Hilbert Modular Threefolds of Arithmetic Genus One

In 1981, Weisser proved that there are exactly four Galois cubic number fields with Hilbert modular threefolds of arithmetic genus one. In this paper, we extend Weisser's work to cover all cubic number fields. Our main result is that there are exactly 33 fields with Hilbert modular threefolds of arithmetic genus one. These fields are enumerated explicitly.     

The Hilbert Scheme Parameterizing Finite Length Subschemes of the Line with Support at the Origin

We introduce symmetrizing operators of the polynomial ring A[x] in the variable x over a ring A. When A is an algebra over a field k these operators are used to characterize the monic polynomials F(x) of degree n in A[x] such that A _k k[x]_(x)/(F(x)) is a free A-module of rank n. We use the characterization to determine the Hilbert scheme parameterizing subschemes of length n of k[x]_(x).     

Reducibility of Punctual Hilbert Schemes of Cone Varieties

In this short note we show that for any pair of positive integers (d, n) with n > 2 and d > 1 or n = 2 and d > 4, there always exist projective varieties X ? ? ^N of dimension n and degree d and an integer s ₀ such that Hilb ^s (X) is reducible for all s ≥ s ₀. X will be a projective cone in ? ^N over an arbitrary projective variety Y ? ? ^N?1. In particular, we show that, opposite to the case of smooth surfaces, there exist projective surfaces with a single isolated singularity which have reducible Hilbert scheme of points.     

Hirzebruch χy genera of the Hilbert schemes of surfaces by localization formula

We use the Atiyah-Bott-Segal-Singer Lefschetz fixed point formula to calculate the Hirzebruch χy genus χy(S[n]), where S[n] is the Hilbert scheme of points of length n of a surface S. Combinatorial interpretation of the weights of the fixed points of the natural torus action on (C2)[n] is used. This is the first step to prove a conjectural formula about the elliptic genus of the Hilbert schemes.     

Hirzebruch Xy genera of the Hilbert schemes of surfaces by localization formula

We use the Atiyah-Bott-Segal-Singer Lefschetz fixed point formula to calculate the Hirzebruch ?_y genus X_y(S^[n]), where S^[n] is the Hilbert scheme of points of length n of a surface S. Combinatorial interpretation of the weights of the fixed points of the natural torus action on ( ²)^[n] is used. This is the first step to prove a conjectural formula about the elliptic genus of the Hilbert schemes.     

Some remarks on tautological sheaves on Hilbert schemes of points on a surface

We study tautological sheaves on the Hilbert scheme of points on a smooth quasi-projective algebraic surface by means of the Bridgeland–King–Reid transform. We obtain Brion–Danila’s Formulas for the derived direct image of tautological sheaves or their double tensor product for the Hilbert–Chow morphism; as an application we compute the cohomology of the Hilbert scheme with values in tautological sheaves or in their double tensor product, thus generalizing results previously obtained for tautological bundles.      

Flag Hilbert schemes and moduli spaces of torsion plane sheaves

We find certain relations between flag Hilbert schemes of points on plane curves and moduli spaces of one-dimensional plane sheaves. We show that some of these moduli spaces are stably rational.     

Hilbert schemes, polygraphs and the Macdonald positivity conjecture

We study the isospectral Hilbert scheme , defined as the reduced fiber product of with the Hilbert scheme of points in the plane , over the symmetric power . By a theorem of Fogarty, is smooth. We prove that is normal, Cohen-Macaulay and Gorenstein, and hence flat over . We derive two important consequences.

(1) We prove the strong form of the conjecture of Garsia and the author, giving a representation-theoretic interpretation of the Kostka-Macdonald coefficients . This establishes the Macdonald positivity conjecture, namely that .

(2) We show that the Hilbert scheme is isomorphic to the -Hilbert scheme of Nakamura, in such a way that is identified with the universal family over . From this point of view, describes the fiber of a character sheaf at a torus-fixed point of corresponding to .

The proofs rely on a study of certain subspace arrangements , called polygraphs, whose coordinate rings carry geometric information about . The key result is that is a free module over the polynomial ring in one set of coordinates on . This is proven by an intricate inductive argument based on elementary commutative algebra.

     

Rigid Curves in Complete Intersection Calabi–Yau Threefolds

Working over the complex numbers, we study curves lying in a complete intersection K3 surface contained in a (nodal) complete intersection Calabi–Yau threefold. Under certain generality assumptions, we show that the linear system of curves in the surface is a connected componend of the the Hilbert scheme of the threefold. In the case of genus one, we deduce the existence of infinitesimally rigid embeddings of elliptic curves of arbitrary degree in the general complete intersection Calabi–Yau threefold.     

Cohomology of Noncommutative Hilbert Schemes

Noncommutative Hilbert schemes, introduced by M. V. Nori, parametrize left ideals of finite codimension in free algebras. More generally, parameter spaces of finite-codimensional submodules of free modules over free algebras are considered. Cell decompositions of these varieties are constructed, whose cells are parametrized by certain types of forests. Asymptotics for the corresponding Poincaré polynomials and properties of their generating functions are discussed. Presented by P. Littleman Mathematics Subject Classifications (2000) Primary: 16G20; secondary: 14D20.     

On Algorithmic Equi-Resolution and Stratification of Hilbert Schemes

Given an algorithm for resolution of singularities that satisfiescertain conditions (‘a good algorithm’), naturalnotions of simultaneous algorithmic resolution, and of equi-resolution,for families of embedded schemes (parametrized by a reducedscheme T) are defined. It is proved that these notions are equivalent.Something similar is done for families of sheaves of ideals,where the goal is algorithmic simultaneous principalization.A consequence is that given a family of embedded schemes overa reduced T, this parameter scheme can be naturally expressedas a disjoint union of locally closed sets T_j, such that theinduced family on each part T_j is equi-resolvable. In particular,this can be applied to the Hilbert scheme of a smooth projectivevariety; in fact, our result shows that, in characteristic zero,the underlying topological space of any Hilbert scheme parametrizingembedded schemes can be naturally stratified in equi-resolvablefamilies. 2000 Mathematics Subject Classification 14E15, 14D99.     

A Few Remarks About the Hilbert Scheme of Smooth Projective Curves

We discuss two conjectures by Francesco Severi and Joe Harris about the irreducibility and the dimension of the Hilbert scheme parameterizing smooth projective curves of given degree and genus.     

Hilbert Schemes of Degree Four Curves

In this paper we determine the irreducible components of the Hilbert schemes H _4,g of locally Cohen-Macaulay space curves of degree four and arbitrary arithmetic genus g: there are roughly (g ²/24) of them, most of which are families of multiplicity structures on lines. We give deformations which show that these Hilbert schemes are connected. For g–3 we exhibit a component that is disjoint from the component of extremal curves and use this to give a counterexample to a conjecture of Aït-Amrane and Perrin.     

Hilbert Scheme of a Pair of Codimension Two Linear Subspaces

We study the component H _n of the Hilbert scheme whose general point parameterizes a pair of codimension two linear subspaces in ? ⁿ for n ≥ 3. We show that H _n is smooth and isomorphic to the blow-up of the symmetric square of 𝔾(n ? 2, n) along the diagonal. Further H _n intersects only one other component in the full Hilbert scheme, transversely. We determine the stable base locus decomposition of its effective cone and give modular interpretations of the corresponding models, hence conclude that H _n is a Mori dream space.     

Hilbert模形式与一类Dirichlet级数的特殊值

在文［２］中，Ｗ．Ｋｏｈｎｎ对权为ｋ和ｌ的任意二个歧点型模形式ｆ和ｇ（其变换群是全模群ＳＬ＿２（Ｚ））定义了一类Ｄｉｒｉｃｈｌｅｔ级数Ｌ＿（ｆ，ｇ，ｎ）（ｓ），利用Ｌ＿（ｆ，ｇ；ｎ）（ｓ）（为整数），可构造一个线性映射Ｗ＿ｇ：Ｓ＿ｋ→Ｓ＿（ｋ－ｌ）．并且讨论了Ｌ＿（ｆ，ｇ；ｎ）的一些特征值．在本文中，我们将［２］中的结果推广到Ｈｉｌｂｅｒｔ模形式的情况，并得到类似的结论．     

Fourier expansion of Eisenstein series on the Hilbert modular group and Hilbert class fields

In this paper we consider the Eisenstein series for the Hilbert modular group of a general number field. We compute the Fourier expansion at each cusp explicitly. The Fourier coefficients are given in terms of completed partial Hecke -series, and from their functional equations, we get the functional equation for the Eisenstein vector. That is, we identify the scattering matrix. When we compute the determinant of the scattering matrix in the principal case, the Dedekind -function of the Hilbert class field shows up. A proof in the imaginary quadratic case was given in Efrat and Sarnak, and for totally real fields with class number one a proof was given in Efrat.