions in which it might be possible to have some `mature' applications of G-function theory, notably to algebraic geometry. In addition there are a number of new results by the author. Let us try to give a summary of the contents. The first part deals with definitions and the introduction of two heights, \(\rho\) (f) and \(\sigma\) (f), of a G- function f. Then an important example (conjecturally the only), namely the case of geometric differential equations is presented. The second part deals with Fuchsian differential systems \(\Lambda\), their formal and arithmetic aspects. Again two heights are introduced, \(\rho\) (\(\Lambda)\) and \(\sigma\) (\(\Lambda)\). A corrected proof of Chudnosky's remarkable theorem: `y cyclic and \(\sigma (y)<\infty\) implies \(\sigma (\Lambda)'\) is presented and finally the main results of this part are assembled on page 125. Part three deals with the arithmetic of values of G-functions. Here the author gives an unusual but original approach to produce linear independence results which is inspired by Gel'fond's method. We also find Bombieri's important idea of global relations in this part. It is on this principle that the author's hope for future applications is based, although up till now I have not seen this hope vindicated by any example. Finally, in part four we find two applications, found by the author, of the previous results to algebraic geometry. One concerns Grothendieck's conjecture on algebraic relations between periods of algebraic varieties. The other gives a bound for the heights of certain abelian varieties with a large endomorphism ring. Although both results apply to very limited situations it is very much worth while to keep these potential application areas for G-functions in mind.
In an appendix we find a new proof of the transcendence of \(\pi\) as a bonus. Unfortunately it contains an error. On line 6 on page 128 it is stated that \(\tau_ v\) belongs to the fundamental domain of SL(2,\({\mathbb{Z}})\) and on line 12 we find that \(\tau_ v\in \{ni,\frac{i+m}{n}| -[\frac{n}{2}]\leq m\leq [\frac{n-1}{2}]\}\). This is a contradiction. Moreover, I am afraid that this error is beyond repair.
To conclude, the book is written on a high level and not easy to read. Sometimes the author has a tendency to impress the reader unnecessarily. The proofs are written in concise, but usually intelligible way. They are not always reliable as is shown by the incorrect transcendence proof of \(\pi\). So the book should be handled with care in this respect. In particular someone ought to check the main theorem of chapter X very carefully. Despite these misgivings I enjoyed studying the book. Because of its originality and the stimulus it gives to provoke new research in the field of G-functions. Gauss-Manin connection; Bombieri-Dwork conjecture; arithmetic results; values of G-functions at algebraic points; applications of G-function theory; geometric differential equations; Fuchsian differential systems; heights; linear independence; global relations; Grothendieck's conjecture; algebraic relations between periods of algebraic varieties; bound for the heights of certain abelian varieties with a large endomorphism ring; transcendence André, Y.: G-functions and Geometry, Aspects of Mathematics, vol. E13. Friedr. Vieweg & Sohn, Braunschweig (1989) Transcendence (general theory), Research exposition (monographs, survey articles) pertaining to number theory, \(p\)-adic differential equations, Research exposition (monographs, survey articles) pertaining to algebraic geometry, Structure of families (Picard-Lefschetz, monodromy, etc.), Hypergeometric integrals and functions defined by them (\(E\), \(G\), \(H\) and \(I\) functions), Ordinary differential equations in the complex domain G-functions and 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 \(V\) be an \(r+1\) dimensional complex linear subspace of \(\mathbb{C}[x]\). The Wronskian of \(V\) is defined as the Wronskian of an ordered basis of \(V\); hence it is defined up to multiplication by a non-zero scalar. The main result of the paper is the proof of the following theorem (earlier B. and M. Shapiro conjecture): If the roots of the Wronskian \(Wr\) of \(V\) are real, then \(V\subset \mathbb{R}[x] \subset \mathbb{C}[x]\). A continuity argument shows that it is enough to prove this theorem in the case when \(Wr\) has simple roots. The authors recall a Schubert calculus argument to find an upper bound for the number of subspaces \(V \subset C[x]\) with a given exponent at infinity (degrees of a basis) and \(Wr=\prod (x-z_i)\), where the \(z_i\)'s are distinct. In Sections 2 and 3 the Bethe ansatz construction for a Gaudin model is studied and used to construct the expected number of subspaces with given exponent and given \(Wr\). A thorough study of the Gaudin Hamiltonians shows that the constructed subspaces \(V\) are all real if the roots of \(Wr\) are all real. This proves the main theorem.
The authors also discuss some related topics. They show that their main theorem implies that if the ramified points of a map \(\phi: \mathbb{C}\mathbb{P}^1 \to \mathbb{C}\mathbb{P}^r\) are contained in a circle in \(\mathbb{C}\mathbb{P}^1\), then \(\phi\) of this circle is contained in an (appropriate) \(\mathbb{R}\mathbb{P}^r \subset \mathbb{C}\mathbb{P}^r\). They also prove a statement about how the transversality of intersections of Schubert cycles in Grassmannians is related to the simplicity of the spectrum of Gaudin Hamiltonians. In Appendix B they show that their results imply the reality of orbits of critical points of master functions for the root systems A, B, and C; and conjecture this statement in general. real Schubert calculus; B. \& M. Shapiro conjecture; Bethe ansatz; Gaudin model \beginbarticle \bauthor\binitsE. \bsnmMukhin, \bauthor\binitsV. \bsnmTarasov and \bauthor\binitsA. \bsnmVarchenko, \batitleThe B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz, \bjtitleAnn. of Math. (2) \bvolume170 (\byear2009), no. \bissue2, page 863-\blpage881. \endbarticle \OrigBibText E. Mukhin, V. Tarasov and A. Varchenko, The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz, Ann. of Math. (2) 170 (2009), no. 2, 863-881. \endOrigBibText \bptokstructpyb \endbibitem Classical problems, Schubert calculus, Applications of Lie algebras and superalgebras to integrable systems, Singularities, monodromy and local behavior of solutions to ordinary differential equations in the complex domain, normal forms, Quantum groups and related algebraic methods applied to problems in quantum theory The B. and M. Shapiro conjecture in real algebraic geometry and the Bethe ansatz | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 this article is to reduce a proof of the conjecture to a statement about principal bundles on affine line over a regular local scheme. This reduction is obtained via a theory of nice triples, which goes back to the ideas of Voevodsky. As an application, an unpublished result due to Gabber is proved. reductive group schemes; principal G-bundles; Grothendieck-Serre's conjecture Group schemes, Linear algebraic groups over adèles and other rings and schemes, Cohomology theory for linear algebraic groups Nice triples and the Grothendieck-Serre conjecture concerning principal G-bundles over reductive group schemes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In this paper, we introduce the notion of a marginal tube, which is a hypersurface foliated by marginal surfaces. It generalises the notion of a marginally trapped tube and several notions of black hole horizons, for example trapping horizons, isolated horizons, dynamical horizons, etc. We prove that if every spacelike section of a marginal tube is a marginal surface, then the marginal tube is null. There is no assumption on the topology of the marginal tube. To prove it, we study the geometry of spacelike surfaces in a four-dimensional spacetime with the help of double null coordinate systems. The result is valid for arbitrary four-dimensional spacetimes, regardless of any energy condition. spacelike surfaces; double null foliations; null hypersurfaces; marginal surfaces; black hole horizons Black holes, Elliptic surfaces, elliptic or Calabi-Yau fibrations, Foliations generated by dynamical systems, Electromagnetic fields in general relativity and gravitational theory, Diffraction, scattering, Special bases (entangled, mutual unbiased, etc.), Spectrum, resolvent Marginal tubes and foliations by marginal 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 Compos. Math. 60, 115-132 (1986; Zbl 0607.13009).]
Let A be an affine normal domain of dimension \(d\geq 2\) over a universal domain. In this paper, a necessary and sufficient condition is obtained for the existence of projective A-modules of rank d which have no direct summand of rank 1. The condition is expressed in two equivalent ways:
(i) the Chow group of zero cycles of A is non-zero;
(ii) the subgroup of \(K_ 0(A)\), the Grothendieck group of projective A- modules, generated by the classes of residue fields of smooth points of Spec(A), is nonzero.
In the course of the proof, the following structure theorem is obtained for the divisor class group of a normal projective variety over an algebraically closed field k: It is an extension of a finitely generated abelian group by the group of k-points of an abelian variety. indecomposable projective modules; Chow group; Grothendieck group; divisor class group of a normal projective variety Picard groups, Projective and free modules and ideals in commutative rings, Parametrization (Chow and Hilbert schemes), Grothendieck groups, \(K\)-theory and commutative rings Indecomposable projective modules on affine domains. 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 S be a quartic surface in \({\mathbb{P}}_ 3\) not containing any line and B the double cover of \({\mathbb{P}}_ 3\) ramified in S. Its intermediate Jacobian \(J=J^ 2(B)\) is a principally polarized abelian variety of dimension 10. Let \(\Theta\) denote a theta divisor of J and \(\Phi: F\to J\) the Abel-Jacobi map, where F is the surface of lines in B. A translate \(F_ u\) of F by a point \(u\in J\) is by definition the surface \(\Phi(F)+u\) in J. The singularities of the theta divisor can be described by restricting the linear system \(| \Theta |_{\Theta}|\) to the translates \(F_ u\) of u contained in \(\Theta\). In this way a component Z of the quadratic singularities \(Sing^ 2(\Theta)\) of \(\Theta\) is constructed, which leads to a geometric interpretation of the projectivized tangent cones of the corresponding singular points: They are the quadrics of rank 5 in \({\mathbb{P}}_ 5\) intersecting the image of the surface S under the Veronese embedding \({\mathbb{P}}_ 3{\mathbb{P}}_ 9\). Since for a general threefold B one can recover the component Z among all components of Sing(\(\Theta\)) in a constructive way (the proof of this fact is not included in the paper), one obtains in this way a constructive Torelli theorem for generic double covers B. 3-fold; double cover of projective 3-space; intermediate Jacobian; theta divisor; Abel-Jacobi map; constructive Torelli theorem Voisin, Claire, Sur la jacobienne intermédiaire du double solide d'indice deux, Duke Math. J., 57, 2, 629-646, (1988) Picard schemes, higher Jacobians, \(3\)-folds, Theta functions and abelian varieties Sur la jacobienne intermédiaire du double solide d'indice deux. (On the intermediate Jacobian of the double solid of index two) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We establish a combinatorial connection between the real geometry and the \(K\)-theory of complex Schubert curves \(\mathcal{S}(\lambda_\bullet)\), which are one-dimensional Schubert problems defined with respect to flags osculating the rational normal curve. The second author [Can. J. Math. 69, No. 1, 143--177 (2017; Zbl 1390.14163)] showed that the real geometry of these curves is described by the orbits of a map \(\omega\) on skew tableaux, defined as the commutator of jeu de taquin rectification and promotion. In particular, the real locus of the Schubert curve is naturally a covering space of \(\mathbb{RP}^1\), with \(\omega\) as the monodromy operator.
We provide a fast, local algorithm for computing \(\omega\) without rectifying the skew tableau, and show that certain steps in our algorithm are in bijective correspondence with \textit{O. Pechenik} and \textit{A. Yong}'s genomic tableaux [Forum Math. Pi 5, Paper No. e3, 128 p. (2017; Zbl 1369.14060)], which enumerate the \(K\)-theoretic Littlewood-Richardson coefficient associated to the Schubert curve. Using this bijection, we give purely combinatorial proofs of several numerical results involving the \(K\)-theory and real geometry of \(\mathcal{S}(\lambda_\bullet)\). Young tableaux; monodromy; Schubert calculus; \(K\)-theory; osculating flag; jeu de taquin Classical problems, Schubert calculus, Combinatorial aspects of representation theory Monodromy and \(K\)-theory of Schubert curves via generalized jeu de taquin | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Descartes' rule of signs and Newton polygons are classical methodes to estimate the number of zeros of a polynomial. The authors use hyperfields to give a unified and conceptual proof for these theorems.
To this end they need to introduce polynomials over hyperfields and define their roots and multiplicities. Given a homomorphism from a field \(K\) into a hyperfield \(F\), the authors prove: The multiplicity of \(b\in F\) as a root of the image of a polynomial~\(p\) over \(K\) is at least the sum of all multiplicity of all preimages of \(b\) as a root of \(p\).
This proposition is then applied to the sign hyperfield and the tropical hyperfield to obtain the two classical theorems.
A rigorous approach to polynomial algebras over hyperfields is discussed in detail in an appendix. hyperfields; polynomials; ordered blueprints; Newton polygons; valuations; tropical algebra Hyperrings, Foundations of tropical geometry and relations with algebra, Generalizations of fields, Polynomials in real and complex fields: location of zeros (algebraic theorems) Descartes' rule of signs, Newton polygons, and polynomials over hyperfields | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 noetherian separated scheme and \(D^b(X)\) be the bounded derived category of coherent sheaves on \(X\). A dualizing complex is an object \(\mathcal{J} \in D^b(X)\) with the property that the functor \(R\underline{\Hom}(-,\mathcal{J})\) from \(D^b(X)^{\text{op}}\) to \(D^b(X)\) is an equivalence of categories. The paper under review is a survey on the recent progress concerning dualizing complexes.
In Section 2 some classical results are recalled. For instance, if \(f: X\rightarrow Y\) is a finite type morphism of noetherian separated schemes and \(Y\) is Gorenstein, that is, the structure sheaf \(\mathcal{O}_Y\) is a dualizing complex, then \(\mathcal{J}=f^!\mathcal{O}_Y\) is a dualizing complex on \(X\). Here, one uses the factorization \(f=g\circ h\) with \(g\) an open immersion and \(h\) proper, and then defines \(f^!\) to be the composition \(g^*\circ h^\#: D^+(\text{QCoh}(Y))\rightarrow D^+(\text{QCoh}(X))\), where we use left-bounded derived categories of quasi-coherent sheaves and \(h^\#\) is the right adjoint of \(Rh_*\). Furthermore, if \(f\) is flat, there is an isomorphism \(\mathcal{J}\cong \Delta^!(\mathcal{J}\boxtimes \mathcal{J})\), where \(\Delta: X\rightarrow X\times_Y X\) is the diagonal map and \(\mathcal{J}\boxtimes \mathcal{J}\) is the external tensor product. A dualizing complex satisfying this additional property is called rigid. In contrast to ordinary dualizing complexes, the rigid ones turn out to be unique.
In the next section the author presents some new results. Let \(X\) be a noetherian affine scheme and consider the homotopy categories \(K(\text{Inj}(X))\) and \(K(\text{Proj}(X))\) of complexes of injective resp.\ projective quasi-coherent sheaves. A complex \(\mathcal{J} \in K(\text{Inj}(X))\) on \(X\) turns out to be dualizing if and only if the functor \(\mathcal{J}\otimes -: K(\text{Proj}(X))\rightarrow K(\text{Inj}(X))\) is an equivalence. Since there are usually no projective sheaves on a non-affine scheme, this result does not globalize as stated. However, there exists a category \(K_m(\text{Proj}(X))\) built as a Verdier quotient from the homotopy category of complexes of flat quasicoherent sheaves, and for a noetherian separated scheme \(X\) there is the following result. An object \(\mathcal{J} \in K(\text{Inj}(X))\) is a dualizing complex if and only if the functor \(\mathcal{J}\otimes -: K_m(\text{Proj}(X)) \rightarrow K(\text{Inj}(X))\) is an equivalence. These results use the relation of dualizing complexes with compactly generated triangulated categories.
The survey concludes with the statement of an open problem. Namely, not every equivalence \(K_m(\text{Proj}(X))\cong K(\text{Inj}(X))\) takes \(\mathcal{O}_X\) to a dualizing complex and the question is whether it is possible to characterize those equivalences which take \(\mathcal{O}_X\) to, for example, rigid dualizing complexes. dualizing complex; Grothendieck duality Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Noncommutative algebraic geometry, Derived categories, triangulated categories Rigid dualizing complexes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 expresses the number of rational points on the characteristic \(p\) Igusa curves \(X_n\) of level \(p^n\) as a sum of class numbers of certain complex quadratic orders. From this the author deduces explicit formulas for the \(L\)-functions associated to the symmetric products of the first étale cohomology groups of the universal elliptic curve over \(X_n\). Finally the Eichler-Selberg trace formula for the group \(\Gamma_1 (p^n)\) is applied and an expression for the \(L\)-functions in terms of the Hecke polynomials associated to the operator \(T_p\) is obtained. \(L\)-functions of symmetric representations; characteristic \(p\) Igusa curves; number of rational points; first étale cohomology groups; Hecke polynomials Amílcar Pacheco, Rational points on Igusa curves and \?-functions of symmetric representations, J. Number Theory 58 (1996), no. 2, 343 -- 360. \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture, Rational points, Cohomology of arithmetic groups, Elliptic curves Rational points on Igusa curves and \(L\)-functions of symmetric 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 Let \(X\) be a non-commutative scheme. A Grothendieck category of quasi-coherent sheaves on \(X\) is called a quasi-scheme and in a certain sense it can be identified with \(X\). Suppose that there is a regularly embedded hypersurface \(Y\subset X\). The authors investigate curves on \(X\) which are in general position with respect to \(Y\). In particular it is shown that the category of quasi-coherent sheaves on such a curve \(C\) is isomorphic to a certain quotient category of graded modules over the \(n\)-dimensional commutative polynomial ring graded by \(\mathbb{Z}^n\), where \(n\) is the number of points of the intersection \(C\cap Y\). The results obtained are applied to some interesting examples of non-commutative schemes such as the enveloping algebra of the two-dimensional non-Abelian Lie algebra, the quantum affine and projective planes, the quantum projective space of \textit{M. Vancliff} [J. Algebra 165, No. 1, 63-90 (1994; Zbl 0837.16023)], and others. non-commutative schemes; quasi-schemes; quasi-coherent sheaves; quantum algebras; quantum planes; curves; Grothendieck categories; graded modules; enveloping algebras S. P. Smith and J. J. Zhang,Curves on quasi-schemes, Algebras and Representation Theory1 (1998), 311--351. Graded rings and modules (associative rings and algebras), Homological dimension in associative algebras, Special algebraic curves and curves of low genus, Relationships between algebraic curves and physics, Grothendieck categories, \(K_0\) of other rings, Rings arising from noncommutative algebraic geometry, Quantum groups (quantized enveloping algebras) and related deformations Curves on quasi-schemes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(F\) be a finite field. A polynomial \(P\in F[X_1,\dots,X_n]\) s called linearized if the induced map \(K^n\longrightarrow K\) is \(F\)-linear for every extension \(K/F\). In this paper, the author studies the properties of linearized polynomials and maps \(K^n\longrightarrow K^n\) defined by an \(n\)-tuple of such polynomials (linearized maps). He starts with the observation that the set of all linearized polynomials is linearly generated by monomials \(X_i^{p^m}\) (\(i=1,2,\dots,n; m=0,1,\dots\)) (Proposition 3.2), and shows that linearized maps are in a one-to-one correspondence with matrices over \(F[X]\) (Remark 3.5). It is shown moreover that an analogue of the Jacobian Conjecture can be established for linearized maps (Corollary 4.1), and all such invertible maps are tame (Theorem 4.3). Corollary 4.5 shows that every linearized polynomial is a coordinate of an invertible map, in Corollary 4.10 it is proved that if \(I\) is an ideal of \(F[X]\) generated by linearized polynomials, then \(F[X]/I\) is a finitely generated \(F\)-algebra if and only if is is a domain, and in Corollary 4.18 an analogue of the Linearization Conjecture is established for linearized maps. In the final section the author considers polynomials without mixed terms and maps defined by them, and shows the truth of the Jacobian Conjecture for such maps. affine space; polynomials over finite fields; linearized polynomial; group of polynomial automorphisms; group of tame automorphisms Joost Berson, Derivations of polynomial rings over a domain, Master's thesis, University of Nijmegen, June 1999. Jacobian problem, Affine spaces (automorphisms, embeddings, exotic structures, cancellation problem), Polynomials over finite fields, Polynomial rings and ideals; rings of integer-valued polynomials, Rational and birational maps Linearized polynomial maps 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 The present paper gives an algorithm for the absolute factorization (factorization over the algebraic closure \(\bar{\mathbb{Q}}\)) of a parametric multivariate polynomial \(F\in \mathbb{Q}[u_1,\dots, u_r][X_0,\dots, X_n]\). The method is a follow-up, with some improvements, of a previous work of the author [Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. 316, 5--29 (2004); translation in J. Math. Sci. 134, 2325--2339 (2006; Zbl 1077.65046)].
Section 1 formalizes the notion of Parametric Absolute Factorization (PAF) of \(F\)\, and stated the main result (theorem 1.5): it is possible to find a finite number of PAFs whose constructible sets form a partition of the parameters space \(\mathcal{P}=\bar{\mathbb{Q}}^r\). Section 2 gathers some necessary tools: Hensel's lemma, the Chistov-Grigoriev method for quantifier elimination [\textit{A. L. Chistov} and \textit{D. Yu. Grigorie}, Mathematical foundations of computer science, Proc. 11th Symp., Praha/Czech. 1984, Lect. Notes Comput. Sci. 176, 17--31 (1984; Zbl 0562.03015)] and an algorithm of the author of the present paper for solving zero-dimensional parametric polynomial systems [PhD Thesis, University of Rennes 1, France (2006)].
The proposed algorithm is detailed in the section 3. Theorem 3.19 includes Theorem 1.5 and gives also bounds for the number of necessary PAFs and the computational complexity of the algorithm. symbolic computation; complexity analysis; polynomial absolute factorization; parametric polynomials Number-theoretic algorithms; complexity, Symbolic computation and algebraic computation, Polynomials in number theory, Polynomials in real and complex fields: factorization, Parametrization (Chow and Hilbert schemes) On factoring parametric multivariate polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In the theory of algebraic surfaces, singularities appear naturally if canonical models are studied (appearance of rational double points), or if degenerations of smooth surfaces of general type are considered: the paper under review studies log-canonical singularities as candidates of a larger class of ``admissible'' singularities for a surface theory.
The first chapters introduce basic notions and review results of Mumford, Brenton, Sakai and Kawamata for later use. -- After that, properties of log-canonical singularities are studied in detail: Lower bounds of the discrepancies are discussed, depending on the resolution graphs. For small values of the index \((\leq 6)\), a complete list of log-canonical singularities is given (by resolution graphs). The main results of the paper are the following: For a log-canonical surface \(X\) with Kodaira dimension \(\kappa (X) \geq 0\), the nonnegativity of the Euler characteristic (known for the smooth case) is proved, together with a generalization, showing that for arbitrary normal singularities this does not remain true. Next, the holomorphic Euler characteristic is considered: The similar result requires its own analysis (contrary to the smooth case which can be deduced from the previous result). More precisely: If \(X\) is a Moishezon surface, having at most quasi-elliptic singularities, then \(\kappa (X) < 0\) only if it is ruled of genus \(\geq 2\). If \(\kappa (X) \geq 0\), then the holomorphic Euler characteristic is \(\geq 0\); it is \(> 0\) if \(\kappa (X) = 2\). This was knows previously for \(X\) Gorenstein and is a classical result for \(X\) smooth. Finally, surfaces \(X\) with \(\kappa (X) = 0\) are considered: Bounds for numerical invariants (holomorphic and topological Euler characteristic, Picard number, index, number of singularities of \(X)\) are given, partially in contrast with the smooth case. This is done using a combination of subtle methods, comparing minimal resolution and rational-double-point- resolution. Some open questions may serve for further investigation. resolution graphs; log-canonical singularities; Euler characteristic; Picard number; rational-double-point-resolution [B1] Blache, R.: Moishezon-Flächen mit log-canonischen Singularitäten. Dissertation, Bochum 1992 Singularities in algebraic geometry, Divisors, linear systems, invertible sheaves, Singularities of surfaces or higher-dimensional varieties, Research exposition (monographs, survey articles) pertaining to algebraic geometry Moishezon- surfaces with log-canonical 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 We construct a presentation for the Grothendieck group of Deligne-Mumford stacks over a field of characteristic zero. The generators for this presentation are smooth, proper Deligne-Mumford stacks and the relations are expressed in terms of stacky blow-ups. In the process we prove a version of the weak factorization theorem for Deligne-Mumford stacks. Grothendieck group of varieties; Deligne-Mumford stack; destackification; weak factorization Generalizations (algebraic spaces, stacks), Rational and birational maps, Motivic cohomology; motivic homotopy theory Weak factorization and the Grothendieck group of Deligne-Mumford stacks | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(X\) be a hyperkähler variety with an anti-symplectic involution \(\iota\). According to Beauville's conjectural ``splitting property'', the Chow groups of \(X\) should split in a finite number of pieces such that the Chow ring has a bigrading. The Bloch-Beilinson conjectures predict how \(\iota\) should act on certain of these pieces of the Chow groups. We verify part of this conjecture for a 19-dimensional family of hyperkähler sixfolds that are ``double EPW cubes'' (in the sense of Iliev-Kapustka-Kapustka-Ranestad [\textit{A. Iliev} et al., J. Reine Angew. Math. 748, 241--268 (2019; Zbl 1423.14220)]). This has interesting consequences for the Chow ring of the quotient \(X/\iota\), which is an ``EPW cube'' (in the sense of Iliev-Kapustka-Kapustka-Ranestad [loc. cit.]). algebraic cycles; Bloch-Beilinson filtration; Bloch's conjecture; Chow groups; (double) EPW cubes; hyperkähler varieties; \(K3\) surfaces; motives; multiplicative Chow-Künneth decomposition; non-symplectic involution; splitting property (Equivariant) Chow groups and rings; motives, Algebraic cycles, Transcendental methods, Hodge theory (algebro-geometric aspects) Algebraic cycles and EPW cubes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 \(X\subset \mathbb{P}^r\) be an integral and non-degenerate variety. Set \(n:= \dim (X)\). Let \(\rho (X)''\) be the maximal integer such that every zero-dimensional scheme \(Z\subset X\) smoothable in \(X\) is linearly independent. We prove that \(X\) is linearly normal if \(\rho (X)''\ge \lceil (r+2)/2\rceil\) and that \(\rho (X)'' < 2\lceil (r+1)/(n+1)\rceil \), unless either \(n=r\) or \(X\) is a rational normal curve. secant varietys \(X\)-ranks zero-dimensional schemes variety with only one ordinary double points Secant varieties, tensor rank, varieties of sums of powers Dependent subsets of embedded projective varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Using the formalism of quantized quadratic Hamiltonians [\textit{A.~B.~Givental}, Mosc. Math. J. 1, No. 4, 551--568 (2001; Zbl 1008.53072)], the authors are able to prove quantum versions of three classical theorems in algebraic geometry; namely, the Riemann-Roch theorem, Serre duality, and the Lefschetz hyperplane section theorem. The key ingredient consists in introducing a notion of twisted Gromov-Witten invariants of a compact projective complex manifold \(X\); the quantum version of the aforementioned theorems can then be seen as relations between the twisted and the nontwisted Gromov-Witten theory of \(X\).
More precisely, let \(X_{g,n,d}\) be the moduli space of genus \(g\), \(n\)-pointed stable maps to \(X\) of degree \(d\), where \(d\) is an element in \(H_2(X;\mathbb{Z})\), and let \(E\) be a holomorphic vector bundle on \(X\). Since a point in \(X_{g,n,d}\) is represented by a pair \((\Sigma,f)\), where \(\Sigma\) is a complex curve and \(f:\Sigma\to X\) a holomorphic map, one can use \(f\) to pull back \(E\) on \(\Sigma\) and then consider the \(K\)-theory Euler character of \(f^*E\), i.e., the virtual vector space \(H^0(\Sigma,f^*E)\ominus H^1(\Sigma,f^*E)\), as the fiber over \([(\Sigma,f)]\) of a virtual vector bundle \(E_{g,n,d}\) over \(X_{g,n,d}\). This intuitive construction is made completely rigorous by considering \(K\)-theory push-pull \(K^0(X)\to K^0(X_{g,n,d})\) along the diagram
\[
\begin{tikzcd} X_{g,n+1,d}\rar["\mathrm{ev}_{n+1}"]\dar["\pi" '] &X\\ X_{g,n,d}\end{tikzcd}
\]
A rational invertible multiplicative characteristic class of a complex vector bundle is an expression of the form
\[
\mathbf{c}(\cdot)=\exp\left(\sum_{k=0}^\infty s_k \text{ch}_k(\cdot)\right),
\]
where \(\text{ch}_k\) are the components of the Chern character, and the \(s_k\) are arbitrary parameters. These data determine a cohomology class \(\mathbf{c}(E_{g,n,d})\) (actually, a formal family of cohomology classes parametrized by the \(s_k\)) in \(H^*(X_{g,n,d};\mathbb{Q})\), and one can define the total \((\mathbf{c},E)\)-twisted descendant potential \(\mathcal{D}_{\mathbf{c},E}^g\) as
\[
\mathcal{D}_{\mathbf{c},E}(t_0,t_1,\dots)=\exp\left(\sum_{g\geq 0}\hbar^{g-1}\mathcal{F}^g_{\mathbf{c},E}(t_0,t_1,\dots)\right),
\]
where
\[
\mathcal{F}_{\mathbf{c},E}^g(t_0,t_1,\dots)=\sum_{n,d}\frac{Q^d}{n!}\int_{[X_{g,n,d}]}\mathbf{c}(E_{g,n,d}) (\sum_{k_1=0}^\infty(\text{ev}_1^*t_k)\psi_1^{k_1}) \cdots (\sum_{k_1=0}^\infty(\text{ev}_n^*t_k)\psi_n^{k_n}).
\]
Here \(Q^d\) is the representative of \(d\) in the semigroup ring of degrees of holomorphic curves in \(X\), \(t_0,t_1,\dots\) are rational cohomology classes on \(X\), and \(\psi_i\) is the first Chern class of the universal cotangent bundle over \(X_{g,n,d}\) corresponding to the \(i\)-th marked point of \(X\). For \(E\) the zero element in \(K^0(X)\), the twisted potential \(\mathcal{D}_{\mathbf{c},E}^g\) reduces to \(\mathcal{D}_X\), the total descendant potential of \(X\).
At this point the formalism of quantized quadratic hamiltonians enters the picture. One considers the symplectic space \(\mathcal{H}=H^*(X;\mathbb{Q})((z^{-1}))\) of Laurent polynomials in \(z^{-1}\) with coefficients in the cohomology of \(X\), endowed with the symplectic form
\[
\Omega(\mathbf{f},\mathbf{g})=\frac{1}{2\pi i}\oint \left(\int_X\mathbf{f}(-z)\mathbf{g}(z)\right)\,dz.
\]
The subspace \(\mathcal{H}_+=H^*(X;\mathbb{Q})[z]\) is a Lagrangian subspace, and \((\mathcal{H},\Omega)\) is identified with the canonical symplectic structure on \(T^*\mathcal{H}_+\). Finally, given an infinitesimal symplectic transformation \(T\) of \(\mathcal{H}\), one can consider the differential operator \(\hat{T}\) of order \(\leq 2\) on functions on \(\mathcal{H}_+\), which is associated by quantization to the quadratic Hamiltonian \(\Omega(T\mathbf{f},\mathbf{f})/2\) on \(\mathcal{H}\). By the inclusion \(\mathcal{H}_+\hookrightarrow H^*(X;\mathbb{Q})[[z]]\), the operator \(\hat{T}\) acts on asymptotic elements of the Fock space, i.e., on functions of the formal variable \(\mathbf{q}(z)=q_0+q_1z+q_2z^2+\cdots\) in \(H^*(X;\mathbb{Q})[[z]]\). By the dilaton shift, i.e., setting \(\mathbf{q}(z)=\mathbf{t}(z)-z\), with \(\mathbf{t}(z)=t_0+t_1z+t_2z^2+\cdots\), the operator \(\hat{T}\) acts on any function of \(t_0,t_1,\dots\), notably on the descendant potentials.
Having introduced this formalism, the authors are able to express the relation between twisted and untwisted Gromov-Witten invariants in an extremely elegant way: up to a scalar factor,
\[
\mathcal{D}_{\mathbf{c},E}=\hat{\Delta}\mathcal{D}_X,
\]
where \(\Delta:\mathcal{H}\to \mathcal{H}\) is the linear symplectic transformation defined by the asymptotic expansion of
\[
\sqrt{\mathbf{c}(E)}\prod_{m=1}^\infty \mathbf{c}(E\otimes L^{-m})
\]
under the identification of the variable \(z\) with the first Chern class of the universal line bundle \(L\). This is the quantum Riemann-Roch theorem; it explicitly determines all twisted Gromov-Witten invariants, of all genera, in terms of untwisted invariants. The result is a consequence of Mumford's Grothendieck-Riemann-Roch theorem applied to the universal family \(\pi:X_{g,n+1,d}\to X_{g,n,d}\). If \(E=\mathbb{C}\) is the trivial line bundle, then \(E_{g,n,d}=\mathbb{C}\ominus \mathbf{E}_g^*\), where \(\mathbf{E}_g\) is the Hodge bundle, and one recovers from quantum Riemann-Roch results of \textit{D. Mumford} [Arithmetic and geometry, Pap. dedic. I. R. Shafarevich, Vol. II: Geometry, Prog. Math. 36, 271--328 (1983; Zbl 0554.14008)] and \textit{C. Faber, R. Pandharipande} [Invent. Math. 139, No.1, 173--199 (2000; Zbl 0960.14031)] on Hodge integrals.
If \(\mathbf{c}^*\) is the multiplicative characteristic class
\[
\mathbf{c}^*(\cdot)=\exp\left(\sum_{k=0}^\infty(-1)^{k+1} s_k\mathrm{ch}_k(\cdot)\right),
\]
then \(\mathbf{c}^*(E^*)=1/\mathbf{c}(E)\), and one the following quantum version of Serre duality:
\[
\mathcal{D}_{\mathbf{c}^*,E^*}(\mathbf{t}^*)=(\mathrm{sdet}\,\mathbf{c}(E))^{-\frac{1}{24}}\mathcal{D}_{\mathbf{c},E}(\mathbf{t}),
\]
where \(\mathbf{t}^*(z)=\mathbf{c}(E)\mathbf{t}(z)+(1-\mathbf{c}(E))z\).
Finally, if \(E\) is a convex vector bundle and a submanifold \(Y\subset X\) is defined by a global section of \(E\), then the genus zero Gromov-Witten invariants of \(Y\) can be expresssed in terms of the invariants of \(X\) twisted by the Euler class of \(E\). These are in turn related to the untwisted Gromov-Witten invariants of \(X\) by the quantum Riemann-Roch theorem, so the authors end up with a quantum Lefschetz hyperplane section principle, expressing genus zero Gromov-Witten invariants of a complete intersection \(Y\) in terms of those of \(X\). This extends earlier results [\textit{V.~V.~Batyrev, I.~Ciocan-Fontanine, B.~Kim} and \textit{D.~van Straten}, Acta Math. 184, No. 1, 1--39 (2000; Zbl 1022.14014); \textit{A.~Bertram}, Invent. Math. 142, No. 3, 487--512 (2000; Zbl 1031.14027); \textit{A.~Gathmann}, Math. Ann. 325, No. 2, 393--412 (2003; Zbl 1043.14016); \textit{B.~Kim}, Acta Math. 183, No. 1, 71--99 (1999; Zbl 1023.14028); \textit{Y.-P.~Lee}, Invent. Math. 145, No. 1, 121--149 (2001; Zbl 1082.14056)], and yields most of the known mirror formulas for toric complete intersections. The idea of deriving mirror formulas by applying the Grothendieck-Riemann-Roch theorem to universal stable maps is not new: according to the authors it can be traced back at least to Kontsevich's investigations in the early 1990s, and to Faber's and Pandharipande's work on Hodge integrals. Gromov-Witten invariants; mirror symmetry; Grothendieck-Riemann-Roch theorem; Lefschetz hyperplane section principle; Serre duality T. Coates, A. Givental, Quantum Riemann-Roch, Lefschetz and Serre. \textit{Ann. of Math}. (2) \textbf{165} (2007), 15-53. MR2276766 Zbl 1189.14063 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Riemann-Roch theorems, Mirror symmetry (algebro-geometric aspects) Quantum Riemann-Roch, Lefschetz and Serre | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 was published twice, one times in the book Zbl 0527.00017.]
This paper is a sequel to the author's paper ''Fifteen characterizations of rational double points and simple critical points'' [Enseign. Math. 25, 132-163 (1979; Zbl 0418.14020)]. The characterizations of that paper are for complex varieties and complex functions, and involve the Dynkin diagrams \(A_ k\), \(D_ k\) and \(E_ k\). It turns out that the missing Dynkin diagrams \(B_ k\), \(C_ k\), and \(F_ 4\) (but not \(G_ 2)\) correspond to real singularities and real functions, and that a smaller number of similar characterizations are true for these as well. The main theorem of this paper contains four such characterizations: Let \(f:({\mathbb{R}}^ 3,0)\rightsquigarrow(\mathbb{R},0)\) be the germ at the origin of a real analytic function. Then the following are equivalent: (1) The germ f is right-left equivalent to one of the germs given in a certain list. (2) The germ f is simple (in the sense of Arnold). (3) The complexified variety \(f^{-1}(0)\) has a rational singularity at the origin. (4) A resolution of the real variety \(f^{-1}(0)\) is given in a certain list. The proof of the theorem proceeds by direct computation, or by referring to the corresponding theorem in the complex case. germ of a real analytic function; singularities of real varieties; rational double points; simple critical points; Dynkin diagrams; real singularities A. Durfee, 14 characterizations of rational double points (to appear). Singularities in algebraic geometry, Real algebraic and real-analytic geometry, Germs of analytic sets, local parametrization, Singularities of differentiable mappings in differential topology, Local complex singularities Four characterizations of real rational double points | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials If \(G\) is a commutative semigroup, one may define, for two subsets \(A\) and \(B\) of \(G\), the sum \(A+B\) and the sum of \(N\) copies of \(A\) denoted by \(N*A\). In the paper, it is shown that if \(A\) and \(B\) are finite sets, then the number of elements of the set \(B + N* A\) with \(N\) large enough is a polynomial in \(N\) whose degree is less than the number of elements of \(A\) (theorem 1). The polynomials in this result and in its generalization (theorem 2) are Hilbert polynomials of certain graded modules over the ring of polynomials in several indeterminates.
When \(G\) is an abelian group without elements of finite order, denoting by \(G(A)\) the subgroup generated by the differences of the elements of \(A\), \(A \subseteq G\), then \(G(A)\) is isomorphic to \(\mathbb{Z}^ n\) where \(n\) is the rank of \(G(A)\) and \(\overline A = A - a\) is included into \(G(A)\) for all \(a \in A\). The reduced Newton polyhedron of \(A \subseteq G\) is the convex hull, in the space \(\mathbb{R}^ n\) containing the lattice \(\mathbb{Z}^ n\), of the image of \(A\) under an isomorphism of the group \(G(A)\) on \(\mathbb{Z}^ n\). The main result on the reduced Newton polyhedron is the following: The number of points in \(B+N*A\) divided by \(N^ n\) tends as \(N \to \infty\), to the product of the volume of the reduced Newton polyhedron of \(A\) and the number \(i(A,B)\) of cosets of the group \(G\) modulo the subgroup \(G(A)\) which contain points of the set \(B\). number of elements; Hilbert polynomials; reduced Newton polyhedron A.G. Khovanskii , Newton polyhedron, Hilbert polynomial, and sums of finite sets. Functional . Anal. Appl. 26 ( 1992 ), 276 - 281 . MR 1209944 | Zbl 0809.13012 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Toric varieties, Newton polyhedra, Okounkov bodies Newton polyhedron, Hilbert polynomial, and sums of finite 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 straightening law of Doubilet-Rota-Stein tells that the standard bitableaux bounded by a pair \(m=(m(1),m(2))\) give a vector space basis of the polynomial algebra in \(m(1)m(2)\) variables. In an enumerative proof of the straightening law Abhyankar enumerated the set \(\text{stab}(2,m,p,a,V)\) of certain standard bitableaux. The Abhyankar formula gives also the Hilbert polynomial of a class of determinantal ideals \(I(p,a)\).
In the paper under review, the author outlines an alternate proof of the Abhyankar formula for the cardinality of \(\text{stab}(2,m,p,a,V)\) using a recent result on nonintersecting lattice paths obtained independently by Modak, Kulkarni and Krattenthaler. The lattice path approach leads also to some other known results on the numerators of the Hilbert-Poincaré series of \(I(p,a)\), the so-called \(h\)-vector of the associated simplicial complex, and gives better bounds for the degree of the numerator of the Hilbert series of \(I(p,a)\) (and in some cases the exact value of the degree). The author also discusses some related problems concerning possible generalizations to higher dimensions. He indicates as well connections between the Hilbert function for the Schubert varieties in Grassmannians and the Abhyankar formula. Stanley-Reisner ring; straightening law; standard bitableaux; Abhyankar formula; Hilbert polynomial; determinantal ideals; nonintersecting lattice paths; Hilbert series; Hilbert function; Schubert varieties DOI: 10.1016/0378-3758(95)00156-5 Combinatorial aspects of representation theory, Determinantal varieties, Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series, Factorials, binomial coefficients, combinatorial functions, Exact enumeration problems, generating functions, Linkage, complete intersections and determinantal ideals, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Grassmannians, Schubert varieties, flag manifolds Young bitableaux, lattice paths and Hilbert functions | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The set \( \pi_{2} \) of all real bivariate algebraic polynomials of total degree at most two is considered. The authors give a full description of the strictly definite extreme points of the unit ball of \( \pi_{2}(\Delta) \). The extreme points are useful for solving extremal problems for polynomials. convexity; extreme points; polynomials L. MILEV, N. NAIDENOV, \textit{Strictly definite extreme points of the unit ball in a polynomial space}, C. R. Acad. Bulgare Sci. 61 (2008), 1393--1400. Real polynomials: analytic properties, etc., Convexity of real functions of several variables, generalizations, Semialgebraic sets and related spaces, Convexity and finite-dimensional Banach spaces (including special norms, zonoids, etc.) (aspects of convex geometry) Strictly definite extreme points of the unit ball in a polynomial space | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The double-base number system (DBNS) uses two bases, 2 and 3, in order to represent any integer \(n\). A double-base chain (DBC) is a special case of a DBNS expansion. DBCs have been introduced to speed up the scalar multiplication \([n]P\) on certain families of elliptic curves used in cryptography. In this context, our contributions are twofold. First, given integers \(n\), \(a\), and \(b\), we outline a recursive algorithm to compute the number of different DBCs with a leading factor dividing \(2^a3^b\) and representing \(n\). A simple modification of the algorithm allows to determine the number of DBCs with a specified length as well as the actual expansions. In turn, this gives rise to a method to compute an optimal DBC representing \(n\), i.e. an expansion with minimal length. Our implementation is able to return an optimal expansion for most integers up to \(2^{60}\) bits in a few minutes. Second, we introduce an original and potentially more efficient approach to compute a random scalar multiplication \([n]P\), based on the concept of controlled DBC. Instead of generating a random integer \(n\) and then trying to find an optimal, or at least a short DBC to represent it, we propose to directly generate \(n\) as a random DBC with a chosen leading factor \(2^{a }3^{b }\) and length \(\ell \). To inform the selection of those parameters, in particular \(\ell \), which drives the trade-off between the efficiency and the security of the underlying cryptosystem, we enumerate the total number of DBCs having a given leading factor \(2^a3^b\) and a certain length \(\ell \). The comparison between this total number of DBCs and the total number of integers that we wish to represent a priori provides some guidance regarding the selection of suitable parameters. Experiments indicate that our new near optimal controlled DBC approach provides a speedup of at least 10\% with respect to the NAF for sizes from 192 to 512 bits. Computations involve elliptic curves defined over \(\mathbb{F}_p\), using the inverted Edwards coordinate system and state of the art scalar multiplication techniques. Double-base number system; elliptic curve cryptography Cryptography, Applications to coding theory and cryptography of arithmetic geometry On the enumeration of double-base chains with applications to elliptic curve cryptography | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 (1) characterize the Schubert varieties that arise as variations of Hodge structure (VHS); (2) show that the isotropy orbits of the infinitesimal Schubert VHS `span' the space of all infinitesimal VHS; and (3) show that the cohomology classes dual to the Schubert VHS form a basis of the invariant characteristic cohomology associated with the infinitesimal period relation (a.k.a. Griffiths' transversality). Schubert variety; variation of Hodge structure; infinitesimal period relation; Griffiths' transversality; Hodge theory; Mumford-Tate group Robles, C., \textit{Schubert varieties as variations of Hodge structure}, Selecta Math. (N.S.), 20, 719-768, (2014) Period matrices, variation of Hodge structure; degenerations, Grassmannians, Schubert varieties, flag manifolds, Variation of Hodge structures (algebro-geometric aspects) Schubert varieties as variations of Hodge structure | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 \(Y\) be a smooth projective surface defined over an algebraically closed field \(k\) with \(\text{char}\;k \not=2\), and let \(\pi:X \to Y\) be a double covering branched along a smooth curve \(B\).
The author proves that if the tangent bundle \(\mathcal T_Y\) is semistable with respect to some ample line bundle \(\mathcal H \in \text{Pic}(Y)\), then \(\mathcal T_X\) is stable with respect to \(\pi^*\mathcal H\). Since the tangent bundle to the projective space is stable, this implies that \(\mathcal T_X\) is stable if \(\pi:X \to \mathbb P^2\) is a double covering as above.
This applies in particular, when \(B\) is a smooth plane sextic, in which case \(X\) is a \(K3\) surface. Therefore, the tangent bundle of a general \(K3\) surface of degree \(2\) is stable, even in positive characteristic \(\not= 2\). Moreover, in the case \(\text{char}\;k=p > 2\), letting \(F\) denote the relative Frobenius morphism over \(k\), the author shows that
\(F_*\) preserves semistability of torsion free sheaves of rank \(1\) on \(X\), provided that \(B\) has degree \(\geq 6\). stability; tangent bundle; double covering; Frobenius morphism Vector bundles on surfaces and higher-dimensional varieties, and their moduli On the stability of tangent bundle on double coverings | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 book the author discusses the Hilbert scheme \(X^{[n]}\) of points on a complex surface \(X\). This object is originally studied in algebraic geometry but, as it has been recently realized, it is related to many other branches of mathematics, such as singularities, symplectic geometry, representation theory, and even to theoretical physics. The book reflects this feature on Hilbert schemes and therefore the subjects are analyzed from various points of view. One sees that \(X^{[n]}\) inherits structures of \(X\), e.g., it is a nonsingular complex manifold, it has a holomorphic symplectic form if \(X\) has one, it has a hyper-Kähler metric if \(X= \mathbb{C}^2\), and so on. A new structure is revealed when one studies the homology group of \(X^{[n]}\). The generating function of Poincaré polynomials has a very nice expression. The direct sum \(\bigoplus_n H_* (X^{[n]})\) is a representation of the Heisenberg algebra.
The book, which is nicely written and well-organized, tries to tell the harmony between different fields rather than focusing attention on a particular one. The reader is assumed to have basic knowledge on algebraic geometry and homology groups of manifolds. Some chapters require more background, say spectral sequences, Riemannian geometry, Morse theory, intersection cohomology. symplectic structure; moment map; hyper-Kähler quotients; Dynkin diagrams; vertex algebra; symmetric products; Hilbert scheme of points; Poincaré polynomials; Heisenberg algebra; Morse theory; intersection cohomology H. Nakajima, \textit{Lectures on Hilbert schemes of points on surfaces}, \textit{University Lecture Series}\textbf{18}, American Mathematical Society, Providence RI U.S.A., (1999). Parametrization (Chow and Hilbert schemes), Vector bundles, sheaves, related construction [See also 18F20, 32Lxx, 46M20], Research exposition (monographs, survey articles) pertaining to algebraic geometry, Singularities of surfaces or higher-dimensional varieties, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Special Riemannian manifolds (Einstein, Sasakian, etc.) Lectures on Hilbert schemes of points on 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 In the article we present a criterion for convergence of the Mellin-Barnes integral for zeros of a system of Laurent polynomials. Also we give a hypergeometric series for these zeros. Mellin-Barnes integrals; hypergeometric series; Laurent polynomials Hypergeometric series and the Mellin-Barnes integrals for zeros of a system of Laurent 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 paper under review is concerned with the general problem of constructing dual objects in the category of projective schemes, meaning that a given scheme should be canonically isomorphic to its associated double dual. Prominent examples in algebraic geometry are provided by the dual of an abelian variety, by S. Mukai's duality theory of polarized \(K3\) surfaces, or by certain moduli spaces of stable vector bundles over a smooth projective curve [à la \textit{M. S. Narasimhan} and \textit{S. Ramanan}, Ann. Math. (2) 101, 391--417 (1975; Zbl 0314.14004)]. With these classical examples in mind, the author's underlying philosophy is that certain stacks of vector bundles should be natural categorical candidates for dual objects in algebraic geometry. Indeed, pursuing this strategy, he is able to construct dual objects for quotient stacks of schemes in the following way.
Let \({\mathcal X}= [X/G]\) be the quotient stack associated with a scheme \(X\) acted on by an affine group scheme \(G\), and denote by \(\text{VB}({\mathcal X})\) the category of all vector bundles on the stack \({\mathcal X}\). For any affine base scheme \(S\), a \(S\)-valued fiber functor on \(\text{VB}({\mathcal X})\) is defined to be an exact additive tensor functor from \(\text{VB}({\mathcal X})\) to \(\text{VB}(S)\), and these fiber functors form a fibered category denoted by \(\text{FIB}(\text{VB}({\mathcal X}))\). The basic technical framework for this construction is explained in the first sections of the present paper. Then it is shown that there is a natural morphism of fibered categories \(D_{{\mathcal X}}:{\mathcal X}\to\text{FIB} (\text{VB}({\mathcal X}))\), and the author's main theorem establishes the following fact:
Assume that the scheme \(X\) is quasi-compact and that there is a line bundle on the quotient stack \({\mathcal X}= [X/G]\) whose underlying line bundle on \(X\) is very ample. Then the associated ``duality functor'' \(D_{{\mathcal X}}:{\mathcal X}\to\text{FIB} ({\mathcal VB}({\mathcal X}))\) is a categorical equivalence. In particular, if the affine group \(G\) is trivial, i.e., if \(G= \text{Spec}(k)\) for a field \(k\), then one is dealing with true schemes, in which case the author's main theorem yields that for a quasi-compact \(k\)-scheme \(X\) admitting an ample line bundle, the duality morphism \(D_X\) provides a duality construction in the ordinary sense. The author's main result may be regarded as a stack-theoretic analogue of the classical Tannaka duality for affine groups in the sense of \textit{P. Deligne} and \textit{J. S. Milne} [in: Hodge cycles, motives, and Shimura varieties, Lect. Notes Math. 900, 101--228 (1982; Zbl 0477.14004)]. On the other hand, the author's result allows to derive a just as interesting and important conclusion, which makes the analogy to the classical duality theorems immediately recognizable. This second main result ensures the existence of another categorical equivalence, the so-called bi-duality morphism, between the quotient stack \({\mathcal X}\) and some tensor category fibered in groupoids. This construction uses the stack of all vector bundles on \({\mathcal X}=[X/G]\) as well as the existing universal bundle for them. In the course of the proofs, the framework of sheaves on fibered tensor categories appears as an essential ingredient. Accordingly, the technical details are thoroughly developed and explained, thereby making this rather advanced work easily accessible even for non-experts. algebraic stacks; group actions on schemes; tensor categories; Grothendieck topologies; duality theory; vector bundles; scheme; fibered categories; Tannakian categories Savin, V, Tannaka duality for quotient stacks, Manuscripta math., 119, 287-303, (2006) Generalizations (algebraic spaces, stacks), Group actions on varieties or schemes (quotients), Algebraic moduli problems, moduli of vector bundles, Étale and other Grothendieck topologies and (co)homologies, Fibered categories, Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) Tannaka duality on quotient stacks | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(A\) be a matrix whose entries are algebraic functions defined on a reduced quasi-projective algebraic set \(X, e.g\). multivariate polynomials defined on \(X:= \mathbb C N\) . The sets \({\mathcal S}_k(A)\), consisting of \(x\epsilon X\) where the rank of the matrix function \(A(x)\) is at most \(k\), arise in a variety of contexts: for example, in the description of both the singular locus of an algebraic set and its fine structure; in the description of the degeneracy locus of maps between algebraic sets; and in the computation of the irreducible decomposition of the support of coherent algebraic sheaves, \(e.g\). supports of finite modules over polynomial rings. In this article we present a numerical algorithm to compute the sets \(\mathcal S_k(A)\) efficiently. rank deficiency; matrix of polynomials; homotopy continuation; irreducible components; numerical algebraic geometry; polynomial system; Grassmannians D.J. Bates, J.D. Hauenstein, C. Peterson, and A.J. Sommese, \textit{Numerical decomposition of the rank-deficiency set of a matrix of multivariate polynomials}, in Approximate Commutative Algebra, Texts Monogr. Symbol. Comput., Springer, Vienna, 2009, pp. 55--77. Computational aspects of higher-dimensional varieties, Grassmannians, Schubert varieties, flag manifolds, Computational aspects and applications of commutative rings, Matrices over function rings in one or more variables, Numerical computation of solutions to systems of equations Numerical decomposition of the rank-deficiency set of a matrix of multivariate 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 \(X\) be a compact Riemann surface of genus \(g\geq 2\) and let \(\mathbb{C}(X)\) denote its function field. The paper gives an upper bound for the number \(k\) of subfunction fields of index 2 of \(\mathbb{C}(X)\) or equivalently the number of inequivalent ramified double coverings \(\pi:X\to Y\) of degree 2. Moreover it is shown that this bound is sharp. In fact there are infinitely many odd values for \(g\) where this bound is taken. For even genus the situation is completely different: There is either 1 or 3 subfield of degree 2 of \(\mathbb{C}(X)\). The proofs use the description of \(X\) as a Fuchsian group. Reviewer's remark: All curves with the maximum number of double coverings are hyperelliptic. double covering Bujalance, E., Gromadzki, G.: On ramified double covering maps of Riemann surfaces. J. Pure Appl. Algebra 146(1), 29--34 (2000) Compact Riemann surfaces and uniformization, Riemann surfaces; Weierstrass points; gap sequences, Coverings of curves, fundamental group On ramified double covering maps of Riemann 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 In [\textit{C. Monical} et al., Transform. Groups 26, No. 3, 1025--1075 (2021; Zbl 1472.05152)], we introduced the notion of a K-crystal, with applications to K-theoretic Schubert calculus and the study of Lascoux polynomials. We conjectured that such a K-crystal structure existed on the set of semistandard set-valued tableaux of any fixed rectangular shape. Here, we establish this conjecture by explicitly constructing the K-crystal operators. As a consequence, we establish the first combinatorial formula for Lascoux polynomials \(L_{w\lambda}\) when \(\lambda\) is a multiple of a fundamental weight as the sum over flagged set-valued tableaux. Using this result, we then prove corresponding cases of conjectures of \textit{C. Ross} and \textit{A. Yong} [Sémin. Lothar. Comb. 74, B74a, 11 p. (2015; Zbl 1328.05200)] and \textit{C. Monical} [ibid. 78B, 78B.35, 12 p. (2017; Zbl 1384.05160)] by constructing bijections with the respective combinatorial objects. Grothendieck polynomial; crystal; Lascoux polynomial; quantum group; set-valued tableau; Kohnert move; skyline tableau Combinatorial aspects of representation theory, Symmetric functions and generalizations, Combinatorial identities, bijective combinatorics, Grassmannians, Schubert varieties, flag manifolds, Quantum groups (quantized enveloping algebras) and related deformations K-theoretic crystals for set-valued tableaux of rectangular shapes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 series of (three) papers is to prove rigorously some enumerative formulae of Schubert for triple contacts of plane curves which move in suitable families. Two smooth plane curves have at least triple contact at a point P iff they have the same tangent line at P and the same second-order data at P. Here the problem is to give a conceptual approach for the second-order data. If C is a smooth plane curve and \(x,y,z\in C\) are three points approaching a given point P, consider the family \(\Sigma(x,y,z)\) of all concis through \(x,y,z.\) Consider \(P=P^ 2\times P^ 2\times P^ 2\) and \v{P}\(=\check P^ 2\times \check P^ 2\times \check P^ 2\) and define \(W=\{(x,y,z;L,M,N)\in P\times \check P|\) \(x,y\in L,\) \(x,z\in M\) and \(y,z\in N\},\) Then one has a rational map \(W\to G(2,5)\) defined by \((x,y,z;L,M,N)\to \Sigma(x,y,z)\) (one thinks of G(2,5) as the parameter variety for 2-dimensional linear families of conics). Let \(W^*\subset W\times G(2,5)\) be the closure of the graph of the above rational map. The variety \(W^*\), called the model of Schubert triangles, is the interesting object to be studied for the problem the authors have in mind. To this end one also considers the blow-up \(\bar W\) of W along the closed subvariety \(X=\{(x,x,x;L,L,L)|\) \(x\in P^ 2,L\in \check P^ 2,x\in L\}\) (X is just the singular locus of W and \(\bar W\) is smooth), and the full-diagonal blow-up B of P. Then one has a commutative diagram
\[
\begin{tikzcd}\bar {W} \ar[r,"p"]\ar[d,"p" '] & W^\ast\ar[d,"qw"] \\ B \ar[r,"p_W" '] & W \end{tikzcd}
\]
in which \(\bar W\) is identified with the blow-up of B along \(X_ B=p_ W^{-1}(X)\), and also with the blow-up of \(W^*\) along \(X^*=q_ W^{-1}(X)\). Both blow-ups have the same exceptional locus \(\bar X\) of \(\bar W.\) In this first part of the paper the authors begin the program aiming to determine the rational equivalence ring \(A^{\bullet}(W^*)\), by establishing some basic properties of B and \(W^*\) and by computing \(Pic(W^*)\). The variety \(X^*\) is called the variety of the second-order data of \(P^ 2.\)
[See also the following two reviews.] enumerative formulae; triple contacts of plane curves; Schubert triangles; variety of the second-order data Roberts-Speiser , '' Enumerative Geometry of Triangles, I '' Comm. in Alg. 12(10) 1213-1255 (1984). Enumerative problems (combinatorial problems) in algebraic geometry, (Equivariant) Chow groups and rings; motives Enumerative geometry of triangles. I | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let X be the fixed point set in the flag variety of a unipotent element u in \(GL_ n\). The irreducible components of X are in bijective correspondence with standard Young tableaux. The intersection properties of these components have connections with the theory of Kazhdan-Lusztig polynomials. The author determines the intersection codimensions in the case that u has two Jordan blocks. He uses the methods of \textit{A. Lascoux} and \textit{M.-P. Schützenberger} [Astérisque 87-88, 249-266 (1981; Zbl 0504.20007)]. fixed point set; flag variety; unipotent element; irreducible components; Young tableaux; Kazhdan-Lusztig polynomials; intersection codimensions; Jordan blocks J.Wolper, Some intersection properties of the fibres of Springer's resolution, Proc. Am. Math. Soc. 91 (1984), 182--188. Representation theory for linear algebraic groups, Classical groups (algebro-geometric aspects), Grassmannians, Schubert varieties, flag manifolds, Global theory and resolution of singularities (algebro-geometric aspects) Some intersection properties of the fibres of Springer's resolution | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This is an awkward book. As the author says it ``is actually a collection of essays by the author since 1977 centered around the subject which has been baptized mathematics mechanization''. In fact, it is a kind of retrospective on the author's scientific work.
Although several chapter headings refer to theorem proving the main theme is rather ``zeros of polynomials''. Surprisingly, there is an unexpected pervading side-theme, viz., ancient Chinese mathematics. Not only does Part I deal with ``historical developments'', the author even traces modern developments back to Chinese roots. It would need an expert to decide whether the explanation in modern terms is a correct description of what the ancient Chinese did. To the reviewer this tends to give the book an idiosyncratic touch. In any case, it is the author's point that Chinese mathematics was mostly computational with less emphasis on theorem proving. This describes quite well the spirit of the present book.
Part II is called ``Principles and Methods''. Here, the author turns to his main theme, the study of the zero set of a set of polynomials. One of the main tools is the author's approach [Syst. Sci. Math. Sci. 4, 193-207 (1991; Zbl 0802.13006)] to Gröbner bases. This part also includes a chapter on computational algebraic geometry. The last part ``Applications and Examples'' is the most heterogeneous part of the book. It starts with numerical solving of polynomial equations (including an unorthodox approach to round-off errors). The next chapter deals with automated proofs in geometry though not in the usual sense: all problems are more or less reduced to polynomial equations, so we are again back at the main problem. The book closes with ``Diverse Applications''. Here, the author has gathered some rather disparate items (ranging from the automated determination of geometric loci and nonlinear optimization to robotics and CAGD) the unifying concept again being polynomial equations.
This review would be incomplete without a word on the publisher's rôle: On the one hand, he expects the reader to pay the horrendous price of 0.41~EUR per page, on the other hand there has obviously neither been any copy editing nor any help of a native speaker. A copy editor should have convinced the author to write complete sentences instead of telegraphic messages, e.g. ``For a variety \(V=\text{Zero}(PS)\), \(PS:=\) A basis of \(V\)''. Moreover, the reader is disturbed by awkward abbreviations, like ``pol'' for polynomial or ``polset'' for set of polynomials. Of course, at most places it is easy to guess what the author intends to say and the awkward English is just a nuisance. There are, however, sentences which reveal their meaning only after some time of intense meditation. So, at first reading, it has not been clear to this reviewer how to understand the following definition (2.6 in chapter VI) (which is typical for the style of the book): ``An exact representation \(c_i=c_{i0}+t_i\) is pathological at stage \(k :=\) for some monomial in \(x_i\)'s in \(P'_k\) of the sequence (2.8)' its coefficient as a pol in the tinies has its part not involving any tiny equal to 0. In this case such a monomial \(:=\) A pathological monomial''. mechanization; automated theorem proving; computer algebra; zeros of polynomials; Gröbner basis; robotics; CAGD Wu W T, \textit{Mathematics Mechanization}, Science Press and Kluwer Academic Publishers, Beijing, 2000. , Introductory exposition (textbooks, tutorial papers, etc.) pertaining to computer science, History of Chinese mathematics, Polynomials in real and complex fields: location of zeros (algebraic theorems), Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Computational aspects of algebraic curves, Elementary problems in Euclidean geometries, Error analysis and interval analysis, Artificial intelligence for robotics, Computer science aspects of computer-aided design, Symbolic computation and algebraic computation Mathematics mechanization. Mechanical geometry theorem-proving, mechanical geometry problem-solving and polynomial equations-solving | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 known that differences of symmetric functions corresponding to various bases are nonnegative on the nonnegative orthant exactly when the partitions defining them are comparable in dominance order. The only exception is the case of homogeneous symmetric functions where it is only known that dominance of the partitions implies nonnegativity of the corresponding difference of symmetric functions. It was conjectured by \textit{A. Cuttler} et al. [Eur. J. Comb. 32, No. 6, 745--761 (2011; Zbl 1229.05267)] that the converse also holds, as in the cases of the monomial, elementary, power-sum, and Schur bases. In this paper we provide a counterexample, showing that homogeneous symmetric functions break the pattern. We use semidefinite programming to find an explicit sums of squares decomposition of the polynomial \(H_{44} - H_{521}\) as a sum of 41 squares. This rational certificate of nonnegativity disproves the conjecture, since a polynomial which is a sum of squares cannot be negative, and since the partitions 44 and 521 are incomparable in dominance order. sums of squares; symmetric polynomials; semidefinite programming Symmetric functions and generalizations, Real algebraic and real-analytic geometry, Semidefinite programming An SOS counterexample to an inequality of symmetric functions | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper deals with partition varieties, i.e., projective varieties associated to some partition \(\lambda=(\lambda_1,\dots,\lambda_n)\). The main result is that the partition variety of \(\lambda\) is a CW-complex which has a cellular decomposition that can be described combinatorially in terms of rook placements of the Ferrers board of \(\lambda\), and that the Poincaré polynomial \(P_\lambda(q)\) for the cohomology equals \(R_\lambda(q^2)\), where \(R_\lambda\) is the rook length polynomial of \(\lambda\). The combinatorial description of the cells also characterizes the geometric situtation, i.e., whether some cell is contained in the closure of another cell. The exposition is self-contained. Many examples and four appendices (concerning Grassmannian manifolds, Schubert varieties, flag manifolds and an embedding theorem) make the paper accessible also to the non-expert in algebraic geometry. Rook length polynomial; projective variety; partition variety; Grassmannian manifold; flag manifold; Schubert cells; Bruhat order K. Ding, Rook placements and cellular decomposition of partition varieties, Discrete Math. 170 (1997), 107-151. Algebraic combinatorics, Grassmannians, Schubert varieties, flag manifolds, Combinatorial aspects of partitions of integers, Permutations, words, matrices Rook placements and cellular decomposition of partition varieties | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The paper gives a new proof of Torelli's theorem for smooth projective curves using the locus of double points of the theta divisor \(\theta\) of the Jacobian variety. To be more precise, the projectivized tangent cone to \(\theta\) at a double point \(t\) is a rank 4 quadric \(Q_ t\) in \(\mathbb{P}(H^ 1(C,{\mathcal O}_ C))\), which contains the canonical model \(\kappa(C)\) of \(C\). Consider the family of \((g-1)\)-secant \((g-3)\)-planes of \(\kappa(C)\) in \(\mathbb{P}_{g-1}\), which are the linear subspaces of maximal dimension contained in the \(\mathbb{Q}_ t\)'s. Projecting from \(g-5\) general points of \(\kappa(C)\) into \(\mathbb{P}_ 4\), one obtains a family of 4-secant 2-planes of the projected curve in \(\mathbb{P}_ 4\), it is shown that if \(C\) is not hyperelliptic, not trigonal and not a plane quintic, then this family determines the curve \(\kappa(C)\) uniquely. In the exceptional cases it is shown that the pair \((\text{Pic}^{g-1}(C)\), \(\text{Sing}(\theta))\) determines the curve \(C\) uniquely using simple geometric description of \(\text{Sing}(\theta)\) in terms of the curve \(C\). Section 1 rewrites the classical results of C. Segre on congruences of planes in a modern language. Torelli's theorem for smooth projective curves; double points; theta divisor; Jacobian variety Ciliberto, C.; Sernesi, E., \textit{singularities of the theta divisor and congruences of planes}, J. Algebr. Geom., 1, 231-250, (1992) Jacobians, Prym varieties, Theta functions and abelian varieties Singularities of the theta divisor and congruences of planes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Schubert varieties are Borel orbits in homogeneous spaces. The study of their geometric properties is old and well developed. Richardson varieties are defined as intersections of Schubert varieties and `opposite' Schubert varieties, they are also key objects in geometry. The authors prove a very general statement claiming that essentially all questions concerning singularities of Richardson varieties reduce to corresponding questions about Schubert varieties. The properties/quantities reduced from Schubert varieties to Richardson varieties include smoothness and the property `Cohen-Macaulay with rational singularities', as well as multiplicity and H-polynomial. The authors give two versions of their elementary proof, one in the language of algebraic groups and one that avoids it. Richardson variety; Schubert variety; singularity A. Knutson, A. Woo, and A. Yong, Singularities of Richardson varieties, \textit{Math. Res. Lett.}, 20 (2013), no. 2, 391--400.Zbl 1298.14053 MR 3151655 Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus Singularities of Richardson 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 article is an expository one based on lectures at the Summer Institute in Algebraic Geometry, Seattle, 2005. It gives a review of the main achievements in the subject.
The first lecture includes a brief review of \(p\)-adic integration, a result of Denef-Loeser giving a formula for the Euler characteristic of a smooth complex variety \(X\) in terms of its log resolution and the independence of the zeta function of a pair from the log resolution. There is also a result of Batyrev that the Betti numbers of two birationally equivalent Calabi-Yau varieties are equal, which led to his conjecture about the equality of their Hodge numbers, and so gave the main motivation for the invention of the motivic integration by Kontsevich.
The second lecture gives the basics of motivic integration. It discusses briefly arc spaces, additive invariants and Grothendieck rings. Then the change of variables formula and some of its major applications are given. Among them is a formula for the class of \(X\) in terms of its log resolution, the proof of the Batyrev conjecture by Kontsevich, the independence of the stringy invariant \(E_{st}(X)\) of a normal terminal \(\mathbb{Q}\)-Gorenstein variety on the log resolution, and the theorem of Mustată giving a formula for the log canonical threshold in terms of codimensions of some jet spaces.
In lecture 3, after introducing briefly the Milnor fiber and the nearby cycle functor, a relation between Euler characteristic and Lefschetz numbers is given. Then the motivic analogue of Igusa's local zeta function is defined, and the monodromy conjecture is formulated. After introducing briefly the Hodge spectrum and convolution product, two versions of Thom-Sebastani theorem are formulated, one for the Hodge spectrum, and another as the motivic version of the theorem. Also, the motivic version of a result of Saito proving a Steenbrink conjecture is given.
In the last lecture a general setting for motivic integration is developed, based on a series of works of the author with R. Cluckers. After an introduction to semialgebraic geometry and some notions from model theory, the Denef-Pas cell decomposition theorem is stated. Using the language of constructible motivic functions, the general motivic measure satisfying some axioms is defined, the general change of variables formula is stated, and the motivic analogues of exponential functions are introduced. In the last part is obtained a general transfer principle, allowing to transfer relations between integrals from \(\mathbb{Q}_p\) to \(\mathbb{F}_p((t))\), and vice versa. As a special case one has the Ax-Kochen-Eršov theorem. motivic integration; arc space; Grothendieck ring Loeser, François, Seattle lectures on motivic integration.Algebraic geometry---Seattle 2005. Part 2, Proc. Sympos. Pure Math. 80, 745-784, (2009), Amer. Math. Soc., Providence, RI Arcs and motivic integration Seattle lectures on motivic integration | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 Schubert class on \(\mathrm{Gr}(k,V)\) where \(V\) is a symplectic vector space of dimension \(2n\), we consider its restriction to the symplectic Grassmannian \(\mathrm{SpGr}(k,V)\) of isotropic subspaces. \textit{P. Pragacz} [J. Algebra 226, No. 1, 639--648 (2000; Zbl 0945.05065)] gave tableau formulae for positively computing the expansion of these \(H^\ast(\mathrm{Gr}(k,V))\) classes into Schubert classes of the target when \(k=n\), which corresponds to expanding Schur polynomials into Q-Schur polynomials. \textit{I. Coşkun} [J. Comb. Theory, Ser. A 125, 47--97 (2014; Zbl 1296.14037)] described an algorithm for their expansion when \(k\le n\). We give a puzzle-based formula for these expansions, while extending them to equivariant cohomology. We make use of a new observation that usual Grassmannian puzzle pieces are already enough to do some 2-step Schubert calculus, and apply techniques from quantum integrable systems (``scattering diagrams''). Schubert calculus; puzzles; Grassmannian; symplectic Grassmannian Grassmannians, Schubert varieties, flag manifolds Restricting Schubert classes to symplectic Grassmannians using self-dual puzzles | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 reader will find an elegant approach to the theory of classical motives, i.e. (sums of) pure motives, with some emphasis on geometric aspects. In particular, special attention is payed to Chow motives, i.e. one starts from smooth projective varieties over some field and takes for the morphisms the cycle groups on products of such varieties modulo rational equivalence, which is the finest adequate equivalence relation for algebraic cycles. The coarsest adequate equivalence is numerical equivalence. The corresponding motives are called Grothendieck motives. Fixing some Weil cohomology one may speak of homological equivalence which is a priori finer than numerical equivalence but coarser than rational equivalence. It was conjectured by Grothendieck that, however, homological and numerical equivalence coincide, thus implying that the theory of motives can be considered as a universal cohomology theory. The paper consists of an introduction, six sections and references.
The first section deals with the definition and formal properties of motives for any adequate equivalence relation on cycles. A motive is defined as a triple \((X,p,m)\), where \(X\) is a smooth projective variety over some field \(k\) assumed to be purely \(d\)-dimensional, \(m\) is an integer, and \(p\) is a projector, i.e. an idempotent in the space of self- correspondence of \(X\) of degree zero \(\text{Corr}^ 0(X,X)\) which is just \(A^ d (X \times X) = {\mathcal Z}^ d (X \times X) \otimes \mathbb{Q}/\sim\) with \(\sim\) any of the aforementioned adequate equivalence relations. Motives form a category \({\mathcal M}_ k\) with morphisms given by \(\text{Hom}_{{\mathcal M}_ k}((X,p,m), (Y,q,n)) = q \circ \text{Corr}^{n-m}(X,Y) \circ p = q \circ A^{d+n-m} (X \times Y) \circ p\). There are two distinguished motives: \(\mathbf{1} = (\text{Spec} k,\text{id},0)\) (the unit motive) and \(\mathbb{L} = (\text{Spec }k,\text{id},- 1)\) (the Lefschetz motive). One defines the direct sum \((X,p,m) \oplus (Y,q,m) = (X\coprod Y,p \oplus q,m)\). For a smooth projective variety \(X\) one defines its associated motive by \(h(X) = (X,\text{id},0)\), where \(\text{id}\) is the class of the diagonal \(\Delta_ X\) in \(A^ d(X \times X)\). E.g. for \(X = \mathbb{P}^ 1\), \(\Delta_ X\) is rationally equivalent to the sum of \(\{x\} \times \mathbb{P}^ 1\) and \(\mathbb{P}^ 1 \times \{x\}\) for any \(x \in \mathbb{P}(k)\), and this gives a decomposition \(\text{id}_{\mathbb{P}^ 1} = p_ 0 + p_ 2\), \(p_ 2 = {}^ t p_ 0\), and a canonical decomposition \(h(\mathbb{P}^ 1) = h^ 0(\mathbb{P}^ 1)\oplus h^ 2 (\mathbb{P}^ 1) = \mathbf{1} \oplus \mathbb{L}\). More generally, \(h(\mathbb{P}^ n) = \mathbf{1} \oplus \mathbb{L} \oplus \cdots \oplus \mathbb{L}^ n\), where \(\mathbb{L}^ i\) means \(\mathbb{L}^{\otimes i}\), \(i \in \mathbb{N}\). One can define the tensor product of two motives by \((X,p,m) \otimes (Y,q,n) = (X \times Y, p \otimes q, m+n)\). The diagonal defines a product structure on \(h(X)\): \(h(X) \otimes h(X) = h(X\times X)\overset {\Delta^*} \longrightarrow h(X)\). One has \((X,p,m) = (X,p,0) \otimes \mathbb{L}^{-m}\), and any motive is a direct factor of some \(h(X) \otimes \mathbb{L}^ n\). This makes it possible to define the direct sum of any two motives \((X,p,m)\) and \((Y,q,n) = (X \times (\mathbb{P}^ 1)^{n-m} \coprod Y, p' \oplus q,n)\), where \(p'\) is a projector of \(X \times (\mathbb{P}^ 1)^{n-m}\) constructed by means of \(p\) and the canonical projector \(p_ 2\) of \(\mathbb{P}^ 1\). One also has a dual motive \((X,p,m)^ \vee = (X,{}^ tp, d - m)\), in particular \(h(X)^ \vee = h(X) \otimes \mathbb{L}^{-d}\) (`Poincaré duality'). For arbitrary motives \(M\), \(N\) and \(P\) one obtains the formula for the Hom's, \(\text{Hom} (M \otimes N,P) = \text{Hom} (M,N^ \vee \otimes P)\) and thus one may define the internal Hom, \(\text{Hom} (M,N):= M^ \vee \otimes N\). Altogether, \({\mathcal M}_ k\) becomes a rigid, pseudo-abelian, \(\mathbb{Q}\)- linear tensor category. Jannsen proved that \({\mathcal M}_ k\) is abelian semi-simple if and only if one takes numerical equivalence in the \(A^*(X \times Y)\). Thus with Grothendieck's conjecture one would have that \({\mathcal M}^{\text{num}}_ k = {\mathcal M}^{\text{hom}}_ k\) is a semi-simple \(\mathbb{Q}\)-linear tannakian category (after suitable definition of commutativity and associativity constraints). On the other hand it is shown in the third section that \({\mathcal M}^{\text{rat}}_ k\) is not abelian in general.
One can recover the cycle class groups \(A^*(X)\) by the formula \(A^ i(X) = \text{Hom}({\mathbb{L}}^ i,h(X))\) which follows from the definition of motives. For an arbitrary motive \(M\) one defines \(A^ i(M) := \text{Hom}({\mathbb{L}}^ i,M)\), and one obtains a contravariant functor from the category of smooth projective varieties over \(k\) to finite dimensional \(\mathbb{Q}\)-vector spaces \(\omega_ M(Y) = A^*(M \otimes h(Y))\). Manin's identity principle says that a morphism \(f : M \to N\) in \({\mathcal M}_ k\) is an isomorphism iff the induced map \(\omega_ f(Y) : A^* (M\otimes h(Y)) \to A^* (N \otimes h(Y))\) is an isomorphism for all smooth projective varieties \(Y\). Two morphisms \(f,g : M \to N\) coincide iff \(\omega_ f(Y) = \omega_ g(Y)\) for all \(Y\). Also, a sequence
\[
0 \rightarrow M' \to M \to M'' \to 0
\]
in \({\mathcal M}_ k\) is exact iff the induced sequence
\[
0 \to \omega_{M'}(Y) \to \omega_ M(Y) \to \omega_{M''}(Y) \to 0
\]
is exact for all \(Y\). As applications the calculation of the motive of a projective bundle and of a blow-up are given. This last result was used in the sixties by Manin to prove the Weil conjectures for three-dimensional unirational varieties over finite fields.
In the third section the relation between the motive of a curve and its Jacobian, due in essence to Weil, is studied. If \(X\) and \(X'\) are smooth projective curves with Jacobians \(J\) and \(J'\), respectively, one has \(\text{Hom}(h^ 1(X), h^ 1(X')) = \text{Hom} (J,J') \otimes \mathbb{Q}\), where one uses the decomposition \(h(X) = h^ 0(X) \oplus h^ 1(X) \oplus h^ 2(X)\) and similarly for \(X'\), where \(h^ 0\) and by transposition \(h^ 2\) are determined by projectors \(p_ 0\) and \(p_ 2\) defined by some zero cycle of positive degree (e.g. a rational point if there are) on \(X\) and then \(h^ 1(X)\) is just \((X,p_ 1,0)\) with \(p_ 1 = \text{id} - p_ 0 - p_ 2 \in \text{Corr}^ 0(X, \overline {X}) = A^ 1 (X \times X)\), and similarly for \(X'\). Furthermore,
\[
\text{Hom}(\mathbb{L},h^ 1 (X)) = \begin{cases} 0 & \text{if \(\sim\) is numerical (or homological) equivalence,}\\ J(k) \otimes \mathbb{Q} & \text{if \(\sim\) is rational equivalence.}\end{cases}
\]
As a corollary one deduces that the category of abelian varieties of \(k\) up to isogeny is equivalent to the full subcategory of \({\mathcal M}_ k\) whose objects are direct summands of motives of the form \(h^ 1(X)\), where \(X\) is of dimension one. Another corollary, already mentioned before, says that \({\mathcal M}^{\text{rat}}_ k\) will not be abelian if \(k\) is not contained in the closure of a finite field. The proof of this fact depends on the existence of an elliptic curve over \(k\) with a \(k\)-rational point of infinite order.
The fourth section is concerned with the construction of Murre's Picard and Albanese's motives (for rational equivalence!) for a smooth projective \(d\)-dimensional variety \(X\) over \(k\). It consists in the explicit construction of projectors \(p_ 1\) and \(p_{2d-1} = {}^ t p_ 1\) such that \(p_ 0\), \(p_ 1\), \(p_{2d-1}\) and \(p_{2d}\) are orthogonal idempotents. One also obtains a Lefschetz type isomorphism and a description of the Chow groups of \(h^ 1(X)\) and \(h^{2d-1}(X)\). In particular, for a surface \(X\) one obtains a complete list of the Chow groups of the constituent parts \(h^ i(X)\), \(i = 0,1,2,3,4\), of the motive \(h(X)\) of \(X\). For \(h^ 2(X) = (X, p_ 2,0)\) one takes \(p_ 2 = \text{id} - p_ 0 - p_ 1 - p_ 3 - p_ 4\).
The fifth section deals with work of Deninger and Murre and of Künnemann on motives of abelian varieties (or, more generally, abelian schemes). For these varieties one has a canonical decomposition in \({\mathcal M}_ k = {\mathcal M}^{\text{rat}}_ k\) of the form
\[
h(X) = \bigoplus_{i = 0}^{2g} h^ i(X)^{\text{can}},
\]
where \(g = \dim X\), and where the \(h^ i (X)\) are the `\(n^ i\)-eigenspaces' for multiplication by \(n\) on \(X\). One has for every \(i \geq 0\) isomorphisms \(\bigwedge^ i h^ 1(X)^{\text{can}}\overset \sim \longrightarrow h^ i(X)^{\text{can}}\). Also, for the class \(\xi\in A^ 1(X)\) of a symmetric line bundle on \(X\), one gets \(h(X) \overset {\xi^{2g - 1}} \longrightarrow h(X) \otimes \mathbb{L}^{1-g}\) and this fits in a commutative diagram with \(h^ i(X) \overset \sim \longrightarrow h^{2g-1} \otimes \mathbb{L}^{1-g}\), where the indexed \(h\)'s map into \(h(X)\). Finally, the canonical projectors defining the \(h^ i(X)^{\text{can}}\) coincide with the \(p_ i\)'s of the fourth section for \(i = 0,1,2g - 1,2g\) whenever these are constructed by means of a symmetric class \(\xi\) of a very ample line bundle on \(X\).
In the last sections several less classical topics are briefly discussed: (i) relative motives; (ii) the general picture englobing numerical, homological and rational equivalence and the respective categories of motives. This is mainly conjectural, depending on Grothendieck's standard conjectures and conjectures due to Beilinson and Murre on a filtration of Chow groups; (iii) the formalism in terms of derived categories: there should exist a derived category of mixed motives containing \({\mathcal M}_ k^{\text{num}}\) as the abelian, semi-simple subcategory of `pure complexes' of any weight concentrated in degree zero, and containing \({\mathcal M}^{\text{rat}}_ k\) as the subcategory of `pure complexes of weight zero'. Manin's identity principle; abelian variety; motives; Chow motives; algebraic cycles; Grothendieck motives A. J. Scholl, ''Classical motives,'' in Motives, Proc. Sympos. Pure Math., Seattle, WA, 1991 (Amer. Math. Soc., Providence, RI, 1994), Vol. 55, Part 1, pp. 163--187. Generalizations (algebraic spaces, stacks), Algebraic cycles, (Co)homology theory in algebraic geometry, (Equivariant) Chow groups and rings; motives Classical 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 [For the entire collection see Zbl 0509.00015.]
Consider the Grassmann variety G(k,n). For each non-increasing sequence of non-negative integers \(a=(a_ 1,...,a_ k)\) with \(n-k\geq a_ 1\), let \(\sigma_ a\) denote the homology class of a corresponding Schubert cycle. The intersection cycle of \(\sigma_ a\) and \(\sigma_ b\) can be written \(\sigma_ a\cdot \sigma_ b=\sum_{c}\delta(a,b;c)\sigma_ c\), where \(\delta\) (a,b;c) denotes the intersection number of \(\sigma_ a\cdot \sigma_ b\) with \(\sigma_{\tilde c}\) and where \(\tilde c=(n-k- c_ k,...,n-k-c_ 1)\). No general formula is known for this intersection number. The purpose of the present paper is to give a recursive formula for \(\delta\) (a,b;c) involving the intersection numbers \(\delta\) (a,b;d) for all \(d<c\) (in a suitable odering). The proof of the formula involves representation theory for the unitary group U(k). Schubert calculus; Grassmann variety; Schubert cycle; representation of unitary group; intersection numbers Grassmannians, Schubert varieties, flag manifolds, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Representation theory for linear algebraic groups A note on 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 Let \(X\) be a smooth projective complex threefold with Néron-Severi group \(NS(X) \simeq \mathbb{Z}\) and \(f\) be a nonconstant morphism from \(X\) to a 3-dimensional quadric \(Q\). The author proves that \(f\) is finite and its degree is bounded in terms of the Chern numbers of \(X\) and of the index \(r\) of \(X\) (i.e. \(K_X \equiv rH\) where \(H\) is the ample generator of \(NS(X))\). In particular, if \(X\) is a smooth cubic threefold there are no nonconstant morphisms from \(X\) to \(Q\).
The main tool used in the proof is a Miyaoka bound for the number of double points of a surface. threefold; Néron-Severi group; 3-dimensional quadric; Chern numbers; Miyaoka bound; number of double points C. Schuhmann, Math. Ann. 306(3), 571 (1996). 3-quadrics \(3\)-folds, Low codimension problems in algebraic geometry Mapping threefolds onto three-dimensional quadrics | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A matroid variety \(X(M)\) in the Grassmannian \(\mathrm{Gr}(k,n)\) is the closure of the set of points for which a particular set \(M\) of Plücker coordinates vanishes. This paper is an attempt to nail down a structural property of the notoriously ill-behaved matroid varieties. The author associates to \(M\), in purely combinatorial fashion, a positive integer \(\text{ec}(M)\), the \textit{expected codimension} of \(M\). In many cases, but not always, it equals the codimension of \(X(M)\) in \(\mathrm{Gr}(k,n)\). The main result states that positroid varieties, the matroid varieties of positrons, have expected codimension. Moreover, the expected codimension is a valuative matroid invariant. expected codimension; Grassmannian; matroid; matroid variety; positroid; positroid variety; Schubert variety; valuativity Nicolas Ford, The expected codimension of a matroid variety, J. Algebraic Combin. 41 (2015), no. 1, 29 -- 47. Classical problems, Schubert calculus, Combinatorial aspects of matroids and geometric lattices, Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.) The expected codimension of a matroid 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 In the present paper, we discuss the Grothendieck conjecture for hyperbolic curves over Kummer-faithful fields. In particular, we prove that every point-theoretic and Galois-preserving outer isomorphism between the étale/tame fundamental groups of affine hyperbolic curves over Kummer-faithful fields arises from a uniquely determined isomorphism between the original hyperbolic curves. This result generalizes results of \textit{A. Tamagawa} [Compos. Math. 109, No. 2, 135--194 (1997; Zbl 0899.14007)], i.e., our main result in the case where the basefields are either finite fields or mixed-characteristic local fields. Grothendieck conjecture; hyperbolic curve; Kummer-faithful field Coverings of curves, fundamental group, Universal profinite groups (relationship to moduli spaces, projective and moduli towers, Galois theory) On the Grothendieck conjecture for affine hyperbolic curves over Kummer-faithful fields | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a complex semisimple simply-connected algebraic group and let \(T\) be a maximal torus of \(G\), \(B\) a Borel subgroup of \(G\) containing \(T\) and \(W\) the Weyl group of \(G\). Let \(F = G/B\) be the full flag variety and let \(X_w\subset F\) be the Schubert variety, for any \(w\in W\). In the paper under review, the authors prove that if every irreducible component of \(G\) is of type \(B_n\) or \(C_n\), and \(w_1, w_2\) are two distinct involutions in \(W\), then the tangent cones at the base point \(eB\in X_{w_i}\) to the corresponding Schubert subvarieties \(X_{w_1} , X_{w_2}\) in \(F\) do not coincide as subschemes of the tangent space \(T_{eB}(F)\). Similarly, they also show that if every irreducible component of \(G\) is of type \(A_n\) or \(C_n\), then the reduced tangent cones to \(X_{w_1} , X_{w_2}\) do not coincide as subvarieties of \(T_{eB}(F)\). Their first result is proved by using (what they call) Kostant-Kumar polynomials (so was the following result by Eliseev and Ignatyev). Their second result is proved by using a connection between the tangent cones of \(X_w\) and the geometry of coadjoint orbits of \(B\). In a previous work [J. Math. Sci., New York 199, No. 3, 289--301 (2014); translation from Zap. Nauchn. Semin. POMI 414, 82--105 (2013; Zbl 1312.14116)], \textit{D. Yu. Eliseev} and \textit{M. V. Ignatyev} had proved that \(X_{w_1} , X_{w_2}\) (for distinct involutions \(w_1, w_2\)) do not coincide as subschemes of the tangent space \(T_{eB}(F)\) when the irreducible components of \(G\) are of type \(A_n\), \(F_4\) and \(G_2\) only (partially confirming an earlier conjecture by Panov in 2011 for any \(G\)). flag variety; Schubert variety; tangent cone; reduced tangent cone; involution in the Weyl group; Kostant-Kumar polynomial Bochkarev, M; Ignatyev, M; Shevchenko, A, Tangent cones to Schubert varieties in types \(A_n\), \(B_n\) and \(C_n\), J. Algebra, 465, 259-286, (2016) Grassmannians, Schubert varieties, flag manifolds, Root systems, Classical groups (algebro-geometric aspects) Tangent cones to Schubert varieties in types \(A_{n}\), \(B_{n}\) and \(C_{n}\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper presents a method to construct curves over finite fields whose number of rational points is close to the bound obtained by Osterlé via optimization of Serre-Weil explicit formulae [see \textit{J.-P. Serre}, Rational points on curves over finite fields; Notes by F. Gouvêa of lectures at Harvard University (1985)]. Polynomials over finite fields; Curves over finite fields with many rational points Deolalikar, V., Extensions of algebraic function fields with complete splitting of all rational places, Comm. Algebra, 30, 6, 2687-2698, (2002) Curves over finite and local fields, Finite ground fields in algebraic geometry, Arithmetic theory of algebraic function fields Extensions of algebraic function fields with complete splitting of all rational places | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Coordinate polynomials are elements of \(P_n= K[x_1, \dots, x_n]\), \(K\) a commutative field, which can be included in a generating set of \(P_n\) of cardinality \(n\). The authors prove that an endomorphism of \(P_2\) which fixes the set of coordinate polynomials is actually an automorphism. Assuming the Jacobian conjecture this result holds for arbitrary \(n\), too. Other results concern so-called test polynomials, for which \(\varphi (p)=p\), \(\varphi\in \text{End} P_n\), implies \(\varphi\) is an automorphism. coordinate polynomials; endomorphism; Jacobian conjecture; test polynomial Essen, A; Shpilrain, V, Some combinatorial questions about polynomial mappings, J. Pure Appl. Algebra, 119, 47-52, (1997) Polynomials over commutative rings, Morphisms of commutative rings, Polynomial rings and ideals; rings of integer-valued polynomials, Automorphisms of curves Some combinatorial questions about polynomial mappings | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Minimal complex surfaces \(X\) of general type with \(\chi(\mathcal O_X)=1\) satisfy \(1\le K^2_X\le 9\), hence they belong to a finite number of families and can in principle be classified. The inequality \(q(X):=h^1(\mathcal O_X)\le 4\) also holds, and the cases \(q(X)=3,4\) are completely classified. On the other hand, for \(q(X)\le 2\) a complete classification seems out of reach at the moment, and therefore the construction of new examples helps to improve our understanding of this class of surfaces.
In this paper the author considers the case \(K^2_X=7\), \(\chi(\mathcal O_X)=1\) and constructs new examples with \(q(X)=0,1,2\). Note that no example with \(K^2_X=7\), \(\chi(\mathcal O_X)=1\), \(q(X)=2\) was previously known.
Although all the examples can be described as double covers of some nodal surface, the examples are constructed as \(\mathbb Z_2^2\)-covers (``bidouble covers'') of rational surfaces \(Y\) with very ad hoc choices of the surface \(Y\) and of the branch data. surface of general type; Albanese map; double covering Surfaces of general type New surfaces with \(K^2 = 7\) and \(p_g = q \leq 2\) | 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 Let \(V\) be an \(N\)-dimensional complex vector space equipped with a symmetric or skew-symmetric bilinear form \(\omega\), which can be either trivial or non-degenerate. The Grassmannians \(I G_{\omega}(m, N)\) of classical Lie type parameterize \(m\)-dimensional isotropic vector subspaces of \(V\). The cohomology ring of an isotropic Grassmannian \(X =I G_{\omega}(m, N)\), or more generally of a homogeneous variety, has an additive basis of Schubert classes represented by Schubert subvarieties \(X_{\lambda}\). One of the central problems of Schubert calculus is to find a manifestly positive formula for the structure constants of the cup product of two Schubert cohomology classes, or equivalently, for the triple intersection numbers of three Schubert subvarieties in general position. Such a positive formula, called a Littlewood-Richardson rule, has deep connections to various subjects, including geometry, combinatorics and representation theory.
An isotropic Grassmannian \(X\) can be written as a quotient of a classical complex simple Lie group \(G\) by a maximal parabolic subgroup \(P\) (with two notable exceptions of Lie type \(D_n\)). Fix a choice of maximal complex torus \(T\) and a Borel subgroup \(B\) with \(T \subset B \subset P\). The Schubert varieties \(X_{\lambda}\) are closures of \(B\)-orbits, and hence are \(T\)-stable. They give a basis \([X_{\lambda}]^T\) for the \(T\)-equivariant cohomology \({H^*}_T (X)\) as a \({H^*}_T (pt)\)-module. The structure coefficients \({N^{\nu}}_{\lambda,\mu}\) in the equivariant product,
\[
[X_{\lambda}]^T \cdot [X_{\mu}]^T=\sum\limits_{\nu}{N^{\nu}}_{\lambda,\mu} [X_{\nu}]^T,
\]
are homogeneous polynomials which satisfy a positivity condition conjectured by \textit{D. Peterson} [Lectures on quantum cohomology of \(G/B\), MIT (1996)] and proved by \textit{W. Graham} [Duke Math. J. 109, No. 3, 599--614 (2001; Zbl 1069.14055)]. In particular, they are Graham-positive, meaning they are polynomials in the negative simple roots, with non-negative integer coefficients. These equivariant structure coefficients carry much more information than the triple intersection numbers of Schubert varieties, and are more challenging to study.
In the present paper, the authors give for the first time an equivariant Pieri rule for Grassmannians of Lie types \(B, C\), and \(D\), as well as a new proof of the Pieri rule in type \(A\). Such a rule concerns products with the special Schubert classes \([X_{p}]^T\) , which are related to the equivariant Chern classes of the tautological quotient bundle, and generate the \(T\) -equivariant cohomology ring. Using geometric methods, they give a manifestly positive formula for the structure coefficients \({N^{\mu}}_{\lambda,p}\) of the equivariant multiplication \([X_{\lambda}]^T \cdot [X_{p}]^T\). isotropic Grassmannian; Schubert calculus; Littlewood-Richardson rule; \(T\)-equivariant cohomology; Graham-positive; equivariant Pieri rule Li, C.; Ravikumar, V.: Equivariant Pieri rules for isotropic grassmannians Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Equivariant homology and cohomology in algebraic topology Equivariant Pieri rules for isotropic Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a simple simply-laced algebraic group of adjoint type over the field \(C\) of complex numbers, \(B\) be a Borel subgroup of \(G\) containing a maximal torus \(T\) of \(G\). In this article, we show that \(\omega_\alpha\) is a minuscule fundamental weight if and only if for any parabolic subgroup \(Q\) containing \(B\) properly, there is no Schubert variety \(X_Q(w)\) in \(G/Q\) such that the minimal parabolic subgroup \(P_\alpha\) of \(G\) is the connected component, containing the identity automorphism of the group of all algebraic automorphisms of \(X_Q(w)\). minuscule weights; co-minuscule roots; Schubert varieties; automorphism groups Grassmannians, Schubert varieties, flag manifolds, Homogeneous spaces and generalizations Minimal parabolic subgroups and automorphism groups 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 We prove a formula for Thom polynomials of \(A_d\) singularities in any codimension. We use a combination of the test-curve model of Porteous, and the localization methods in equivariant cohomology. Our formulas are independent of the codimension, and are computationally effective up to \(d=6\). contact singularities; equivariant localization; Thom polynomials Bérczi, G.; Szenes, A., Thom polynomials of morin singularities, Ann. of Math. (2), 175, 567-629, (2012) Global theory of singularities, Singularities of differentiable mappings in differential topology, Enumerative problems (combinatorial problems) in algebraic geometry Thom polynomials of Morin 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 Let G be a reductive algebraic group defined over an algebraically closed field. Moreover let Q be a parabolic subgroup. At least three proofs of the normality of the Schubert subvarieties in G/Q are known [\textit{H. H. Andersen}, Invent. Math. 79, 611-618 (1985; Zbl 0591.14036), \textit{S. Ramanan} and \textit{A. Ramanathan}, Invent. Math. 79, 217-224 (1985; Zbl 0553.14023), \textit{C. S. Seshadri}, Proc. Bombay Colloquium on vector bundles 1984)]. In the present article it is shown that the normality is an easy consequence of the fact that the Schubert varieties are Frobenius split proved by \textit{V. B. Mehta} and \textit{A. Ramanathan} in Ann. Math., II. Ser. 122, 27-40 (1986; Zbl 0601.14043)] and the following lemma:
Let \(f: Y\to X\) be a proper surjective morphism of irreductible varieties in characteristic p. Suppose that Y is normal, the fibres of f are connected and that X is Frobenius split. Then X is normal. normality of the Schubert subvarieties in homogeneous space; Schubert varieties are Frobenius split V. B. Mehta and V. Srinivas, Normality of Schubert varieties, Amer. J. Math. 109 (1987), 987--989. JSTOR: Grassmannians, Schubert varieties, flag manifolds, Homogeneous spaces and generalizations Normality 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 \(M(l\times m)\) be the affine space of matrices over a field with \(l\) rows and \(m\) columns. Denote by \(V\) the algebraic set consisting of those pairs \((A_1, A_2)\) in \(M(n_2\times n_1) \times M(n_3\times n_2)\) such that \(A_2A_1=0\). Then each irreducible component of \(V\) is isomorphic to the opposite cell in a Schubert variety \(SL(n_1+n_2+n_3)/Q\) where \(Q\) is a parabolic group. In this article the singular locus of each irreducible component of \(V\) is determined. A conjecture of Lakshmibai and Sandhya on how to write the singular locus of the associated Schubert variety \(X(\nu) =\overline{B\nu B} \pmod B\) of \(C\) as a union of varieties \(X(\lambda)\) is also proved. singular loci; varieties of complexes; Schubert varieties V. Lakshmibai. Singular loci of varieties of complexes. \textit{J. Pure Appl. Algebra} 152 (2000), 217--230. Grassmannians, Schubert varieties, flag manifolds, Linear algebraic groups over arbitrary fields Singular loci of varieties of complexes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a connected, simply-connected, semisimple algebraic group defined over an algebraically closed field \(k\) of characteristic 0, \(B\subset G\) a fixed Borel subgroup, \(T\subset B\) a maximal torus and \(W = N_G(T)/T\) the Weyl group of \(G\). Consider the complete flag variety \(G/B\). The fixed points of the action from the left of \(T\) on \(G/B\) are \(e_w := wB\), \(w\in W\). The Schubert variety \(X_w\) is defined as the Zariski closure of \(Be_w\) in \(G/B\). If \(B\) acts from the right (resp., left) on a variety \(X\) (resp., \(Y\)) then \(B\) acts from the right on the product \(X\times Y\) by : \((x,y)b := (xb,b^{-1}y)\) and one defines \(X{\times}^BY\) as \(X\times Y/B\). If \(V\) is a rational representation of \(B\) one gets a vector bundle \({\mathcal L}(V):=G{\times}^BV\) over \(G/B\). In particular, if \(\lambda \in X^{\ast}(B)\simeq X^{\ast}(T)\) is a character and if \(k_{\lambda}\) is the representation associated to \(\lambda \) then \({\mathcal L}(\lambda ):={\mathcal L}(k_{\lambda})\) is a line bundle on \(G/B\).
In this paper, the author gives explicitly the vanishing and non-vanishing of the cohomology modules \(\text{H}^i(X_w,{\mathcal L}(\lambda ))\) for \(w\in W\) and ``most'' \(\lambda \in X^{\ast}(T)\) when \(G\) has rank 2 (i.e., it is of type \(A_2\), \(B_2\), or \(G_2\)). For his computations, the author uses the Bott-Samelson-Demazure-Hansen desingularizations of \(X_w\), the fact that the cohomology of \({\mathcal L}(\lambda )\) on \(X_w\) is the same as the cohomology of its pull-back on the BSDH desingularisation, the realization of the BSDH desingularisations as towers of \({\mathbb P}^1\)-bundles starting with \({\mathbb P}^1\) and the Leray spectral sequence.
The author's explicit tables reveal nice patterns. For example, the author conjectures that, for \(G\) of arbitrary rank, if \(\text{H}^m(X_w,{\mathcal L}(\lambda ))\neq 0\) and \(\text{H}^n(X_w,{\mathcal L}(\lambda ))\neq 0\) for some \(m\leq n\) then \(\text{H}^i(X_w,{\mathcal L}(\lambda ))\neq 0\) for all \(i\) with \(m\leq i\leq n\). Schubert variety; cohomology of line bundles; semisimple algebraic group; root system Paramasamy, K.: Cohomology of line bundles on Schubert varieties: the rank two case, Proc. indian acad. Sci. 114, No. 4, 345-363 (2004) Grassmannians, Schubert varieties, flag manifolds, Divisors, linear systems, invertible sheaves, Classical groups (algebro-geometric aspects) Cohomology of line bundles on Schubert varieties: The rank two case | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials This paper deals with the problem of conjugacy of Cartan subalgebras for affine Kac-Moody Lie algebras. Unlike the methods used by Peterson and Kac, our approach is entirely cohomological and geometric. It is deeply rooted on the theory of reductive group schemes developed by Demazure and Grothendieck, and on the work of J. Tits on buildings. affine Kac-Moody Lie algebra; conjugacy; reductive group scheme; torsor; Laurent polynomials; non-abelian cohomology Chernousov, V.; Gille, P.; Pianzola, A.; Yahorau, U.: A cohomological proof of Peterson-Kac's theorem on conjugacy of Cartan subalgebras for affine Kac-Moody Lie algebras. J. algebra 399, 55-78 (2014) Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras, Galois cohomology of linear algebraic groups, Group schemes, Coverings in algebraic geometry A cohomological proof of Peterson-Kac's theorem on conjugacy of Cartan subalgebras for affine Kac-Moody Lie algebras | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The article under review presents a comprehensive study of preorderings on matrix rings of the form \(M_n(R)\), where \(R\) is an arbitrary unital commutative ring, with special emphasis given to matrix polynomial rings \(M_n({\mathbb R}[X_1,\ldots,X_g])\).
Given a set of symmetric matrices \(G\subseteq M_n(R)\), the author defines the preordering \(T_G\) generated by \(G\) to be the smallest quadratic module containing \(G\), whose scalar elements are closed under multiplication. This allows the author to give abstract intersection theorems which are then used to give an Artin-Lang homomorphism theorem for matrix polynomials, and various Positivstellensätze. The latter unify several important existing results, e.g. \textit{J. L. Krivine}'s [J. Anal. Math. 12, 307--326 (1964; Zbl 0134.03902)] and the Gondard-Ribenboim matrix version [\textit{D. Gondard} and \textit{P. Ribenboim}, Bull. Sci. Math., II. Ser. 98, 49--56 (1974; Zbl 0298.12104)] of Hilbert's 17th problem.
This carefully written paper contains many instructive examples illustrating the theory developed. matrix polynomials; positive polynomials; preordering; sum of squares; Positivstellensatz Cimprič, J., Real algebraic geometry for matrices over commutative rings, J. Algebra, 359, 89-103, (2012) Real algebra, Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones), Semialgebraic sets and related spaces Real algebraic geometry for matrices over commutative rings | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We introduce a new perspective on the \(K\)-theory of exact categories via the notion of a \textit{CGW-category}. CGW-categories are a generalization of exact categories that admit a Quillen \(Q\)-construction, but which also include examples such as finite sets and varieties. By analyzing Quillen's proofs of dévissage and localization we define \textit{ACGW-categories}, an analogous generalization of abelian categories for which we prove theorems akin to dévissage and localization. In particular, although the category of varieties is not quite ACGW, the category of reduced schemes of finite type is; applying dévissage and localization allows us to calculate a filtration on the \(K\)-theory of schemes of finite type. As an application of this theory we construct a comparison map showing that the two authors' definitions of the Grothendieck spectrum of varieties are equivalent. Algebraic K-Theory, Dévissage, Localization, Grothendieck ring of varieties, motives Dévissage and localization for the Grothendieck spectrum of 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 main aim of the present article is to define and study polynomials that the authors propose as type \(B\), \(C\) and \(D\) double Schubert polynomials. For the general linear group the corresponding objects are the double Schubert polynomials of Lascoux and Schützenberger. These type \(A\) polynomials possess a series of remarkable properties and the authors propose a theory with as many of the analogous properties as possible. They succeed in obtaining several properties which are desirable both from the geometric and combinatorial points of view.
When restricted to maximal Grassmannian elements of the Weyl group, the single versions of the polynomials are the \(\widetilde P\)- and \(\widetilde Q\)-polynomials of \textit{P. Pragacz} and \textit{J. Ratajski} [J. Reine Angew. Math. 476, 143--189 (1996; Zbl 0847.14029)]. The latter polynomials play, in some sense, the role in types \(B\), \(C\) and \(D\) analogous to that of Schur's \(S\)-functions in type \(A\). The utility of the \(\widetilde P\)- and \(\widetilde Q\)-polynomials in the description of Schubert calculus and degeneracy loci was studied by \textit{P. Pragacz} and \textit{J. Ratajski} [Compos. Math. 107, No. 1, 11--87 (1997; Zbl 0916.14026)], and according to the authors [see Compos. Math. 140, No. 2, 482--500 (2004; Zbl 1077.14083) and J. Reine Angew. Math. 516, 207--223 (1999; Zbl 0934.14018)] the multiplication of \(\widetilde Q\)-polynomials describes both the arithmetic and quantum Schubert calculus on the Lagrangian Grassmannian. Thus the double Schubert polynomials in the present article are closely related to natural families of representing polynomials.
In many cases the authors obtain an analogue of the determinantal formula for Schubert cycles in Grassmannians and they answer a question of \textit{W. Fulton} and \textit{P. Pragacz} [Schubert varieties and degeneracy loci. Lect. Notes Math. 1689 (1998; Zbl 0913.14016)]. The formulas generalize those obtained by Pragacz and Ratajski [loc. cit.].
The main ingredients in the proofs are the geometric work of \textit{W. Fulton} [Duke Math. J. 65, No. 3, 381--420 (1992; Zbl 0788.14044) and Isr. Math. Conf. Proc. 9, 241--262 (1996; Zbl 0862.14032)] and \textit{W. Graham} [J. Differ. Geom. 45, 471--487 (1997; Zbl 0935.14015)] and the algebraic tools developed by \textit{A. Lascoux} and \textit{P. Pragacz} [Adv. Math. 140, No. 1, 1--43 (1998; Zbl 0951.14035) and Mich. Math. J. 48, Spec. Vol., 417--441 (2000; Zbl 1003.05106)]. determinantal formula; Weyl groups; Grassmannians; Lagrangian; Schubert cycles; Chern classes A. Kresch and H. Tamvakis, ''Double Schubert Polynomials and Degeneracy Loci for the Classical Groups,'' Ann. Inst. Fourier 52(6), 1681--1727 (2002). Classical problems, Schubert calculus, Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Double Schubert polynomials and degeneracy loci for the classical 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 This comprehensive memoir provides a profound contribution to the currently developing topic of homotopical and higher categorical structures in algebraic geometry. Its main purpose is to generalize the concept of affine schemes to an adequate construction in homotopical algebraic geometry, namely to the concept of affine stacks, and to show how these new objects can be used to treat several questions in rational and \(p\)-adic homotopy theory from a novel point of view.
One of the motivating ideas of the present work can be traced back to A. Grothendieck's monumental, visionary manuscript ``Pursuing Stacks'' (unpublished) from the early 1980s. In this famous program, Grothendieck sketched a problem which he called the ``schematization problem for homotopy types'' stating that for any affine scheme \(\text{Spec\,}A\) there should be an appropriate notion of ``\(\infty\)-stack in groupoids over \(\text{Spec\,}A\)'' generalizing all sheaves and stacks in groupoids over the grand fpqc-site of \(\text{Spec\,}A\). Later on, it was shown by \textit{A. Joyal} (Letter to Grothendieck, 1984, unpublished) and by \textit{J. F. Jardine} [Homology Homotopy Appl. 3, No. 2, 361--384 (2001; Zbl 0995.18006)] that an adequate model for a theory of \(\infty\)-stacks in groupoids would be given by the theory of simplicial presheaves. According to this fundamental insight, the word ``stack'' in the present memoir, is used for an object in the homotopical category of simplicial sheaves (à la A. Joyal and J. F. Jardine).
After a thorough introduction to the contents of the present treatise, including a historical sketch of the developments leading to its subject, explanations of the basic conceptual framework, and an, outline of the links to related works by other researchers in the field. Section 1 recalls the fundamentals from the theory of simplicial presheaves on a Grothendieck site, that is from the theory of general stacks in the sense made precise above. Apart from an introduction to the basic definitions and results, homotopic limits, Postnikov decompositions, the cohomology of simplicial presheaves, and schemes in affine groups are the main topics of this preparatory section.
Section 2 introduces the first of the two novel fundamental notions of the memoir under review: affine stacks. These objects appear as a homotopic version of ordinary affine schemes, obtained from a model category with simplicial structure over the category of co-simplicial \(A\)-algebras. This model category is then used to define a derived functor of the Spec-functor, which in turn induces a certain functor from the homotopical category of co-simplicial \(A\)-algebras to the homotopical category of simplicial sheaves over the site \((\text{Aff}/A)_{\text{fpgc}}\). The category of affine stacks is then defined to be the essential image category of the latter functor. In the sequel, it is proved that the category of affine stacks is equivalent to the opposite category of the homotopical category of co-simplicial \(A\)-algebras, thereby generalizing the analoguous property of the category of ordinary affine schemes.
Finally, the author invents another important construction, namely that of the ``affinization of a simplicial presheaf'', which appears to be significant for the study of homotopy sheaves, and he gives a concrete criterion for the existence of affinizations. In the special case of a base scheme \(\text{Spec\,}k\), where \(k\) is a field, the affine stacks are completely characterized, and analogues of the standard theorems on rational and \(p\)-adic homotopy of algebraic varieties are deduced by using affine stacks.
Section 3 deals with the second crucial novelty provided by the author's work. More precisely, he introduces the notion of so-called ``affine \(\infty\)-gerbes'' and the concept of ``schematic homotopy type''. The underlying idea is to use affine stacks in order to define a homotopical version of affine gerbes in the Tannakian formalism (à la P. Deligne). This is done by glueing affine stacks to obtain so-called ``\(\infty\)-geometric stacks'', which may be seen as a generalization of ordinary algebraic stacks assed in algebraic moduli theory and non-abelian Hodge theory (à la C. Simpson). This framework is then applied to give two different solutions to A. Grothendieck's ``schematization problem for homotopy types of algebraic varieties'' mentioned above. In this context, homotopy types with respect to various cohomology theories (Betti, de Rham, crystalline, \(\ell\)-adic, etc.) are described in greater detail.
The concluding Section 4 is exclusively devoted to the study of ``\(\infty\)-geometric stacks'' over a ground field \(k\), with applications to the construction of analogues of some moduli stacks in this extended homotopical context.
In an appendix of the present memoir, the author recalls A. Grothendieck's ``schematization problem for homotopy types'', together with his interpretation and reflection of Grothendieck's vague sketches.
Without any doubt, this memoir is of utmost fundamental and propelling character with regard to further developments in homotopical algebraic geometry. A wealth of significant new concepts, methods, and applications is presented in a very detailed and lucid manner, and an important new point of view towards Grothendieck's program of pursuing stacks is strikingly exhibited. homotopical algebraic geometry; gerbes; Grothendieck topologies; cohomology theories; simplicial sheaves; model categories Toën, Bertrand, Champs affines, Selecta Math. (N.S.), 1022-1824, 12, 1, 39-135, (2006) Generalizations (algebraic spaces, stacks), Classical real and complex (co)homology in algebraic geometry, Homotopy theory and fundamental groups in algebraic geometry, Étale and other Grothendieck topologies and (co)homologies, Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects), Grothendieck topologies and Grothendieck topoi, Topoi Affine stacks | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Consider the moduli space \({\mathcal M}(\sigma;n)\) of plane curves with a given degree \(n\) and having prescribed set of finite singularities \(\sigma\). Let \(C\), \(C'\in{\mathcal M}\). The pair of curves \((C,C')\) is called a Zariski pair if the pairs of spaces \((\mathbb{P}^2,C)\) and \((\mathbb{P}^2,C')\) are not homeomorphic. A Zariski pair \((C,C')\) is called Alexander-equivalent if their generic Alexander polynomials coincide. The first example of an Alexander-equivalent Zariski pair \((C,C')\) for irreducible plane curves was given by the author in an earlier paper [Geom. Dedicata 75, 67--100 (1999; Zbl 0952.14020)]. They are plane curves of degree 12 with 27 cusps. Here \(C\) is a generic \((3,3)\)-covering of a three cuspidal quartic and \(C'\) is constructed using a six cuspidal non-conical sextic. The purpose of this note is to construct an Alexander-equivalent Zariski pair \((D,D')\) of irreducible curves of degree 8 with 12 cusps. Alexander polynomials; plane curves M Oka, A new Alexander-equivalent Zariski pair, Acta Math. Vietnam. 27 (2002) 349 Coverings of curves, fundamental group, Singularities of curves, local rings A new Alexander-equivalent Zariski pair. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(f = (f_1, \dots, f_n)\) be a system of \(n\) complex homogeneous polynomials in \(n\) variables of degree \(d\). We call \(\lambda \in \mathbb{C}\) an eigenvalue of \(f\) if there exists \(v \in \mathbb{C}^n \backslash \{ 0 \}\) with \(f(v) = \lambda v\), generalizing the case of eigenvalues of matrices (\(d = 1\)). We derive the distribution of \(\lambda\) when the \(f_i\) are independently chosen at random according to the unitary invariant Weyl distribution and determine the limit distribution for \(n \to \infty\). tensors; eigenvalues; eigenvalue distribution; random polynomials; computational algebraic geometry P. Breiding and P. Bürgisser, \textit{Distribution of the eigenvalues of a random system of homogeneous polynomials}, Linear Algebra Appl., 497 (2016), pp. 88--107, . Eigenvalues, singular values, and eigenvectors, Computational aspects in algebraic geometry, Geometric probability and stochastic geometry, Polynomials and rational functions of one complex variable, Other generalizations of function theory of one complex variable, Multilinear algebra, tensor calculus Distribution of the eigenvalues of a random system of homogeneous polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials In the first part of this paper [J. Am. Math. Soc. 12, No. 4, 1055--1090 (1999; Zbl 0944.05097)] the authors studied the cone \(\text{BDRY}(n)\) which is the set of triples of weakly decreasing \(n\)-tuples \((\lambda,\mu,\nu)\in ({\mathbb R}^n)^3\) satisfying the three conditions (1) regarding \(\lambda,\mu,\nu\) as spectra of \(n\times n\) Hermitian matrices, there exist three Hermitian matrices with those spectra whose sum is the zero matrix; (2) regarding \(\lambda,\mu,\nu\) as dominant weights of \(\text{GL}_n({\mathbb C})\), the tensor product \(V_{\lambda}\otimes V_{\mu}\otimes V_{\nu}\) of the corresponding irreducible modules contains a \(\text{GL}_n({\mathbb C})\)-invariant vector; (3) regarding \(\lambda,\mu,\nu\) as possible boundary data on a honeycomb, there exist ways to complete it to a honeycomb.
These conditions were proved to be equivalent. A sufficient list of inequalities for this cone was given due to the efforts of several authors: Klyachko, Helmke and Rosenthal, Totaro, and Belkale in terms of Schubert calculus on Grassmannians. In the present, second, part of the paper the authors introduce new combinatorial objects called puzzles, which are certain kinds of diagrams in the triangular lattice in the plane, composed from unit equilateral triangles and unit rhombi, with edges labeled by 0 and 1. Puzzles are used to compute Grassmannian Schubert calculus, and have much interest in their own right. In particular, the authors get new, puzzle-theoretic, proofs of results of Horn and the above-mentioned authors.
The authors also characterize ``rigid'' puzzles and use them to prove a conjecture of Fulton which states that if the irreducible module \(V_\nu\) appears exactly once in \(V_\lambda \otimes V_\mu\), then for all \(N\in{\mathbb N}\), \(V_{N\lambda}\) appears exactly once in \(V_{N\lambda}\otimes V_{N\mu}\). honeycombs; symmetric functions; Littlewood-Richardson rule; puzzles; Hermitian matrix; eigenvalue problems; Schubert calculus; Grassmannian A. Knutson, T. Tao and C. Woodward, The honeycomb model of GLn tensor products II: Puzzles determine facets of the Littlewood-Richardson cone. \textit{Journal of the American Mathematical Society }17 (2004), 19--48. arXiv:math/0107011.Zbl 1043.05111 MR 2015329 Symmetric functions and generalizations, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Grassmannians, Schubert varieties, flag manifolds, Classical problems, Schubert calculus, Inequalities involving eigenvalues and eigenvectors, Representation theory for linear algebraic groups, Special polytopes (linear programming, centrally symmetric, etc.), Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Combinatorial aspects of representation theory The honeycomb model of \(\text{GL}_n({\mathbb C})\) tensor products. II: Puzzles determine facets of the Little\-wood-Richardson 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 The authors generalize the notion of ladder determinantal varieties, which was introduced by Abhyankar, by allowing ideals of minors of different size of a matrix of indeterminates. Then they explore the relation between these mixed ladder determinantal varieties and Schubert varieties. Next they show that, up to product by affine spaces, each of these varieties is a basic open set in a classical ladder determinantal variety and that it contains as a basic open set another classical ladder determinantal variety. mixed ladder determinantal varieties; Schubert varieties Gonciulea, N.; Miller, C., Mixed ladder determinantal varieties, \textit{J. Algebra}, 231, 1, 104-137, (2000) Determinantal varieties, Grassmannians, Schubert varieties, flag manifolds Mixed ladder determinantal 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 classical Borel-Weil-Bott theorem describes the cohomology of line bundles over flag varieties. In this paper this theorem is generalized for the case of \textit{wonderful varieties of minimal rank}.
Wonderful varieties were introduced by [\textit{C. De Concini} and \textit{C. Procesi}, Complete symmetric varieties. Invariant theory, Proc. 1st 1982 Sess. C.I.M.E., Montecatini/Italy, Lect. Notes Math. 966, 1--44 (1983; Zbl 0581.14041)]; later \textit{D. Luna} [Transform. Groups 1, No. 3, 249--258 (1996; Zbl 0912.14017)] showed that they all are spherical. These are embeddings of a \(G\)-homogeneous space \(G/H\), where \(G\) is a reductive algebraic group, into a smooth complete \(G\)-variety \(X\) satisfying a number of good properties on the boundary \(X-(G/H)\) (in particular, the boundary should have codimension 1). The degree \(r\) of the boundary divisor is called the \textit{rank} of \(X\). For example, wonderful varieties of rank 0 are exactly the partial flag varieties \(G/P\). There is always an inequality
\[
r\geq \mathrm{rk} G -\mathrm{rk}H,
\]
where rank of an algebraic group is the dimension of its maximal torus. A wonderful variety is said to be \textit{of minimal rank} if this inequality turns into an equality.
In this paper A. Tchoudjem computes the cohomology groups of invertible sheaves on such varieties. The answer is formulated in terms of spherical data for the variety \(X\).
The technique used for this computation is as follows: the author considers a Bialynicki-Birula decomposition of \(X\), then for a given invertible sheaf computes the groups of cohomology with support given by the cells of this decomposition, and finally passes from the cohomology with support to the usual cohomology using the Grothendieck-Cousin complex. Borel-Weil-Bott theorem; cohomology with support; spherical varieties; wonderful varieties; flag varieties; Grothendieck-Cousin complex; cohomology of line bundles; Verma modules Tchoudjem, A., Cohomologie des fibrés en droites sur les variétés magnifiques de rang minimal, Bull. Soc. Math. France, 135, 2, 171-214, (2007) Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials, Compactifications; symmetric and spherical varieties, Vanishing theorems in algebraic geometry, Other algebraic groups (geometric aspects), Grassmannians, Schubert varieties, flag manifolds Cohomology of line bundles over wonderful varieties of minimal rank | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 total Chern (\(c(E)\)), Todd (\(\mathrm{td}(E)\)) and Thom-Hirzebruch L-class (\(L(E)\)) of a holomorphic vector bundle \(E\) live on the cohomology ring of its base space. These three characteristic classes are recovered from the \textit{Hirzebruch cohomology class} \(T_y(E):=\prod\limits_{i=1}^{\operatorname{rank} E}\left(\frac{\alpha_i}{1-e^{-\alpha_i(1+y)}}-\alpha_iy\right)\) for \(y=-1,0,1\) respectively, where \(\alpha_1,\ldots,\alpha_{\operatorname{rank} E}\) are the Chern roots of \(E\). \(T_y\) was initially defined by Hirzebruch to prove the generalized Hirzebruch-Riemann-Roch theorem, which recovers the Gauss-Bonnet, the Hirzebruch-Riemann-Roch and the Hirzebruch's signature theorem for \(y=-1,0,1\) respectively.
Given a smooth algebraic variety, the Chern, Todd and L-classes of the variety are defined as the Chern, Todd and L-classes of its tangent bundle. These have \textit{homological} generalizations to singular varieties (which do not have a tangent bundle), namely the Chern-Schwartz-MacPherson class [\textit{R. D. MacPherson}, Ann. Math. (2) 100, 423--432 (1974; Zbl 0311.14001)], Baum-Fulton-MacPherson's Todd class [\textit{P. Baum} et al., Publ. Math., Inst. Hautes Étud. Sci. 45, 101--145 (1975; Zbl 0332.14003)] and Cappell-Shaneson's L-class [\textit{S. E. Cappell} and \textit{J. L. Shaneson}, J. Am. Math. Soc. 4, No. 3, 521--551 (1991; Zbl 0746.32016)], which are Poincaré dual to the corresponding cohomology classes in the smooth case. More precisely, these have a \textit{natural transformation} description, and they are characterized as the unique natural transformations \[ \begin{array}{c} c_*:F(-)\rightarrow H_*(-)\\
\mathrm{td}_*:G_0(-)\rightarrow H_*(-)\otimes\mathbb{Q}\\
L_*: \Omega(-)\rightarrow H_*(-)\otimes\mathbb{Q}, \end{array} \] such that \(c_*(\mathbb{1}_X)=c(TX)\cap[X]\) and \(\mathrm{td}_*(\mathcal{O}_X)=\mathrm{td}(TX)\cap [X]\) for every smooth variety \(X\), and \(L_*([\mathbb{Q}_X[\dim X]])=L(TX)\cap[X]\) for every smooth compact variety \(X\), where
\begin{itemize}
\item \(F(X)\) is the set of constructible functions on \(X\),
\item \(G_0(X)\) is the Grothendieck group of coherent sheaves on \(X\),
\item \(\Omega(X)\) is the group of self dual constructible complexes of sheaves on \(X\), and
\item \(H_*(-)\) is the Borel-Moore homology functor.
\end{itemize}
This paper surveys different developments related to the unification of the natural transformations \(c_*,\mathrm{td}_*\) and \(L_*\). That is, it discusses defining and studying a natural transformation homology analogue of the Hirzebruch cohomology class.
Let \(\mathcal{V}\) be the category of complex algebraic varieties, and let \(K_0(\mathcal{V}/X)\) be the Grothendieck group of complex algebraic varieties over the variety \(X\). \(K_0(\mathcal{V}/-)\) provides the appropriate framework for the desired unification as follows:
\underline{Theorem}: (Theorems 4.2 and 5.4 in the survey under review, [\textit{J.-P. Brasselet} et al., J. Topol. Anal. 2, No. 1, 1--55 (2010; Zbl 1190.14009)])
\begin{enumerate}
\item There exists a unique natural transformation \[T_{y*}:K_0(\mathcal{V}/-)\rightarrow H_*(-)\otimes\mathbb{Q}[y]\] such that \(T_{y*}([X\xrightarrow{\mathrm{id}_X} X])\) is the Poincaré-dual of \(T_y(TX)\) for every smooth variety \(X\).
\item There exist natural transformations \(\mathrm{const}\), \(coh\) and \(sd\) from \(K_0(\mathcal{V}/-)\) to \(F(-)\), \(G_0(-)\) and \(\Omega(-)\) respectively such that \(T_{-1*}=\left(c_*\otimes\mathbb{Q}\right)\circ \mathrm{const}\), \(T_{0*}=\mathrm{td}_*\circ coh\), and \(T_{1*}=L_*\circ sd\).
\end{enumerate}
\(T_{y*}\) is called the \textit{motivic Hirzebruch class}, and \(T_{y*}(X):=T_{y*}([X\xrightarrow{\mathrm{id}_X} X])\) is called the \textit{motivic Hirzebruch class of \(X\)} for every algebraic variety \(X\).
The paper also surveys different developments related to \(T_{y*}(X)\), such as a formula for its zeta function (Theorem 6.2 in the survey under review, [\textit{S. E. Cappell} et al., J. Reine Angew. Math. 728, 35--63 (2017; Zbl 1400.14024); \textit{S. Cappell} et al., Geom. Topol. 17, No. 2, 1165--1198 (2013; Zbl 1318.14008); \textit{S. Yokura}, IRMA Lect. Math. Theor. Phys. 20, 285--343 (2012; Zbl 1317.14016)]), as well as bivariant-theoretic (Theorem 7.3 in the survey under review, [\textit{J. Schürmann} and \textit{S. Yokura}, Adv. Math. 250, 611--649 (2014; Zbl 1295.14005)]) and Donaldson-Thomas type invariant versions of it. motivic characteristic classes; singular space; Hirzebruch class; Grothendieck group; bivariant theory; genus; Riemann-Roch; Donaldson-Thomas invariants Motivic cohomology; motivic homotopy theory, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry, Riemann-Roch theorems, Topological properties in algebraic geometry, Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Mixed Hodge theory of singular varieties (complex-analytic aspects), Bordism and cobordism theories and formal group laws in algebraic topology Topics of motivic characteristic classes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(S_n\) denote the symmetric group on \(n\) symbols and \(V\) the natural permutation module with dimension \(n\). Then for any integer \(m\), \(S_n\) acts on \(V^{\oplus m}\) (the direct sum of \(m\) copies of \(V\)) by simultaneous permutation. The ring of multisymmetric polynomials is the ring of vector invariants \(R_{n,m}:= \mathbb{C}[V^{\oplus m}]^{S_n}\). For arbitrary \(m\) and \(n\), it is known that the polarisations of elementary symmetric polynomials or power sums give a minimal system of \(\mathbb{C}\)-algebra generators for \(R_{n,m}\), but the relations between these are not so well understood.
There are two main results in this article. Firstly, an explicit relation of degree \(2n\), which is never contained in the ideal of relations generated by lower degree elements, is described. Thus, the ideal of relations cannot be generated in degree less than \(2n\).
In the special case \(n=3\), the authors show that (for \(m \geq 3\)) the polarisations of of three relations minimally generate the ideal of relations. These are the relation of degree \(2n = 6\) given above, along with a relation of degree 5 and a relation of degree 6, all explicitly stated. For \(m=2\) the second two are sufficient.
The main tool used here is the representation theory of \(\mathrm{GL}_m(\mathbb{C})\). There is an action of \(\mathrm{GL}_n(\mathbb{C})\) on \(V^{\oplus m}\) which commutes with the action of \(S_n\), and so \(R_{n,m}\) can be decomposed into irreducible \(\mathrm{GL}_m(\mathbb{C})\)-modules. The authors point out a connection between indecomposable relations and highest weight vectors of \(\mathrm{GL}_m(\mathbb{C})\)-modules. In the case \(n=3\), classical results are used to establish that no indecomposable relations connected to modules associated with partitions with more than four parts can occur, thereby reducing the theorem to the cases \(m=2,3,4\). These cases are then worked out in detail. multisymmetric polynomials; ideal of relations; highest weight vectors 11.Domokos, M., Puskás, A.: Multisymmetric polynomials in dimension three. J. Algebra 356, 283-303 (2012) Actions of groups on commutative rings; invariant theory, Group actions on varieties or schemes (quotients), Representation theory for linear algebraic groups Multisymmetric polynomials in dimension 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 The intermediate Jacobians of Fano threefolds have turned out to be highly interesting objects of study in algebraic geometry. Especially the geometry of their theta divisors is of greatest significance, above all in view of various concrete applications and constructions and the detailed study of manageable examples has been a popular topic of research in complex algebraic geometry over the past thirty years. One of the most beautiful examples, in this regard, is the so-called quartic double solid, i.e., a double cover of \(\mathbb{P}^3\) ramified in a quartic surface \(W\subset \mathbb{P}^3\). Since 1981, when \textit{G. Welters} [Abel-Jacobi isogenies for certain types of Fano threefolds, Math. Centre Tracts, 141, Mathematisch Centrum, Amsterdam (1981; Zbl 0474.14028)] exhibited a family of curves parametrizing the intermediate Jacqbian \(J(X)\) of a quartic double solid \(X\), many authors have contributed to a refined study of this example. In the paper under review, the authors give an improvement of previous results concerning the parametrization of the theta divisor \(\Theta\) of \(J(X)\), together with important applications to the study of certain moduli spaces of stable vector bundles on Fano threefolds. In 2003, \textit{A. S. Tikhomirov} [Acta Appl. Math. 75, 271--279 (2003; Zbl 1075.14044)] found a simple parametrization of that theta divisor \(\Theta\) by the family \(C^1_5(X)\) of elliptic quintics in \(X\). The advantage of this parametrization is that the elliptic quintics define stable vector bundles on \(X\) (via the Serre construction), by which the associated Abel-Jacobi map \(C_5^1(X)\to J(X)\) admits a factorization through the moduli space \(M(2;0,3)\) of the quartic double solid \(X\). The paper under review provides a detailed account of this previous result of Tikhomirov, whose proof was only sketched in the foregoing paper, and it even gives a substantial refinement of the precise statement. Namely, the main result of the present paper establishes the fact that the Abel-Jacobi map sends the family \(C^1_5(X)\) of elliptic quintics in the quartic double solid \(X\) onto an open subset of a translate of the theta divisor \(\Theta\) in \(J(X)\), and that the Serre construction defines a factorization
\[
C_5^1(X) @>\text{Serre}>> M@>g>> \Theta + \text{const}
\]
through a component \(M\) of the moduli space \(M(2;0,3)\) such that the map \(g\) is generically finite of degree 84. Moreover, it is shown that the family \(C_5^1(X)\) is indeed irreducible (within the corresponding Hilbert scheme). The fine analysis carried out in this beautiful, extremely comprehensive and lucid paper also demonstrates the enormous power of G. Welters's methods developed more than twenty years ago, and the fascinating appeal of concrete algebraic geometry likewise. intermediate Jacobians; Fano varieties; threefolds; theta divisors; families of curves; quartic double solids; moduli spaces of stable vector bundles; Abel-Jacobi map Markushevich, D.G., Tikhomirov, A.S.: A parametrization of the theta divisor of the quartic double solid. Int. Math. Res. Not. \textbf{2003}(51), 2747-2778 (2003) Picard schemes, higher Jacobians, \(3\)-folds, Fano varieties, Vector bundles on surfaces and higher-dimensional varieties, and their moduli, Families, moduli of curves (algebraic), Divisors, linear systems, invertible sheaves A parametrization of the theta divisor of the 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 Let \(G\) be a complex semisimple simply-connected algebraic group and let \(P\) be a parabolic subgroup. Consider the flag variety \(X_:=G/P\) and the Schubert subvarieties \(X_w:= \overline{BwP/P}\subset G/P\) for any \(w\in W/W_P\), where \(W\) is the Weyl group of \(G\), \(W_P\) is the Weyl group of \(P\) and \(B\) is a Borel subgroup of \(G\) contained in \(P\). Then, the classes \([{\mathcal O}_{X_w}]\) of structure sheaves of \(X_w\) form a \(\mathbb{Z}\)-basis of the \(K\)-group \(K(X)\) of \(X\). Write, under the product in \(K(X)\), for any \(u,\,v\in W/W_P\):
\[
[{\mathcal O}_{X_u}]= \sum_{w\in W/W_P} c^w_{u,v}[{\mathcal O}_{X_w}],
\]
for some \(c^w_{u,v}\in \mathbb{Z}\).
Then, the main result of the paper under review asserts that
\[
c^w_{u,v}(-1)^{\text{codim\,}X_u+ \text{codim\,}X_v+ \text{codim\,}X_w}\geq 0.
\]
This was conjectured by \textit{A. Buch} [Acta Math. 189, 37--78 (2002; Zbl 1090.14015)] and proved by him for the Grassmannians.
The author, in fact, proves the following more general result asked by \textit{W. Graham} [Duke Math. J. 102, 599--614 (2001; Zbl 1069.14055)]: Let \(Y\subset X\) be a closed subvariety with rational singularities. Express
\[
[{\mathcal O}_Y]= \sum_{w\in W/W_P} c^w_Y[{\mathcal O}_{X_w}],\text{ for }c^w_Y\in \mathbb{Z}.
\]
Then, \((-1)^{\text{codim\,}X_w+ \text{codim\,}Y} c^w_Y\geq 0\). Schubert calculus \beginbarticle \bauthor\binitsM. \bsnmBrion, \batitlePositivity in the Grothendieck group of complex flag varieties, \bjtitleJ. Algebra \bvolume258 (\byear2002), no. \bissue1, page 137-\blpage159. \endbarticle \OrigBibText Michel Brion, Positivity in the Grothendieck group of complex flag varieties , J. Algebra 258 (2002), no. 1, 137-159. \endOrigBibText \bptokstructpyb \endbibitem Grassmannians, Schubert varieties, flag manifolds, Homogeneous spaces and generalizations, Grothendieck groups, \(K\)-theory and commutative rings, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry Positivity in the Grothendieck group of complex flag varieties. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let G be a semi-simple algebraic group over an algebraically closed field of characteristic \(p>1.\) Let T be a maximal torus in G and B a Borel subgroup, \(B\supset T\). Let \(W\) (\(=N(T)/T,\) \(N(T)\) the normalizer of T in G) be the Weyl group of G. For \(w\in W\), let \(X(w)\) \((=\overline{BwB}(\mod B))\) be the Schubert variety in \(G/B\) associated to w. Using Frobenius-splitting Ramanathan proved that \(X(w)\) is Cohen-Macaulay. The authors give a short proof of Ramanthan's result, using two simple lemmas. Cohen-Macaulayness of Schubert variety; characteristic p; Frobenius-splitting Mehta, V.; Srinivas, V.: A note on Schubert varieties in G/B. Math. ann. 284, 1-5 (1989) Grassmannians, Schubert varieties, flag manifolds, Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) A note on Schubert varieties in G/B | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 takes into consideration that ``the saturation theorem of \textit{A. Knutson} and \textit{T. Tao} [J. Am. Math. Soc. 12, No. 4, 1055--1090 (1999; Zbl 0944.05097)] concerns the nonvanishing of Littlewood-Richardson coefficients''. Further, ``in combination with work of \textit{A. A. Klyachko} [Sel. Math., New Ser. 4, No. 3, 419--445 (1998; Zbl 0915.14010)], it implies \textit{A. Horn}'s conjecture [Pac. J. Math. 12, 225--241 (1962; Zbl 0112.01501)] about eigenvalues of sums of Hermitian matrices''. Then ``the common features between these two eigenvalue problems and their connection to Schubert calculus of Grassmannians'' are illustrated. The main result of this paper deals with an extension of Schubert calculus presenting a Schubert calculus interpretation of Friedland's problem, via equivariant cohomology of Grassmannians, and it derives a saturation theorem for this setting. eigenvalue problem; equivariant cohomology; Schubert calculus; Littlewood-Richardson coefficients; Hermitian matrices; Grassmannians Inequalities involving eigenvalues and eigenvectors, Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80], Equivariant algebraic topology of manifolds Eigenvalues of Hermitian matrices and equivariant cohomology of Grassmannians | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We use a theorem of Chow [\textit{W.-L. Chow}, Ann. Math. (2) 50, 32--67 (1949; Zbl 0040.22901)] on line-preserving bijections of Grassmannians to determine the automorphism group of Grassmann codes. Further, we analyze the automorphisms of the big cell of a Grassmannian and then use it to settle an open question of \textit{P. Beelen}, \textit{S. R. Ghorpade} and \textit{T. Høholdt} [Affine Grassmann codes, IEEE Trans. Inf. Theory 56, No. 7, 3166--3176 (2010; Zbl 1365.94579)] concerning the permutation automorphism groups of affine Grassmann codes. Finally, we prove an analogue of Chow's theorem for the case of Schubert divisors in Grassmannians and then use it to determine the automorphism group of linear codes associated to such Schubert divisors. In the course of this work, we also give an alternative short proof of MacWilliams theorem concerning the equivalence of linear codes and a characterization of maximal linear subspaces of Schubert divisors in Grassmannians. Grassmann variety; Schubert divisor; linear code; automorphism group; Grassmann code; affine Grassmann code Ghorpade S.R., Kaipa K.V.: Automorphism groups of Grassmann codes. Finite Fields Appl. \textbf{23}, 80-102 (2013). Linear codes (general theory), Geometric methods (including applications of algebraic geometry) applied to coding theory, Grassmannians, Schubert varieties, flag manifolds, Finite automorphism groups of algebraic, geometric, or combinatorial structures Automorphism groups of Grassmann codes | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials The author calls a tensor category virtually indecomposable, if its Grothendieck ring (considered over \(\mathbb Z\)) has no nontrivial central idempotents, and proves that a number of tensor categories, such as tensor categories with the Chevalley property, representation categories of affine (super)group schemes and of formal (super)groups, and symmetric tensor categories of exponential growth in characteristic zero, are virtually indecomposable. Among other, this answers affirmatively a question of Serre about connectedness of the spectrum of the Grothendieck ring of a Tannakian category. tensor category; Grothendieck ring; J.-P. Serre Monoidal categories (= multiplicative categories) [See also 19D23], Formal groups, \(p\)-divisible groups, Group schemes, Representation theory of associative rings and algebras, Hopf algebras and their applications, Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Groupoids, semigroupoids, semigroups, groups (viewed as categories), Grothendieck groups (category-theoretic aspects), Representation theory of groups Virtually indecomposable tensor categories | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 explores mirror symmetry for del Pezzo surfaces with cyclic quotient singularities and state a number of conjectures that together allow one to classify a broad class of such surfaces. The conjectures relate mutation-equivalence classes of Fano polygons with \(\mathbb Q\)-Gorenstein deformation classes of del Pezzo surfaces. As evidence, the authors show that their conjectures hold true in the smooth case and the case of simplest residual singularity \(\frac{1}{3}(1,1)\). orbifold Del Pezzo surfaces; cyclic quotient singularities; Fano polygons; mutation-equivalence classes; \(\mathbb Q\)-Gorenstein deformation classes; mirror symmetry; mutable Laurent polynomials M. Akhtar, T. Coates, A. Corti, L. Heuberger, A. Kasprzyk, A. Oneto, A. Petracci, T. Prince, K. Tveiten, Mirror symmetry and the classification of orbifold del Pezzo surfaces. \textit{Proc. Amer. Math. Soc}. \textbf{144} (2016), 513-527. MR3430830 Zbl 06548665 Mirror symmetry (algebro-geometric aspects), Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry), Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects), Rational and ruled surfaces, Fano varieties Mirror symmetry and the classification of orbifold 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 Let X be a double cover of \({\mathbb{P}}^ 3\) branched along a sextic surface S. Using a method of Clemens and Letizia, in this paper we show that, for general X, the Abel-Jacobi map associated to the surface F of curves contained in X which are preimages of conics ''totally tangents'' to S, induces an isomorphism between the Albanese variety of F and the intermediate Jacobian of X. threefold; double cover of projective 3-space branched along a sextic surface; Abel-Jacobi map; Albanese variety; intermediate Jacobian \(3\)-folds, Projective techniques in algebraic geometry The Abel-Jacobi isomorphism for the sextic 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 The paper under review is one of a series of papers devoted to the general theory of convex bodies associated to algebraic varieties; see [\textit{K.~Kaveh} and \textit{A. G.~Khovanskii}, Mosc. Math. J. 10, No. 2, 343--375 (2010; Zbl 1287.14001); J. Fixed Point Theory Appl. 7, No. 2, 401--417 (2010; Zbl 1205.14059); Ann. Math. (2) 176, No. 2, 925--978 (2012; Zbl 1270.14022)] and further publications.
Let \(G\) be a complex reductive algebraic group. An algebraic subgroup \(H\) of \(G\) is said to be horospherical if it contains a maximal unipotent subgroup. In this case the homogeneous space \(G/H\) is spherical, i.e., a Borel subgroup \(B\) of \(G\) acts on \(G/H\) with an open orbit. In particular, every irreducible \(G\)-module appears in the algebra of regular functions \(\mathbb{C}[G/H]\) with multiplicity at most one.
The authors describe the Grothendieck semigroup of \(G\)-invariant subspaces of \(\mathbb{C}[G/H]\) as a semigroup of convex polytopes. This leads to a formula for the number of solutions of a system of equations \(f_1=\ldots=f_n=0\) of \(G/H\), where \(n=\dim G/H\) and each \(f_i\) is a generic element from an invariant subspace \(L_i\) of \(\mathbb{C}[G/H]\). The formula is given in terms of the mixed volume of polytopes associated to the \(L_i\). More precisely, it is represented as the mixed integral of an explicitly defined homogeneous polynomial over the so-called moment polytopes of the subspaces \(L_i\). Alternatively, the authors construct larger polytopes over the moment polytopes such that their mixed volume is equal to the above mixed integral. This generalizes the Bernstein- Kushnirenko formula from toric toric geometry. Similar results for the intersection numbers of invariant linear systems on \(G/H\) are obtained. In this context, the Gelfand-Cetlin polytopes are used.
The last section illustrates the obtained results in the case \(G=\text{GL}_n(\mathbb{C})\). reductive group; moment polytope; Newton polytope; horospherical variety; Bernstein-Kushnirenko theorem; Grothendieck group K. Kaveh and A. G. Khovanskii, ''Newton polytopes for horospherical spaces,'' Mosc. Math. J., vol. 11, iss. 2, pp. 265-283, 407, 2011. Homogeneous spaces and generalizations, Toric varieties, Newton polyhedra, Okounkov bodies Newton polytopes for horospherical 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 We investigate the line arrangement that results from intersecting \(d\) complete flags in \(\mathbb{C}^n\). We give a combinatorial description of the matroid \(\mathcal T_{n,d}\) that keeps track of the linear dependence relations among these lines.
We prove that the bases of the matroid \(\mathcal T_{n,3}\) characterize the triangles with holes which can be tiled with unit rhombi. More generally, we provide evidence for a conjectural connection between the matroid \(\mathcal T_{n,d}\), the triangulations of the product of simplices \(\Delta_{n-1}\times\Delta_{d-1}\), and the arrangements of d tropical hyperplanes in tropical \((n-1)\)-space.
Our work provides a simple and effective criterion to ensure the vanishing of many Schubert structure constants in the flag manifold, and a new perspective on Billey and Vakil's method for computing the non-vanishing ones [cf. in: Algorithms in algebraic geometry. Based on the workshop, Minneapolis, MN, USA, September 18--22, 2006. New York, NY: Springer. The IMA Volumes in Mathematics and its Applications 146, 21--54 (2008; Zbl 1132.14044)]. Matroids; permutation arrays; Schubert calculus; tropical hyperplane arrangements; Littlewood-Richardson coefficients Ardila, F.; Billey, S., \textit{flag arrangements and triangulations of products of simplices}, Adv. Math., 214, 495-524, (2007) Classical problems, Schubert calculus, Relations with arrangements of hyperplanes, , Arrangements of points, flats, hyperplanes (aspects of discrete geometry) Flag arrangements and triangulations of products of simplices | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 generalise results of Deligne and Laumon ([\textit{G. Laumon}, Astérisque 82--83, 173--219 (1981; Zbl 0504.14013)], 2.1.1) for relative smooth and separated curves to the case of smooth and separated morphisms of finite type. The main theorem is:
Theorem 4.3. Let \(k\) be an algebraically closed field of characteristic \(p > 0\), \(S\) be a \(k\)-scheme of finite type, \(f: X \to S\) a separated and smooth \(k\)-morphism of finite type, \(D\) an effective Cartier divisor on \(X\) relative to \(S\), \(U = X \setminus D\), \(j: U \hookrightarrow X\) the canonical open immersion and \(\mathcal{F}\) a locally constant and constructible sheaf of \(\mathbb{F}_\ell\)-modules on \(U\). Let \(\{S_\alpha\}_{\alpha \in J}\) be the set of irreducible components of \(S\), and \(f_\alpha: X_\alpha := X \times_S S_\alpha \to S_\alpha\), assume that for all \(\alpha \in J\) the base change \(D_\alpha:= D \times_S S_\alpha\) is the sum of effective divisors \(D_{\alpha i}\), \(i \in I_\alpha\), relative to \(S_\alpha\) such that each \(D_{\alpha i}\) is irreducible and \(f|_{D_{\alpha i}}: D_{\alpha i} \to S_\alpha\) is smooth. For each geometric point \(\bar{s} \to S\), denote by \(\rho_{\bar{s}}: X_{\bar{s}} \to X\) the base change of \(\bar{s} \to S\) by \(f\). For each \(\alpha \in J\), denote by \(\bar{\eta}_\alpha\) a geometric generic point of \(S_\alpha\). Define the total dimension divisor of \(\mathcal{F}\) by
\[
\mathrm{DT}(j_!\mathcal{F}|_{X_{\bar{\eta}_\alpha}}) = \sum_{i \in I}\mathrm{dimtot}_{E_i}(j_!\mathcal{F}|_{X_{\bar{\eta}_\alpha}}) \cdot E_i
\]
with \(\{E_i\}_{i \in I}\) the set of irreducible components of \(D_\alpha\). For a geometric point \(\xi_i\) of \(E_i\) and \(\bar{\xi}_i\) a geometric point above \(\xi_i\), \(\eta_i\) the generic point of the strict localisation \(X_{(\bar{\xi}_i)}\) and \(\bar{\eta}_i\) a geometric point above \(\eta_i\), \(\mathcal{F}|_{\eta_i}\) associated to a finite \(\mathbb{F}_\ell\)-module, denote by \(\mathrm{dimtot}_{E_i}(j_!\mathcal{F}|_{X_{\bar{\eta}_\alpha}})\) the total dimension
\[
\mathrm{dimtot}_K M = \sum_{r \geq 1}r \cdot \mathrm{dim}_{\mathbb{F}_\ell}M^{(r)}
\]
of \(\mathcal{F}|_{X_{\bar{\eta}_\alpha}}\) for \(K\) a complete discrete valued field, \(M\) a finitely generated \(\mathbb{F}_\ell\)-module on which the wild inertia subgroup of \(G_K\) acts through a finite quotient and \(M = \bigoplus_{r \in \mathbb{Q}}M^{(r)}\) the slope decomposition. For \(\alpha \in J\), define the generic total dimension divisor \(\mathrm{GDT}_{f_\alpha}(j_!\mathcal{F}|_{X_\alpha})\) of on \(X_\alpha\) as the unique Cartier divisor on \(X_\alpha\) supported on \(D_\alpha\) such that \(\rho^*_{\bar{\eta}_\alpha}(\mathrm{GDT}_{f_\alpha}(j_!\mathcal{F}|_{X_\alpha})) = \mathrm{DT}(j_!\mathcal{F}|_{X_{\bar{\eta}_\alpha}})\). For each geometric point \(\bar{s} \to S\), denote by \(J_{\bar{s}}\) the subset of \(J\) such that \(\bar{s}\) maps to \(S_\alpha\). Then there is a dense open subset \(V \subseteq S\) such that:
(i) For any geometric point \(\bar{s} \to V\) and any \(\alpha \in J_{\bar{s}}\), one has
\[
\rho^*_{\bar{s}}(\mathrm{GDT}_{f_\alpha}(j_!\mathcal{F}|_{X_\alpha})) = \mathrm{DT}_{X_{\bar{s}}}(j_!\mathcal{F}|_{X_{\bar{s}}});
\]
(ii) For any geometric point \(\bar{t} \to S \setminus V\) and any \(\alpha \in J_{\bar{t}}\), the difference
\[
\rho^*_{\bar{t}}(\mathrm{GDT}_{f_\alpha}(j_!\mathcal{F}|_{X_\alpha})) - \mathrm{DT}_{X_{\bar{t}}}(j_!\mathcal{F}|_{X_{\bar{t}}})
\]
is an effective Cartier divisor on \(X_{\bar{t}}\). étale and other Grothendieck topologies and cohomologies; Ramification and extension theory Hu, H., Yang, E.: Semi-continuity for total dimension divisors of étale sheaves. Int. J. Math. \textbf{28}(1), 1750001, 1-21 (2017) Étale and other Grothendieck topologies and (co)homologies, Ramification and extension theory Semi-continuity for total dimension divisors of étale sheaves | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For the entire collection see Zbl 0509.00008.]
Let C be the rational normal curve of degree n in \(P^ n\), and consider the osculating flag \(F(C,p)=\{F^ 0(C,p)\subset...\subset F^{n- 1}(C,p)\}\) to C at the point \(p\in C\), where \(F^ k(C,p)\) is the intersection of all hyperplanes of \(P^ n\) meeting \(C\geq k+1\) times (it turns out that \(F^ k(C,p)\) is a k-plane). If \(n-k\geq a_ 0\geq...\geq a_ k\geq 0\), then define the Schubert cycle \(\sigma_{a_ 0,...,a_ k}(p)\) by \(\{all\quad k-planes\quad V\subset P^ n| \dim(V\cap F^{n-k-a_ i+i}(C,p)\geq i \quad for\quad i=0,1,...,k\}.\) In this note the author proves the following result obtained jointly with J. Harris [and used by them in their investigation of the Brill-Noether problem via rational curves with ordinary cusps (instead of the former approach via rational curves with ordinary nodes of Griffiths and Harris); see the author and \textit{J. Harris} [Invent. Math. 74, 371-418 (1983; Zbl 0527.14022)]]. - Theorem. Let k be an integer such that 0\(\leq k\leq n-1\), let \(p_ 1,...,p_ g\) be g points of C and for \(i=1,...,g\) let \(\Sigma_ i\) be any Schubert cycle \(\sigma_{a_ 0^{(i)},...,a_ k^{(i)}}(p_ i)\) defined with the reference to the osculating flag \(F(C,p_ i)\) as above. If the points \(p_ 1,...,p_ g\) are distinct, then either \(co\dim(\cap^{g}_{i=1}\Sigma_ i)=\sum^{g}_{i=1}co\dim(\Sigma_ i)\), or \(\cap^{g}_{i=1}\Sigma_ i=\emptyset,\) the latter case being precisely when the product of the corresponding cohomology classes is 0. A dual form of this result was obtained in 1973 by \textit{A. Iarrobino} (but never published). As an immediate consequence of the above theorem one gets that the dimension of the space of nondegenerate rational curves of degree n in \(P^ r\) having \(\geq d\) ordinary cusps is \((r+1)(n+1)-4- (r-1)d\) if \(d\leq(n-r)(r+1)/r,\) and the space is empty otherwise. moduli of curves; rational curves with cusps; osculating flag; Schubert cycle; Brill-Noether problem Singularities of curves, local rings, Families, moduli of curves (algebraic), Enumerative problems (combinatorial problems) in algebraic geometry, Rational and unirational varieties, Singularities in algebraic geometry Rational curves with cusps | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 survey on some \({\mathcal D}\)-modules. To a differential system on \(X \approx \mathbb{C}^ n\) one associates a coherent \({\mathcal D}\)-module \(M\) (provided with a finite free resolution \(\dots{\mathcal D}^{N_ 1} @>P_ 0>> {\mathcal D}^{N_ 0} \to M \to 0)\). For a complex of sheaves \({\mathcal F}\) (like holomorphic functions, hyperfunctions, microfunctions...) one studies the complex \(\mathbb{R} \Hom_{\mathcal D} (M, {\mathcal F})\) of \({\mathcal F}\)- solutions of the system. The author first makes an algebraic discussion in case of one variable. Then he introduces holonomic \({\mathcal D}\)-modules and Bernstein polynomials. The latter are also studied in further detail when the singularity is isolated. \({\mathcal D}\)-modules; Bernstein polynomials P. Maisonobe, \(\scr D\)-modules: an overview towards effectivity , Computer algebra and differential equations (1992), London Math. Soc. Lecture Note Ser., vol. 193, Cambridge Univ. Press, Cambridge, 1994, pp. 21-55. Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs, Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials \({\mathcal D}\)-modules: An overview towards effectivity | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 regular scheme of finite type over a field. By Grothendieck we have surjections \(Ch^ m(X)\to F^ mK_ 0(X)/F^{m+1}K_ 0(X)\), where \(Ch^ m\) is the codimension-m Chow group and \(F^ m\) a canonical filtration on \(K_ 0\), the Grothendieck group of vectorbundles on X. But there are also ``higher'' K-groups \(K_ i\). So \textit{S. Bloch} [Adv. Math. 61, 27-304 (1986; Zbl 0608.14004)] invented ``higher'' Chow groups \(Ch^ m(X,n)\) and gave a relation analogous to higher K-groups after tensoring all with \({\mathbb{Q}}\). The purpose of the paper under review is to get rid of that ``\(\otimes {\mathbb{Q}}''\). The author constructs the \(Ch^ m(X,n)\) in another way (and prefers to name them Pre Ch\({}^ m(X,n))\) and defines surjections Pre Ch\({}^ m_{\infty}(X,n)\to F^ mK_ n(X)/F^{m+1}K_ n(X)\), where Pre Ch\({}^ m_{\infty}(X,n)\) is a certain subgroup of Pre Ch\({}^ m(X,n).\)
The notion of ``derived categories'' and Milnors ``patching'' method are used as essential tools. higher Chow groups; Grothendieck group; derived categories; patching S. E. Landsburg, Relative cycles and algebraic \(K\)-theory , preprint, 1985. JSTOR: Parametrization (Chow and Hilbert schemes), Grothendieck groups (category-theoretic aspects), Applications of methods of algebraic \(K\)-theory in algebraic geometry Relative cycles and algebraic K-theory | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(Gr_{d,n}\) denote the Grassmannian of \(d\)-dimensional vector subspaces of \(K^n\), \(K\) an algebraically closed field. If \(\alpha\), \(\gamma\) are \(d\)-element subsets of \(\{1,\dots,n\}\) and \((e_1,\dots,e_n)\) is the standard basis of \(K^n\), \(e_\alpha\) denotes the span of \(e_{\alpha_1},\dots,e_{\alpha_d}\). The Richardson variety \(X^\gamma_\alpha\) is then defined as the intersection of the Schubert variety \(X^\gamma\), the closure of the orbit of \(e_\alpha\) under the action of upper triangular matrices, and the opposite Schubert variety \(X_\alpha\), the orbit of \(e_\alpha\) under the action of lower triangular matrices. The point \(e_\beta\) belongs to \(X^\gamma_\alpha\) if and only if \(\alpha\leq \beta\leq\gamma\). In the article under review the author computes the degree of the tangent cone of \(X^\gamma_\alpha\) at \(e_\beta\), ie the multiplicity of \(X^\gamma_\alpha\) at \(e_\beta\). This follows as corollary from the explicit computation of a Gröbner basis \(G^\gamma_{\alpha,\beta}\) for a suitable monomial order of the ideal of \(Y^\gamma_{\alpha,\beta}=X^\gamma_\alpha\cap\mathcal O_\beta\), where \(\mathcal O_\beta\) is a special affine open subset of \(Gr_{d,n}\). The proof relies on a generalization of the Robinson-Schensted-Knuth correspondence, introduced by the author and called bounded RSK correspondence.
This paper extends previous results of \textit{V. Kodiyalam} and \textit{K. N. Raghavan} [J. Algebra 270, No. 1, 28--54 (2003; Zbl 1083.14056)] and \textit{V. Kreiman} and \textit{V. Lakshmibai} [in: Algebra, arithmetic and geometry with applications. Papers from Shreeram S. Ahhyankar's 70th birthday conference, Purdue University, West Lafayette, IN, USA, July 19--26, 2000. (Berlin): Springer. 553--563 (2003; Zbl 1092.14060)] on Schubert varieties. Schubert variety; Grassmannian; multiplicity; Robinson-Schensted-Knuth correspondence S. Upadhyay, \textit{Initial ideals of tangent cones to Richardson varieties in the orthogonal Grassmannian via a orthogonal-bounded-RSK-correspondence}, preprint, http://arxiv.org/abs/0909.1424. Grassmannians, Schubert varieties, flag manifolds, Singularities in algebraic geometry, Group actions on varieties or schemes (quotients), Computational aspects of higher-dimensional varieties, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) Local properties of Richardson varieties in the Grassmannian via a bounded Robinson-Schensted-Knuth correspondence | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials A well-known theorem of Quillen says that if \(r(z,\bar z)\) is a bihomogeneous polynomial on \(\mathbb C^n\) positive on the sphere, then there exists \(d\) such that \(r(z,\bar z)\| z\|^{2d}\) is a squared norm. We obtain effective bounds relating this \(d\) to the signature of \(r\). We obtain the sharp bound for \(d=1\), and for \(d>1\) we obtain a bound that is of the correct order as a function of \(d\) for fixed \(n\). The current work adds to an extensive literature on positivity classes for real polynomials. The classes \(\Psi_d\) of polynomials for which \(r(z,\bar z)\| z\|^{2d}\) is a squared norm interpolate between polynomials positive on the sphere and those that are Hermitian sums of squares. Hermitian sums of squares; Hilbert's 17th problem; positivity classes; Hermitian symmetric polynomials Halfpap, Jennifer; Lebl, Jiří, Signature pairs of positive polynomials, Bull. Inst. Math. Acad. Sin. (N.S.), 2304-7909, 8, 2, 169-192, (2013) Fields related with sums of squares (formally real fields, Pythagorean fields, etc.), Semialgebraic sets and related spaces, Quadratic and bilinear forms, inner products, Holomorphic functions of several complex variables Signature pairs of positive polynomials | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We prove a conjecture in [\textit{G. Lusztig}, Bull. Inst. Math., Acad. Sin. (N.S.) 7, No. 3, 355-404 (2012; Zbl 1283.20045)] stating that certain polynomials \(P^\sigma_{y,w}(q)\) introduced in [\textit{G. Lusztig} and \textit{D. A. Vogan} jun., Bull. Inst. Math., Acad. Sin. (N.S.) 7, No. 3, 323-354 (2012; Zbl 1288.20006)] for twisted involutions in an affine Weyl group give \((-q)\)-analogues of weight multiplicities of the Langlands dual group \(\check G\). We also prove that the signature of a naturally defined Hermitian form on each irreducible representation of \(\check G\) can be expressed in terms of these polynomials \(P^\sigma_{y,w}(q)\). Coxeter systems; generalized Kazhdan-Lusztig polynomials; twisted involutions; affine Weyl groups; weight multiplicities; Langlands dual groups; Hermitian forms; irreducible representations George Lusztig and Zhiwei Yun, A (-\?)-analogue of weight multiplicities, J. Ramanujan Math. Soc. 28A (2013), 311 -- 340. Representation theory for linear algebraic groups, Reflection and Coxeter groups (group-theoretic aspects), Hecke algebras and their representations, Geometric Langlands program (algebro-geometric aspects) A \((-q)\)-analogue of weight multiplicities. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 use enumerative geometry to simplify the formula for the roots of the general one-variable polynomial of degree \(n\), for all \(n\). More precisely, let \(\mathrm{RD}(n)\) denote the minimum \(d\) for which there exists a formula for the roots of the general degree \(n\) polynomial using only algebraic functions of \(d\) or fewer variables. In 1927, Hilbert sketched how the 27 lines on a cubic surface could be used to construct a 4-variable formula for the general degree 9 polynomial (implying \(\mathrm{RD}(9)\le 4)\). In this paper, we turn Hilbert's sketch into a general method. We show this method produces best-to-date upper bounds on \(\mathrm{RD}(n)\) for all \(n\), improving earlier results of Hamilton, Sylvester, Segre and Brauer. resolvent degree; polynomials; lines on cubic surfaces Global ground fields in algebraic geometry, Polynomials in number theory, Equations in general fields Tschirnhaus transformations after Hilbert | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a reductive algebraic group over \(\mathbb{C}\) and let \(N\) be a \(G\)-module. Choose a principal nilpotent \(X\) in \(\text{Lie\,}G\). For any subspace \(M\) of \(N\), the kernels of the powers of \(X\) define the Brylinski-Kostant filtration on \(M\). This filtration is related to a \(q\)-analog of weight multiplicity due to Lusztig.
The author generalizes this filtration to the case when \(X\) is a Richardson nilpotent of a parabolic and shows that this generalized filtration is related to ``parabolic'' versions of Lusztig's \(q\)-analog of weight multiplicity. For this he also needs to generalize results of Broer on cohomology vanishing of vector bundles on cotangent bundles of partial flag varieties. He concludes by computing some explicit examples. Brylinski-Kostant filtrations; cotangent bundle cohomology; flag varieties; Kazhdan-Lusztig polynomials; \(sl_2\)-triples; nilpotent elements Hague C.: Cohomology of flag varieties and the Brylinski-Kostant filtration. J. Algebra 321, 3790--3815 (2009) Cohomology theory for linear algebraic groups, Representation theory for linear algebraic groups, Grassmannians, Schubert varieties, flag manifolds Cohomology of flag varieties and the Brylinski-Kostant filtration. | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) 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 notion of the Gröbner cell for the Hilbert scheme of points in the plane, as well as that of the punctual Hilbert scheme is comprehensively defined. An explicit parametrization of the Gröbner cells in terms of minors of a matrix is recalled. The main core of this paper shows that the decomposition of the Punctual Hilbert scheme into Grönber cells induces that of the compactified Jacobians of plane curve singularities. As an important application of this decomposition, the topological invariance of an analog of the compactified Jacobian and the corresponding motivic superpolynomial for families of singularities is concluded. Hilbert schemes; affine plane; Grothendieck-Deligne map; Gröbner cells; zeta functions; plane curve singularities Parametrization (Chow and Hilbert schemes), Singularities of curves, local rings, Plane and space curves, Exact enumeration problems, generating functions, Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases), Singularities in algebraic geometry, Jacobians, Prym varieties, Hecke algebras and their representations, Combinatorial aspects of representation theory, Braid groups; Artin groups, Basic orthogonal polynomials and functions associated with root systems (Macdonald polynomials, etc.) Gröbner cells of punctual Hilbert schemes in dimension two | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Consider the BGG category \({\mathcal O}\) associated to the general linear Lie algebra \({\mathfrak gl}_n\) and its principal block \({\mathcal O}_0\). Given a pair of (orthogonal) parabolic subalgebras of \({\mathfrak gl}_n\), the author associates a Serre quotient of a Serre subcategory of \({\mathcal O}_0\). In related work of the author [Sel. Math., New Ser. 22, No. 2, 669--734 (2016; Zbl 1407.17012)], these subquotients of \({\mathcal O}_0\) are used to categorify some representations of the quantized Lie superalgebra \({\mathfrak gl}(1|1)\). In the paper under review, these subquotient categories are shown to be equivalent to (graded) module categories of certain diagram algebras. This equivalence is then used to compute the endomorphism rings of two special functors.
The diagram algebras are explicitly constructed following the ideas used by \textit{J. Brundan} and \textit{C. Stroppel} [Mosc. Math. J. 11, No. 4, 685--722 (2011; Zbl 1275.17012)] to construct generalized Khovanov algebras. It is directly shown that these diagram algebras are graded cellular and properly stratified.
The approach used by Brundan and Stroppel to define the multiplication in the diagram algebras will not work in this setting. The author overcomes this obstruction by making use of morphisms between Soergel modules, which are modules for the complex polynomial ring \({\mathbb C}[x_1, \dots, x_n]\). Each module is associated to a permutation (of \(n\) objects), but they are not well understood in general. The computation of the relevant Soergel modules is aided by the fact that they are all cyclic, that is, the corresponding Schubert variety is rationally smooth in the full flag variety. A key portion of the paper (of independent interest) is a study of these particular Soergel modules. Each relevant Soergel module is shown to be isomorphic to \({\mathbb C}[x_1,\dots, x_n]/I\) for an explicit ideal \(I\) generated by symmetric polynomials. From this, the author obtains a description of the morphisms between the relevant Soergel modules.
The ideas in the paper are nicely developed with a thorough discussion of the tools and combinatorics being used, along with numerous examples. diagram algebra; symmetric polynomials; category \({\mathcal O}\); Soergel modules; Khovanov algebra; general linear Lie algebra; Serre category; categorification Sartori, A.: A diagram algebra for Soergel modules corresponding to smooth Schubert varieties. Trans. Amer. Math. Soc. (2013). 10.1090/tran/6346 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights), Symmetric functions and generalizations, Polynomial rings and ideals; rings of integer-valued polynomials, Grassmannians, Schubert varieties, flag manifolds, Graded rings and modules (associative rings and algebras) A diagram algebra for Soergel modules corresponding to smooth 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 [For the entire collection see Zbl 0719.00018.]
Let the algebraic torus \(T\) act on a projective variety \(X\) over \(\mathbb{C}\). Let \(L\) be an ample, \(T\)-linearized line bundle on \(X\). Denote by \(X^{ss}(L)\) the corresponding set of semistable points of \(X\), and by \(Y(L)\) its Mumford quotient by \(T\). Then \(Y(L)\) is projective, and \(L\) defines an ample, \(\mathbb{Q}\)- Cartier divisor \([L]\) on \(Y(L)\).
We study the behaviour of \(Y(L)\) and \([L]\), when \(L\) is replaced by some positive power \(L^ n\), twisted by a character \(\chi\) of \(T\). Then \(Y(L^ n\otimes\chi)\) only depends on \(\chi/n=p\); denote it by \(Y_ p\). There are only finitely many \(Y_ p\)'s, and for different ``regular'' values of \(p\), they are related by sequences of blowing-ups and their inverses. Furthermore, the classes \([L^ n\otimes\chi]/n=L_ p\) on \(Y_ p\) can be glued to a piecewise affine, continuous function. --- We apply our results to the asymptotic behaviour of multigraded modules. In particular, we get a geometric interpretation of Joseph polynomials, in terms of the above construction. Finally, we relate this construction to the symplectic reduction of \(X\) with respect to the maximal compact subtorus of \(T\). action of algebraic torus; semistable points; Mumford quotient; Cartier divisor; multigraded modules; Joseph polynomials M. Brion and C. Procesi, Action d'un tore dans une variété projective, Operator algebras, unitary representations, enveloping algebras, and invariant theory (Paris 1989), Progr. Math. 92, Birkhäuser, Boston (1990), 509-539. Group actions on varieties or schemes (quotients), Homogeneous spaces and generalizations Action d'un tore dans une variété projective. (Torus action on a projective 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 The author studies some hypersurfaces in \(\mathbb{P}^{4}\), whose normalizations is a Calabi-Yau threefold. Then, the lines on these normalizations are examined. More explicitly, the author extends the constructions of Ciliberto to the canonical surfaces of degree \(d=6\) in \(\mathbb{P}^{4}\). The author also includes some work on \(d=10\) case considering quasi étale double coverings of general determinantal symmetric quintics. Calabi-Yau threefold; Clemen's conjecture; étale double covering; rigid Classical problems, Schubert calculus, \(3\)-folds, Calabi-Yau manifolds (algebro-geometric aspects) Lines on some Calabi-Yau threefolds in \(\mathbb {P}^4\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials An integral basis of the cohomology of a flag manifold \(G/B\) is the Schubert classes \(\mathfrak S_w\) indexed by the elements \(w\) of the Weyl group of \(G\). Hence there are, for all pairs of elements \(u,v\) in the Weyl group, integers \(c_{uv}^w\) such that
\[
\mathfrak S_u\mathfrak S_v =\sum_wc_{uv}^w \mathfrak S_w,
\]
where the summation is over the elements in the Weyl group. A Pieri type formula is a formula that describes the structure constants \(c_{uv}^w\) when \(\mathfrak S _v\) is a special Schubert class pulled back from the projection \(G/B\to G/P\), where \(P\) is a maximal parabolic subgroup. When \(G=\text{Gl}_n(\mathbb{C})\) the classical Pieri formula gives such a description. For other \(G\) there are formulas by \textit{H. Hiller} and \textit{B. Boe} [Adv. Math. 62, 49-67 (1986; Zbl 0611.14036)] and by \textit{P. Pragacz} and \textit{J. Ratajski} [J. Reine Angew. Math. 476, 143-189 (1996; Zbl 0847.14029); C. R. Acad. Sci., Paris, Sér. I 317, 1035-1040 (1993; Zbl 0812.14034); Manuscr. Math. 79, 127-151 (1993; Zbl 0789.14041)]. When \(G=\text{Gl}_n(\mathbb{C})\) an interpretation of the structure constants \(c_{uv}^w\) in Pieri's formula, in terms of chains in the Bruhat order, was conjectured by \textit{N. Bergeron} and \textit{S. Billey} [Exp. Math. 2, 257-269 (1993; Zbl 0803.05054)] and given algebraic, geometric and combinatorial proofs by \textit{A. Postnikov} [Prog. Math. 172, 371-383 (1999; Zbl 0944.14019)], \textit{F. Sottile} [Ann. Inst. Fourier 46, 89-110 (1996; Zbl 0837.14041)], and \textit{M. Kogan} and \textit{A. Kumar} [Proc. Am. Math. Soc. 130, 2525-2534 (2002; Zbl 1001.05121)], respectively. The main result of the present article are analogous Pieri type formulas when \(G\) is \(\text{Sp}_{2n}(\mathbb{C})\) and \(\text{SO}_{2n+1}(\mathbb{C})\). One of the techniques used is to explicitly determine triple intersections of Schubert varieties. \textit{F. Sottile} has used this technique with success earlier [see, e.g., Colloq. Math. 82, 49-63 (1999; Zbl 0977.14023)], and shows that the coefficients in the Pieri type formulas are the intersection number of a linear space with a collection of quadrics, and thus are either \(0\) or a power of \(2\). special Schubert classes; Schubert varieties; Bruhat order; Pieri type formulas; Weyl groups; parabolic groups; isotropic flag manifolds; cohomology \beginbarticle \bauthor\binitsN. \bsnmBergeron and \bauthor\binitsF. \bsnmSottile, \batitleA Pieri-type formula for isotropic flag manifolds, \bjtitleTrans. Amer. Math. Soc. \bvolume354 (\byear2002), no. \bissue7, page 2659-\blpage2705 \bcomment(electronic). \endbarticle \OrigBibText ----, A Pieri-type formula for isotropic flag manifolds , Trans. Amer. Math. Soc. 354 (2002), no. 7, 2659-2705 (electronic). \endOrigBibText \bptokstructpyb \endbibitem Classical groups (algebro-geometric aspects), Enumerative problems (combinatorial problems) in algebraic geometry, Symmetric functions and generalizations, Combinatorics of partially ordered sets, Grassmannians, Schubert varieties, flag manifolds, Combinatorial problems concerning the classical groups [See also 22E45, 33C80] A Pieri-type formula for isotropic flag manifolds | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(f\in {\mathbb{F}}_{q^{\kappa}}[X_ 1,...,X_ n]\) and \(\deg_{X_ i}(f)<r\). Set size \(L_ 1(f)=r^ n \kappa \log_ 2q\). An algorithm is suggested factoring f within the polynomial in \(L_ 1(f)\), q time (theorem 1.4).
Let \(f_ 0,...,f_ k\in F[X_ 1,...,X_ n]\), denote by \(L_ 2\) the size of polynomials \(f_ 0,...,f_ k\), degrees \(\deg (f_ i)<d\), and either F is finite or \(F={\mathbb{Q}}\) for simplicity. An algorithm is proposed finding the irreducible components of the variety of common roots of the system \(f_ 0=...=f_ k=0\) within time polynomial in \(L_ 2\), \(d^{n^ 3}\), q (theorem 2.4). factorization of polynomials; polynomial complexity; algorithm; irreducible components; variety D. Yu. Grigor'ev, ''Decomposition of polynomials over a finite field and solution of systems of algebraic equations,'' Zap. Nauchn. Semin. Leningr. Otdel. Mat. Inst.,137, 20--79 (1984). Equations in general fields, Polynomials over finite fields, Analysis of algorithms and problem complexity, Varieties and morphisms, Arithmetic problems in algebraic geometry; Diophantine geometry, Polynomials in general fields (irreducibility, etc.), Polynomials in real and complex fields: factorization Factorization of polynomials over a finite field and solution of a system of algebraic equations | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials [For the entire collection see Zbl 0719.00022.]
Let \(F_ n\) denote the space of complete flags in \(k^ n\) (k\(\in \{{\mathbb{R}},{\mathbb{C}}\})\) and let \(c_{\sigma}\) be the cell of the standard Schubert cell decomposition \(Sch_ f\) of \(F_ n\), corresponding to a complete flag \(f\in F_ n\), whose cells are enumerated by permutations \(\sigma \in S_ n\). Define the link of \(c_{\sigma}\) to be the manifold \(A_{\sigma}=B\setminus Tn_ f\), where B is a sufficiently small (n(n- 1)/2)-dimensional (over k) ball centered at some point of \(c_{\sigma}\) and \(Tn_ f\) is the train of the flag f. Define the Euler characteristic of \(A_{\sigma}\) to be \(\chi_{\sigma}=\sum_{\ell}(-1)^{\ell}\dim (H^{\ell}(A_{\sigma})) \). Finally, define the number \(\chi_{\sigma}^{pq}=\sum_{\ell}(-1)^{\ell}\dim (Gr^ p_ FGr^ w_{p+q}H^{\ell}(A_{\sigma})) \) where \(Gr^ W\) (resp. \(Gr_ F)\) are the associated graded objects of the weight (resp. Hodge filtration). The authors show how to reduce the calculation of \(\chi_{\sigma}\) and \(\chi_{\sigma}^{pq}\) for \(F_ n\) to those for \(F_{n-1}\) and then do the calculation for low dimensions. They also find a relation between \(\chi_{\sigma}\) and \(\chi_{\sigma}^{pq}\). Euler characteristics for links of Schubert cells; Hodge filtrations; train of the flag; weight Shapiro, B. Z.; Vainshtein, A. D.: Euler characteristics for links of Schubert cells in the space of complete flags. Adv. sov. Math. 1, 273-286 (1990) Grassmannians, Schubert varieties, flag manifolds, Stratified sets Euler characteristics for links of Schubert cells in the space of complete flags | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials We describe the structure of the Chow-Arakelov ring of Grassmannians, and deduce from this the rationality of the height of their Plücker embedding. Schubert calculus; Chow-Arakelov ring; Grassmannians; height; Plücker embedding Grassmannians, Schubert varieties, flag manifolds, Arithmetic varieties and schemes; Arakelov theory; heights, (Equivariant) Chow groups and rings; motives, Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry An arithmetic 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 In the paper under review, the authors introduce and develop the theory of Waring-like decompositions of polynomials. Let us present the main idea behind this project. Let \(R = \bigoplus_{i\geq 0} R_{i} = \in k[x_{1}, \dots,x_{n}]\) with \(k\) algebraically closed and \(n \geq 2\). The Waring decomposition of \(f \in R_{d}\) is
\[
f = \sum_{i=1}^{s} L_{i}^{d},
\]
where \(L_{i}\)'s are linear forms, and this is a classical subject of research. In the paper, the authors define a certain variation on the Waring decomposition. Fix an integer \(d\), we denote by \(\mathcal{P}_{r} = \mathbb{Z}[Z_{1},\dots,Z_{r}]\), and by \(\mathcal{M}_{r,d}\) the subset of all monomials such that
\[
M= Z_{1}^{d_{1}} \cdots Z_{r}^{d_{r}}
\]
and
\((\bullet)\) \(d_{i} > 0\) for all \(i\);
\((\bullet \bullet)\) \(d_{1} + \cdots + d_{r} = d.\)
Of course, the above conditions imply that \(r \leq d\).
Let \(M\) be as above, and let \(f \in R_{d}\).
\((\bullet)\) An \(M\)-decomposition of \(f\) having length \(s\) is an expression of the form
\[
f = \sum_{j=1}^{s} L_{1,j}^{d_{1}} \cdot L_{2,j}^{d_{2}} \cdots L_{r,j}^{d_{r}},
\]
where \(L_{i,j}\)'s are linear forms.
\((\bullet \bullet)\) The \(M\)-rank of \(f\) is the least integer \(s\) such that \(f\) has an \(M\)-decomposition of length \(s\).
Observe that if \(M = Z_{1}^{d}\), then the \(M\)-rank of \(f\) is known as the Waring rank of \(f\), and it can be shown that every \(f \in R_{d}\) has an \(M\)-decomposition of finite length for any choice of \(M\)
There is a nice geometric way of considering the problem of finding \(M\)-rank of a polynomial \(f \in R_{d}\) with \(M \in \mathcal{M}_{r,d}\). Let \(\mathbb{P}(R_{1})\) be the projective space based on the \(k\)-vector space \(R_{1}\). We define the morphism
\[
\phi_{M} : (\mathbb{P}(R_{1}))^{r} \rightarrow \mathbb{P}(R_{d}) \simeq \mathbb{P}^{{ d + n - 1 \choose n-1} - 1},
\]
where \((\mathbb{P}(R_{1}))^{r}\) denotes the cartesian product of \(r\) copies of \(\mathbb{P}(R_{1})\), by
\[
\phi_{M}([L_{1}],\dots,[L_{d}]) = [L_{1}^{d_{1}}L_{2}^{d_{2}} \cdots L_{r}^{d_{r}}],
\]
and denote the image of \(\phi_{M}\) by \(\mathbb{X}_{M}\).
If \(X \subset \mathbb{P}^{t}\) is any projective variety, then
\[
\sigma_{s}(X) := \overline{ \{ P \in \mathbb{P}^{t} : P \in \langle P_{1}, \dots, P_{s} \rangle , P_{i} \in X \}}
\]
is defined as the \(s\)-secant variety of \(X\). A natural question behind this definition focuses on the dimension of \(\sigma_{s}(X)\) if \(s \geq 2\). It is easy to see that
\[
\dim \sigma_{s}(X) \leq \min \{ s \dim X + s - 1,t\},
\]
and if the above inequality is satisfied for some \(s\), then we say that \(\sigma_{s}(X)\) has the expected dimension, and if the above inequality is strict for some \(s\), then we say that \(\sigma_{s}(X)\) is defective. The difference between
\[
\min \{ s \dim X + s - 1,t\} - \dim \sigma_{s}(X)
\]
is called the \(s\)-defect of \(X\).
Now we are ready to formulate the main results of the paper. The first one is a natural variation on Terracini's Lemma.
Theorem 1. Let \(R = k[x_{1},\dots,x_{n}]\), \(M = Z_{1}^{d_{1}} \cdots L_{r}^{d_{r}} \in \mathcal{M}_{r,d}\), and let \(L_{1},\dots, L_{r}\) be general linear forms in \(R_{1}\) so that \(P = [L_{1}^{d_{1}} \cdots L_{r}^{d_{r}}]\) is a general point of \(\mathbb{X}_{M}= \phi_{M}((\mathbb{P}^{n-1})^{r})\). If \(F = L_{1}^{d_{1}} \cdots L_{r}^{d_{r}}\) and \(I_{p} = ( F/L_{1},\dots, F / L_{r}) = \bigoplus_{j \geq 0} (I_{P})_{j}\), then
\[
T_{P}(\mathbb{X}_{M}) = \mathbb{P}((I_{P})_{d}).
\]
For binary forms there is a nice classification result.
Theorem 2. Let \(R = k[x,y] = \bigoplus_{j\geq 0} R_{j}\) and let \(M = Z_{1}^{d_{1}} \cdots Z_{r}^{d_{r}} \in \mathcal{M}_{r,d}\) for any \(r\) and any \(d\) with \(r \leq d\). Then \(\sigma_{s}(\mathbb{X}_{M})\) has the expected dimension for every \(s\), i.e.,
\[
\dim \sigma_{s}(\mathbb{X}_{M}) = \min \{ s \dim \mathbb{X}_{M} + (s-1),d\} = \min \{sr + s - 1,d\}
\]
for every \(s\) and every \(M\).
The last result is devoted to secant line varieties to \(\mathbb{X}_{M}\) with \(n\geq 3\) variables.
Theorem 3. Let \(R = k[x_{1},\dots,x_{n}]\), \(M \in \mathcal{M}_{r,d}\) with \(n \geq 3\), \(r \geq 2\), \(d \geq 3\), and
\[
M = Z_{1}^{d_{1}} \cdots Z_{r}^{d_{r}}.
\]
Then \(\sigma_{2}(\mathbb{X}_{M})\) is not defective, except for \(M = Z_{1}^{2}Z_{2}\) and \(n=3\). For this last case, \(\mathbb{X}_{M}\) has \(2\)-defect equal to \(1\). Waring problems; polynomials; secant varieties M. V. Catalisano, L. Chiantini, A. V. Geramita, and A. Oneto Waring-like decompositions of polynomials, 1, Linear Algebra Appl. 533 (2017), 311--325. Projective techniques in algebraic geometry, Computational aspects of higher-dimensional varieties, Effectivity, complexity and computational aspects of algebraic geometry, Polynomials, factorization in commutative rings, Special varieties Waring-like decompositions of polynomials. 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 Cohomological descent was invented by Deligne in the early 60's. I recall memories about this and discuss related developments and applications that arose since then, pertaining to cotangent complex and deformation theory, mixed Hodge theory, de Jong alterations, \(p\)-adic Hodge theory. Galois descent; cohomological descent; hypercovering; simplicial space; Grothendieck's six operations; cotangent complex; first order deformation; algebraic stack; mixed Hodge theory; de Jong alteration; rigid cohomology; \(p\)-adic étale cohomology; \(p\)-adic Hodge theory; independence of \(\ell\) History of algebraic geometry, History of mathematics in the 20th century, (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.), Deformations and infinitesimal methods in commutative ring theory, Transcendental methods, Hodge theory (algebro-geometric aspects), Étale and other Grothendieck topologies and (co)homologies, \(p\)-adic cohomology, crystalline cohomology, Simplicial sets, simplicial objects (in a category) [See also 55U10] \(\ll\) The Galois descent \(\dots \gg\) | 0 |
In the paper under review, the author introduces the quantum double Grothendieck polynomials \(\widetilde G_w(x,y)\) and also the quantum double dual Grothendieck polynomials \(\widetilde H_w(x,y)\) and investigates some of their properties. It may be recalled that the quantum double Schubert polynomials \(\widetilde {\mathfrak G}_w(x,y)\) were introduced and studied by Kirillov-Maeno. Grothendieck polynomials; Schubert polynomials; double Schubert polynomials Grassmannians, Schubert varieties, flag manifolds, Algebraic combinatorics Quantum Grothendieck polynomials Let \(G\) be a simple algebraic group over an algebraically closed field of arbitrary characteristic, and \(B\subset G\) a Borel subgroup. Recall that for every element \(w\) of the Weyl group of \(G\), the Schubert variety \(X(w)\subset G/B\) is defined as the Zariski closure of the Bruhat cell \(BwB/B\subset G/B\). A connection between Schubert varieties and toric varieties has been studied extensively, in particular, flat toric degenerations of Schubert varieties were constructed in [\textit{P. Caldero}, Transform. Groups 7, No. 1, 51--60 (2002; Zbl 1050.14040)].
The present paper describes all Schubert varieties that are already toric. Namely, the author proves that \(X(w)\) is a toric variety if and only if \(w\) is a product of pairwise distinct simple reflections (in particular, the dimension of \(X(w)\) does not exceed the rank of \(G\)). Schubert variety; toric variety Karuppuchamy, P., On Schubert varieties, Commun. Algebra, 41, 4, 1365-1368, (2013) Grassmannians, Schubert varieties, flag manifolds, Toric varieties, Newton polyhedra, Okounkov bodies 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 It has been known since Segre's work [\textit{B. Segre}, Q. J. Math., Oxf. Ser. 14, 86--96 (1943; Zbl 0063.06860)] that a smooth quartic complex surface in \(\mathbb P^3\) contains at most 64 lines (although Segre's arguments contained some gaps which were filled in [\textit{S. Rams} and \textit{M. Schütt}, Math. Ann. 362, No. 1--2, 679--698 (2015; Zbl 1319.14042)] while at the same time extending the result to any field of characteristic \(\neq 2,3\)). The present paper generalizes these results to the situation where the quartic may admit isolated rational double points as siingularities (so that the minimal resolution is again a \(K3\) surface, hence the title `\(K3\) quartic surface'). (Worse singularities have been treated by \textit{V. González-Alonso} and \textit{S. Rams} in [Taiwanese J. Math. 20, No. 4, 769--785 (2016; Zbl 1357.14052)].)
The main theorem of the paper is that Segre's bound of 64 lines continues to persist in the presence of isolated ADE singularities. In fact, non-smooth quartics with isolated ADE singularities tend to have substantially fewer lines. The current record seems to be 52 outside characteristics \(2,3,5\) (due to Degtyarev abstractly, and to Veniani with explicit equations [private correspondence, 2016]).
Unlike previous work, the present paper does not use the flecnodal divisor at all, but it makes heavy use of elliptic fibrations. The main new technical ingredients of the paper are certain configurations of lines with rich geometry (which the author calls `twin lines'), and a mostly lattice-theoretic investigation of the situation where the surface does not admit a hyperplane splitting into 4 lines (called the `triangle-free' case, following Degtyarev). \(K3\) surface; quartic surface; rational double point; line Veniani, D. C.: The maximum number of lines lying on a\textit{K}3 quartic surface. \textit{Math. Z.}285 (2017), no. 3-4, 1141--1166. \(K3\) surfaces and Enriques surfaces, Hypersurfaces and algebraic geometry, Enumerative problems (combinatorial problems) in algebraic geometry, Varieties of low degree The maximum number of lines lying on a \(K3\) quartic surface | 0 |
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