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In this article we compare different conditions on abelian schemes with real multiplication which occur in the integral models of the Hilbert-Blumenthal Shimura variety considered by Rapoport, Deligne, Pappas and Kottwitz. We show that the models studied by Deligne/Pappas and Kottwitz are isomorphic over Spec Z_(p). We also examine the associated local models and prove that they are equal.
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We determine the number of irreducible components of the reduction mod p of any Hilbert-Blumenthal moduli space with a parahoric level structure, where p is unramified in the totally real field.
The purpose of this article is to construct a Hecke-equivariant Chow motive whose realizations equal interior (or intersection) cohomology of Hilbert-Blumenthal varieties with non-constant algebraic coefficients.
Flach system on Quaternionic Hilbert--Blumenthal surfaces and distinguished periods
本文研究四元数 Hilbert--Blumenthal 曲面上由基变换得到的 Hilbert 模形式的区分周期，证明其与四元数 Peterson 范数之比的整性，并通过 Flach 的 Euler 系统方法给出 Selmer 群子空间的界限。
Let $O_F$ be the ring of integers of a totally real field $F$ of degree $g$. We study the reduction of the moduli space of separably polarized abelian $O_F$-varieties of dimension $g$ modulo $p$ for a fixed prime $p$. The invariants and related conditions for the objects in the moduli space are discussed. We construct a scheme-theoretic stratification by $a$-numbers on the Rapoport locus and study the relation with the slope stratification. In particular, we recover the main results of Goren and Oort [GO, J. Alg. Geom. 2000] on the stratifications when $p$ is unramified in $O_F$. We also prove…
We prove a "height-free" effective isogeny estimate for abelian varieties of $\mathrm{GL}_2$-type. More precisely, let $g\in \mathbb{Z}^+$, $K$ a number field, $S$ a finite set of places of $K$, and $A,B/K$ $g$-dimensional abelian varieties with good reduction outside $S$ which are $K$-isogenous and of $\mathrm{GL}_2$-type over $\overline{\mathbb{Q}}$. We show that there is a $K$-isogeny $A\to B$ of degree effectively bounded in terms of $g$, $K$, and $S$ only. We deduce among other things an effective upper bound on the number of $S$-integral $K$-points on a Hilbert modular variety.
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 unirational.
We study the existence of components with the expected number of moduli of the Hilbert scheme of integral nodal curves $C \subset \mathbb {P}^r$ with prescribed degree, arithmetic genus and number of singular points.
We investigate the moduli space of sheaves supported on space curves of degree 4 and having Euler characteristic 1. We give an elementary proof of the fact that this moduli space consists of three irreducible components.
We construct the Hilbert compactification of the universal moduli space of semistable vector bundles over smooth curves. The Hilbert compactification is the GIT quotient of some open part of an appropriate Hilbert scheme of curves in a Grassmannian. It has all the properties asked for by Teixidor.
We construct five families of two-dimensional moduli spaces of parabolic Higgs bundles (respectively local systems) by taking the equivariant Hilbert scheme of a certain finite group acting on the cotangent bundle of an elliptic curve. We show that the Hilbert scheme of m points of these surfaces is again a moduli space of parabolic Higgs bundles (respectively local systems), confirming a conjecture of Boalch in these cases and extending a result of Gorsky--Nekrasov--Rubtsov. Using the McKay correspondence we establish the autoduality conjecture for the derived categories of the moduli spaces …
We classify all Gieseker semi-stable sheaves on the complex projective plane that have dimension 1, multiplicity 6 and Euler characteristic 3. We show that their moduli space is birational to the blow-up at a special point of a certain moduli space of semi-stable Kronecker modules.
We compute the Hodge numbers of the moduli space of semi-stable sheaves on the complex projective plane supported on quintic curves and having Euler characteristic 3. For this purpose we study the fixed-point set for a certain torus action on the moduli space.
We study the Simpson moduli space of semi-stable sheaves on the complex projective plane that have dimension 1, multiplicity 6 and Euler characteristic 2. We describe concretely these sheaves as cokernels of morphisms of locally free sheaves and we stratify the moduli space according to the types of sheaves that occur.
Hilbert schemes of K3 surfaces are dense in moduli
本文证明射影K3曲面的n点Hilbert概型轨迹，在对应不可约全纯辛流形模空间中稠密，并将结论推广到广义Kummer流形。
Moduli spaces of generalised tautological bundles on Hilbert schemes
本文在代数曲面点的 Hilbert 概形上构造了由模空间连通分支参数化的新稳定向量丛，推广了先前关于重言丛及 O'Grady 的近期研究。
We obtain effective results for the global generation of pluritheta line bundles on moduli spaces of vector bundles on curves. The main ingredient is an independent result giving an upper bound on the dimension of the Hilbert scheme of coherent quotients of a given vector bundle.
Refined Verlinde formulas for Hilbert schemes of points and moduli spaces of sheaves on K3 surfaces
该研究计算了曲面上点的希尔伯特概形上取值于线丛的椭圆亏格生成函数，并进一步得到K3曲面上层的模空间上取值于行列式线丛的椭圆亏格生成函数。
Moduli of non-pure sheaves and contractions of Hilbert schemes
本文定义光滑射影簇上秩一层栈并证明其具有恰当良好模空间，进而通过非GIT穿墙构造手术图，并利用精细模解释证明Kawamata DK假设的实例。
In this paper, we investigate the apparent singularities and the dual parameters of rank 2 parabolic connections on $\mathbb{P}^1$ and rank 2 (parabolic) Higgs bundle on $\mathbb{P}^1$. Then we obtain explicit descriptions of Zariski open sets of the moduli space of the parabolic connections and the moduli space of the Higgs bundles. For $n=5$, we can give global descriptions of the moduli spaces in detail.
We introduce the good Hilbert functor and prove that it is algebraic. This functor generalizes various versions of the Hilbert moduli problem, such as the multigraded Hilbert scheme and the invariant Hilbert scheme. Moreover, we generalize a result concerning formal GAGA for good moduli spaces.
Let $C$ be a hyperelliptic curve embedded in its Jacobian $J$ via an Abel-Jacobi map. We compute the scheme structure of the Hilbert scheme component of $\textrm{Hilb}_J$ containing the Abel-Jacobi curve as a point. We relate the result to the ramification (and to the fibres) of the Torelli morphism $\mathcal M_g\rightarrow \mathcal A_g$ along the hyperelliptic locus. As an application, we determine the scheme structure of the moduli space of Picard sheaves (introduced by Mukai) on a hyperelliptic Jacobian.
Let $(C,\bt)$ ($\bt=(t_1,...,t_n)$) be an $n$-pointed smooth projective curve of genus $g$ and take an element $\blambda=(λ^{(i)}_j)\in\C^{nr}$ such that $-\sum_{i,j}λ^{(i)}_j=d\in\mathbf{Z}$. For a weight $\balpha$, let $M_C^{\balpha}(\bt,\blambda)$ be the moduli space of $\balpha$-stable $(\bt,\blambda)$-parabolic connections on $C$ and let $RP_r(C,\bt)_{\ba}$ be the moduli space of representations of the fundamental group $π_1(C\setminus\{t_1,...,t_n\},*)$ with the local monodromy data $\ba$ for a certain $\ba\in\C^{nr}$. Then we prove that the morphism $\RH:M_C^{\balpha}(\bt,\blambda)\righ…
The aim of this paper is to determine a bound of the dimension of an irreducible component of the Hilbert scheme of the moduli space of torsion-free sheaves on surfaces. Let $X$ be a non-singular irreducible complex surface and let $E$ be a vector bundle of rank $n$ on $X$. We use the $m$-elementary transformation of $E$ at a point $x \in X$ to show that there exists an embedding from the Grassmannian variety $\mathbb{G}(E_x,m)$ into the moduli space of torsion-free sheaves $\mathfrak{M}_{X,H}(n;c_1,c_2+m)$ which induces an injective morphism from $X \times M_{X,H}(n;c_1,c_2)$ to $Hilb_{\, \ma…
The Hilbert scheme $S^{[n]}$ of points on an algebraic surface $S$ is a simple example of a moduli space and also a nice (crepant) resolution of singularities of the symmetric power $S^{(n)}$. For many phenomena expected for moduli spaces and nice resolutions of singular varieties it is a model case. Hilbert schemes of points have connections to several fields of mathematics, including moduli spaces of sheaves, Donaldson invariants, enumerative geometry of curves, infinite dimensional Lie algebras and vertex algebras and also to theoretical physics. This talk will try to give an overview over …
Let $M(n,ξ)$ be the moduli space of stable vector bundles of rank $n\geq 3$ and fixed determinant $ξ$ over a smooth projective algebraic curve $X$ over $\mathbb{C}$ of genus $g\geq 4.$ We use the gonality of the curve and $r$-Hecke morphisms to describe a smooth open set and to compute the dimension of a component of the Hilbert scheme $Hilb_{M(n,ξ)}$, of the scheme of morphisms $Mor(\mathbb{G},M(n,ξ))$ and of the moduli space $ M_{X \times \mathbb{G}}$ of stable bundles over $X\times \mathbb{G},$ where $\mathbb{G}$ is the Grassmannian $\mathbb{G}(n-r,\mathbb{C}^n)$. In particular, we prove th…
This paper applies methods of Van der Put and Van derPut-Saito to the fourth Painlevé equation. One obtains a Riemann--Hilbert correspondence between moduli spaces of rank two connections on $\mathbb{P}^1$ and moduli spaces for the monodromy data. The moduli spaces for these connections are identified with Okamoto--Painlevé varieties and the Painlevé property follows. For an explicit computation of the full group of Bäcklund transformations, rank three connections on $\mathbb{P}^1$ are introduced, inspired by the symmetric form for ${\rm PIV}$ as was studied by M. Noumi and Y. Yamada.
Denote by $\mathcal{H}_{d,g,r}$ the Hilbert scheme of smooth curves, that is the union of components whose general point corresponds to a smooth irreducible and non-degenerate curve of degree $d$ and genus $g$ in $\mathbb P^r$. A component of $\mathcal{H}_{d,g,r}$ is rigid in moduli if its image under the natural map $π:\mathcal{H}_{d,g,r} \dashrightarrow \mathcal{M}_{g}$ is a one point set. In this note, we provide a proof of the fact that $\mathcal{H}_{d,g,r}$ has no components rigid in moduli for $g > 0$ and $r=3$, from which it follows that the only smooth projective curves embedded in $\m…
The $\mathcal{D}$-Geometric Hilbert Scheme -- Part II: Hilbert and Quot DG-Schemes
本文为关联于对合代数PDEs的微分理想层构建导出模空间，证明导出$\mathcal{D}$-Quot函子可由微分分级$\mathcal{D}$-流形表示，并分析其有限性等性质，奠定导出形变理论基础。
Let $X$ be a compact connected Riemann surface of genus at least two and $G$ a connected reductive complex affine algebraic group. The Riemann--Hilbert correspondence produces a biholomorphism between the moduli space ${\mathcal M}_X(G)$ parametrizing holomorphic $G$--connections on $X$ and the $G$--character variety $${\mathcal R}(G):= \text{Hom}(π_1(X, x_0), G)/\!\!/G\, .$$ While ${\mathcal R}(G)$ is known to be affine, we show that ${\mathcal M}_X(G)$ is not affine. The scheme ${\mathcal R}(G)$ has an algebraic symplectic form constructed by Goldman. We construct an algebraic symplectic for…
We investigate the geometry of the Simpson moduli space M of stable sheaves on P_3 with Hilbert polynomial H(m)=3m+1 and describe explicitly the two smooth, rational components, their 11-dimensional smooth, transversal intersection and the relation between M and the Hilbert scheme Hilb_H(P_3) of cubic space curves. We also compute the Betti numbers of the Chow groups of the moduli space M.
We study birational maps among 1) the moduli space of semistable torsion sheaves of Hilbert polynomial $4m+2$ on a smooth quadric surface, 2) the moduli space of semistable torsion sheaves of Hilbert polynomial $m^{2}+3m+2$ on $\mathbb{P}^{3}$, 3) Kontsevich's moduli space of genus zero stable maps of degree 2 to Grassmannian $Gr(2, 4)$. A regular birational morphism from 1) to 2) is described in terms of Fourier-Mukai transform. The map from 3) to 2) is Kirwan's partial desingularization. Also we investigate several geometric properties of 1) by using the variation of moduli spaces of stable …
As a continuation of the work of Freiermuth and Trautmann, we study the geometry of the moduli space of stable sheaves on $\mathbb{P}^3$ with Hilbert polynomial $4m+1$. The moduli space has three irreducible components whose generic elements are, respectively, sheaves supported on rational quartic curves, on elliptic quartic curves, or on planar quartic curves. The main idea of the proof is to relate the moduli space with the Hilbert scheme of curves by wall crossing. We present all stable sheaves contained in the intersections of the three irreducible components. We also classify stable sheav…
In this paper, we study the birational geometry of the Hilbert scheme of n points on P^2. We discuss the stable base locus decomposition of the effective cone and the corresponding birational models. We give modular interpretations to the models in terms of moduli spaces of Bridgeland semi-stable objects. We construct these moduli spaces as moduli spaces of quiver representations using G.I.T. and thus show that they are projective. There is a precise correspondence between wall-crossings in the Bridgeland stability manifold and wall-crossings between Mori cones. For n at most 9, we explicitly …
We study closures of GL_2(R)-orbits on the total space of the Hodge bundle over the moduli space of curves under the assumption that they are algebraic manifolds. We show that, in the generic stratum, such manifolds are the whole stratum, the hyperelliptic locus or parameterize curves whose Jacobian has additional endomorphisms. This follows from a cohomological description of the tangent bundle to strata. For non-generic strata similar results can be shown by a case-by-case inspection. We also propose to study a notion of 'linear manifold' that comprises Teichmueller curves, Hilbert modular s…
The aim of this paper is two--fold. We first strongly improve our previous main result Theorem 3.1 in Arxiv 1702.00918v3 12Feb2018 ("Brill-Noether loci of rank two vector bundles on a general $ν$-gonal curve"), concerning classification of irreducible components of the Brill--Noether locus parametrizing rank 2 semistable vector bundles of suitable degrees $d$, with at least $d-2g+4$ independent global sections, on a general $ν$--gonal curve $C$ of genus $g$. We then uses this classification to study several properties of the Hilbert scheme of suitable surface scrolls in projective space, which…
Moduli spaces of one dimensional sheaves on log surfaces and Hilbert schemes
本文研究了光滑射影有理曲面上的一维层模空间，证明其在对应曲面奇点的Hilbert概型乘积上光滑，从而得出该模空间具有辛奇点并存在唯一辛解消。
Given an infinite reductive group G acting on an affine scheme X over C and a Hilbert function h: Irr G \to N_0, we construct the moduli space M_θ(X) of θ-stable (G,h)-constellations on X, which is a generalization of the invariant Hilbert scheme after Alexeev and Brion and an analogue of the moduli space of θ-stable G-constellations for finite groups introduced by Craw and Ishii. Our construction of a morphism M_θ(X) \to X//G makes this moduli space a candidate for a resolution of singularities of the quotient X//G.
We regard the pre-hyperkähler structure on the moduli space of framed flags of sheaves $\mathcal{F}(1,3,1)$ on the projective plane $\mathbb{P}^2$ via an adaptation of the ADHM construction of framed sheaves. Then, we study and categorize the degenerate points of the holomorphic pre-symplectic form presented in the moduli space.
We examine a sequence of examples of pairs of moduli spaces of sheaves on $\mathbb P^2$ where Le Potier's strange duality is expected to hold. One of the moduli spaces in these pairs is the Hilbert scheme of two points. We compute the sections of the relevant theta bundle as a representation of $SL(X)$, where $\mathbb P^2=\mathbb P(X)$. For the higher rank space, we construct a moduli space using the resolution of exceptional bundles from Coskun, Huizenga, and Woolf. We compute a subspace of the sections of the theta bundle which is dual to the sections on the Hilbert scheme. In the second par…
In this paper we study the Hilbert scheme of smooth, linearly normal, special scrolls under suitable assumptions on degree, genus and speciality.
We construct a moduli scheme for semistable pre-$\D$-modules with prescribed singularities and numerical data on a smooth projective variety. These pre-$\D$-modules are to be viewed as regular holonomic $\D$-modules with `level structure'. We also construct a moduli scheme for perverse sheaves on the variety with prescribed singularities and other numerical data, and represent the de Rham functor (which gives the Riemann-Hilbert correspondence) by an analytic morphism between the two moduli schemes.
We show that the Hilbert scheme of curves and Le Potier's moduli space of stable pairs with one dimensional support have a common GIT construction. The two spaces correspond to chambers on either side of a wall in the space of GIT linearisations. We explain why this is not enough to prove the "DT/PT wall crossing" conjecture relating the invariants derived from these moduli spaces when the underlying variety is a 3-fold. We then give a gentle introduction to a small part of Joyce's theory for such wall crossings, and use it to give a short proof of an identity relating the Euler characteristic…
Let $X$ be a projective K3 surfaces. In two examples where there exists a fine moduli space $M$ of stable vector bundles on $X$, isomorphic to a Hilbert scheme of points, we prove that the universal family $\mathcal{E}$ on $X\times M$ can be understood as a complete flat family of stable vector bundles on $M$ parametrized by $X$, which identifies $X$ with a smooth connected component of some moduli space of stable sheaves on $M$.
The main result of this article is the fact that the currents defined by Levin give a description of the polylogarithm of an abelian scheme at the topological level. This result was a conjecture of Levin. This provides a method to explicit the Eisenstein classes of an abelian scheme at the topological level. These classes are of special interest since they have a motivic origin by a theorem of Kings. In a forthcoming article, we use the main result of this paper to prove that the Eisenstein classes of the universal abelian scheme over an Hilbert-Blumenthal variety degenerate at the boundary of…
We study semistable pairs on elliptic K3 surfaces with a section: we construct a family of moduli spaces of pairs, related by wall crossing phenomena, which can be studied to describe the birational correspondence between moduli spaces of sheaves of rank 2 and Hilbert schemes on the surface. In the 4-dimensional case, this can be used to get the isomorphism between the moduli space and the Hilbert scheme described by Friedman.
It is a well established fact, that any projective algebraic variety is a moduli space of representations over some finite dimensional algebra. This algebra can be chosen in several ways. The counterpart in algebraic geometry is tautological: every variety is its own Hilber scheme of sheaves of length one. This holds even scheme theoretic. We use Beilinson's equivalence to get similar results for finite dimensional algebras, including moduli spaces and quiver grassmannians. Moreover, we show that several already known results can be traced back to the Hilbert scheme construction and Beilinson'…
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…
This is a survey article on moduli of affine schemes equipped with an action of a reductive group. The emphasis is on examples and applications to the classification of spherical varieties.
Let $k$ be an algebraically closed field. Fix integers $n$ and $b$ with $n\geq 3$ and $1\leq b\leq n-1.$ Let $T^d_k$ be the moduli space of hypersurfaces $[F]$ in $\mathbb{P}^n_k$ of degree $l$ whose singular locus contains a subscheme of dimension $b$ with Hilbert polynomial among the Hilbert polynomials of $b$-dimensional integral closed subschemes of $\mathbb{P}^n$ of degree $d$. We prove that when $l$ is sufficiently large and $2\leq d\leq \frac{l+1}{2},$ any irreducible component $Z$ of $T^d_k$ satisfies $Z=T^1_k$ or $\dim Z<\dim T^1_k.$
Hilbert-Mumford stability on algebraic stacks and applications to $\mathcal{G}$-bundles on curves
本文以代数栈重述 Hilbert-Mumford 判据，给出保证稳定对象子栈存在分离粗模空间的几何条件，并应用于构造曲线上 parahoric 群概形下 torsor 的粗模空间。
The space of smooth rational cubic curves in projective space $\PP^r$ ($r\ge 3$) is a smooth quasi-projective variety, which gives us an open subset of the corresponding Hilbert scheme, the moduli space of stable maps, or the moduli space of stable sheaves. By taking its closure, we obtain three compactifications $\bH$, $\bM$, and $\bS$ respectively. In this paper, we compare these compactifications. First, we prove that $\bH$ is the blow-up of $\bS$ along a smooth subvariety which is the locus of stable sheaves which are planar (i.e. support is contained in a plane). Next we prove that $\bS$ …
Let $k$ be a commutative ring and let $R$ be a commutative $k-$algebra. The aim of this paper is to define and discuss some connection morphisms between schemes associated to the representation theory of a (non necessarily commutative) $R-$algebra $A. $ We focus on the scheme $\ran//\GL_n$ of the $n-$dimensional representations of $A, $ on the Hilbert scheme $\Hilb_A^n$ parameterizing the left ideals of codimension $n$ of $A$ and on the affine scheme Spec $Γ_R^n(A)^{ab} $ of the abelianization of the divided powers of order $n$ over $A. $ We give a generalization of the Grothendieck-Deligne no…
Results on stability of tautological sheaves on Hilbert schemes of points are extended to higher dimensions and transferred to abelian surfaces and to the restriction of tautological sheaves to generalised Kummer varieties. This provides a big class of new examples of stable sheaves on higher dimensional irreducible symplectic manifolds. Some aspects of deformations of tautological sheaves are studied.
In this paper, we show that the family of moduli spaces of $\balpha'$-stable $(\bt, \blambda)$-parabolic $ϕ$-connections of rank 2 over $\BP^1$ with 4-regular singular points and the fixed determinant bundle of degree -1 is isomorphic to the family of Okamoto--Painlevé pairs introduced by Okamoto \cite{O1} and \cite{STT02}. We also discuss about the generalization of our theory to the case where the rank of the connections and genus of the base curve are arbitrary. Defining isomonodromic flows on the family of moduli space of stable parabolic connections via the Riemann-Hilbert correspondences…
We construct a local model for Hilbert-Siegel moduli schemes with $Γ_1(p)$-level bad reduction over $\text{Spec }\mathbb{Z}_{q}$, where $p$ is a prime unramified in the totally real field and $q$ is the residue cardinality over $p$. Our main tool is a variant over the small Zariski site of the ring-equivariant Lie complex $_A\underline{\ell}_G^{\vee}$ defined by Illusie in his thesis, where $A$ is a commutative ring and $G$ is a scheme of $A$-modules. We use it to calculate the $\mathbb{F}_{q}$-equivariant Lie complex of a Raynaud group scheme, then relate the integral model and the local mode…
We establish a new moduli theory for divisors, that interpolates between the Hilbert scheme and the Cayley-Chow variety. This completes the last step in the construction of a good moduli theory for stable pairs $(X,Δ)$.
Eisenstein-Kronecker classes on Hilbert moduli spaces and a new construction of Katz's $p$-adic measure
将Kings-Sprang的Eisenstein-Kronecker类推广到Hilbert模空间上的万有交换概形，并与Katz的Eisenstein级数比较，构造插值这些类的$p$-进测度，恢复Katz的$p$-进Eisenstein测度。
Hilbert scheme of smooth projective curves of unexpected dimension \& existence of a component with less than the expected number of moduli
本文研究了光滑射影曲线的希尔伯特方案的维度问题，展示了与猜想相悖的例子，并证明了存在具有少于预期模数数量的组件。
The geometry of double nested Hilbert schemes of points on curves
本文研究曲线上点的双嵌套希尔伯特概形的几何，证明其连通、约化且具有纯维数，给出分支的组合刻画与消解，并得到 motivic 类的闭公式。
We study multiview moduli problems that arise in computer vision. We show that these moduli spaces are always smooth and irreducible, in both the calibrated and uncalibrated cases, for any number of views. We also show that these moduli spaces always admit open immersions into Hilbert schemes for more than two views, extending and refining work of Aholt-Sturmfels-Thomas. We use these moduli spaces to study and extend the classical twisted pair covering of the essential variety.
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