Linda Chen
Hyperquot schemes are generalizations of Grothendieck's Quot scheme to partial flags. Using a Bialynicki-Birula decomposition, we obtain combinatorial data for the Betti numbers, and collect this information into the form of rational generating functions for the Poincare polynomials.
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Hyperquot schemes on curves: virtual class and motivic invariants
本文在曲线上的 Hyperquot 概形上构造完美障碍理论，给出光滑性判据，并利用动机 zeta 函数确定其动机配分函数。
By using Oprea's Bialynicki-Birula decomposition for the stack of genus zero stable maps to flag manifolds. We calculate the Poincaré polynomial of the moduli space in degree one and degree two.
We will show that the duality for regular weight systems introduced by K. Saito can be interpreted as the duality for orbifoldized Poincare polynomials.
Given a curve over a finite field, we compute the number of stable bundles of not necessarily coprime rank and degree over it. We apply this result to compute the virtual Poincare polynomials of the moduli spaces of stable bundles over a curve. A similar formula for the virtual Hodge polynomials and motives is conjectured.
We obtain a combinatorial proof that the number of points of the space $\overline{{\mathcal M}_{0,n}}({\mathbb F}_q)$ satisfies the recurrent formula for Poincare polynomials of the space $\overline{{\mathcal M}_{0,n}}({\mathbb C})$.
We single out a large class of semisimple singularities with the property that all roots of the Poincaré polynomial of the Lie algebra of derivations of the corresponding suitably (not necessarily quasihomogeneously) graded moduli algebra lie on the unit circle; for a still larger class there might occur exactly four roots outside the unit circle.
We compute Betti numbers of both the components of the moduli space of rank 2 semi-stable torsion-free sheaves with fixed determinant over a reducible nodal curve with two smooth components intersecting at a node. We also compute the intersection Betti numbers of the moduli space.
Virtual Poincare polynomial of moduli space of semistable sheaves of rank two on reducible curves
本文研究由两条光滑曲线在一点粘合而成的稳定曲线上秩二半稳定层的模空间，证明其不可约且与分量曲线上稳定向量丛模空间的射影丛双有理，并计算其虚拟庞加莱多项式。
Let $\mathbf{M}^α(d,χ)$ be the moduli space of $α$-stable pairs $(s,F)$ on the projective plane $\mathbb{P}^2$ with Hilbert polynomial $χ(F(m))=dm+χ$. For sufficiently large $α$ (denoted by $\infty$), it is well known that the moduli space is isomorphic to the relative Hilbert scheme of points over the universal degree $d$ plane curves. For the general $(d,χ)$, the relative Hilbert scheme does not have a bundle structure over the Hilbert scheme of points. In this paper, as the first non trivial such a case, we study the wall crossing of the $α$-stable pairs space when $(d,χ)=(5,2)$. As a direc…
Optimality of Curtiss Bound on Poincare Multiplier for Positive Univariate Polynomials
本文证明对于正单变量多项式，在所有仅依赖于非实根辐角的乘子次数上界中，柯蒂斯界是最优的。
We factor the virtual Poincare polynomial of every homogeneous space G/H, where G is a complex connected linear algebraic group and H is an algebraic subgroup, as $t^{2u} (t^2-1)^r Q_{G/H}(t^2)$ for a polynomial $Q_{G/H}$ with non-negative integer coefficients. Moreover, we show that $Q_{G/H}(t^2)$ divides the virtual Poincaré polynomial of every regular embedding of G/H, if H is connected.
We prove an explicit formula for the Poincaré polynomials of parabolic character varieties of Riemann surfaces with semisimple local monodromies, which was conjectured by Hausel, Letellier and Rodriguez-Villegas. Using an approach of Mozgovoy and Schiffmann the problem is reduced to counting pairs of a parabolic vector bundles and a nilpotent endomorphism of prescribed generic type. The generating function counting these pairs is shown to be a product of Macdonald polynomials and the function counting pairs without parabolic structure. The modified Macdonald polynomial $\tilde H_λ[X;q,t]$ is i…
The closure of a generic torus orbit in the flag variety $G/B$ of type $A$ is known to be a permutohedral variety and its Poincare polynomial agrees with the Eulerian polynomial. In this paper, we study the Poincare polynomial of a generic torus orbit closure in a Schubert variety in $G/B$. When the generic torus orbit closure in a Schubert variety is smooth, its Poincare polynomial is known to agree with a certain generalization of the Eulerian polynomial. We extend this result to an arbitrary generic torus orbit closure which is not necessarily smooth.
In this paper we prove that the generating series of the Poincare polynomials of quasihomogeneous Hilbert schemes of points in the plane has a beautiful decomposition into an infinite product. We also compute the generating series of the numbers of quasihomogeneous components in a moduli space of sheaves on the projective plane. The answer is given in terms of characters of the affine Lie algebra $\hat{sl}_m$.
We show that the Poincare polynomial associated with the orbifold cell decomposition of the moduli space of smooth algebraic curves with distinct marked points satisfies a topological recursion formula of the Eynard-Orantin type. The recursion uniquely determines the Poincare polynomials from the initial data. Our key discovery is that the Poincare polynomial is the Laplace transform of the number of Grothendieck's dessins d'enfants.
Poincare Polynomials of Heavy-Light Hassett Spaces
本文研究Hassett的重轻模空间的庞加莱多项式，给出其递推公式、显式公式及有序贝尔数的推广与递推关系，丰富了稳定曲线模空间的组合拓扑研究。
We study the BPS invariants for local del Pezzo surfaces, which can be obtained as the signed Euler characteristic of the moduli spaces of stable one-dimensional sheaves on the surface $S$. We calculate the Poincare polynomials of the moduli spaces for the curve classes $β$ having arithmetic genus at most 2. We formulate a conjecture that these Poincare polynomials are divisible by the Poincare polynomials of $((-K_S).β-1)$-dimensional projective space. This conjecture motivates upcoming work on log BPS numbers.
We perform a study of the moduli space of framed torsion-free sheaves on Hirzebruch surfaces by using localization techniques. We discuss some general properties of this moduli space by studying it in the framework of Huybrechts-Lehn theory of framed modules. We classify the fixed points under a toric action on the moduli space, and use this to compute the Poincare polynomial of the latter. This will imply that the moduli spaces we are considering are irreducible. We also consider fractional first Chern classes, which means that we are extending our computation to a stacky deformation of a Hir…
We study intersection cohomology of moduli spaces of semistable vector bundles on a complex algebraic surface. Our main result relates intersection Poincare polynomials of the moduli spaces to Donaldson-Thomas invariants of the surface. In support of this result, we compute explicitly intersection Poincare polynomials for sheaves with rank two and three on ruled surfaces.
We give the explicit algorithm computing the motivic generalization of the Poincare series of the plane curve singularity introduced by A. Campillo, F. Delgado and S. Gusein-Zade. It is done in terms of the embedded resolution of the curve. The result is a rational function depending of the parameter q, at q=1 it coincides with the Alexander polynomial of the corresponding link. For irreducible curves we relate this invariant to the Heegard-Floer knot homologies constructed by P. Ozsvath and Z. Szabo. Many explicit examples are considered.
A Fourier transform technique is introduced for counting the number of solutions of holomorphic moment map equations over a finite field. This in turn gives information on Betti numbers of holomorphic symplectic quotients. As a consequence simple unified proofs are obtained for formulas of Poincare polynomials of toric hyperkahler varieties, Poincare polynomials of Hilbert schemes of points and twisted ADHM spaces of instantons on C^2 and Poincare polynomials of all Nakajima quiver varieties. As an application, a proof of a conjecture of Kac on the number of absolutely indecomposable represent…
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