Recovering the good component of the Hilbert scheme

A Hilbert-Mumford criterion for SL_2-actions

Let the special linear group $G := SL_{2}$ act regularly on a $Q$-factorial variety $X$. Consider a maximal torus $T \subset G$ and its normalizer $N \subset G$. We prove: If $U \subset X$ is a maximal open $N$-invariant subset admitting a good quotient $U \to U // N$ with a divisorial quotient space, then the intersection $W(U)$ of all translates $g \dot U$ is open in $X$ and admits a good quotient $W(U) \to W(U) // G$ with a divisorial quotient space. Conversely, we obtain that every maximal open $G$-invariant subset $W \subset X$ admitting a good quotient $W \to W // G$ with a divisorial qu…

The dimension of the Hilbert scheme of special threefolds

The Hilbert scheme of projective 3-folds of codimension 3 or more that are linear scrolls over the projective plane or over a smooth quadric surface or that are quadric or cubic fibrations over the projective line is studied. All known such threefolds of degree from 7 to 11 are shown to correspond to smooth points of an irreducible component of their Hilbert scheme, whose dimension is computed. A relationship with the locus of good determinantal subschemes is investigated

非纯层的模空间与希尔伯特概型的收缩

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。

从对数希尔伯特方案到超kähler品种的退化

From logarithmic Hilbert schemes to degenerations of hyperkähler varieties

本文提出通过构造超kähler品种的退化例子，研究其对数希尔伯特方案与退化结构之间的关系，并计算了对偶复形。

点希尔伯特概型上 tautological 层的容斥原理

An inclusion-exclusion principle for tautological sheaves on Hilbert schemes of points

本研究证明双点退化纤维上点希尔伯特概型tautological层的欧拉特征满足容斥型等式，推导出光滑真代数空间上对应欧拉特征的通用多项式存在性，并将Zhou的相关猜想约简至射影空间乘积情形，主要工具为优态退化与代数配边。

On Algorithmic Equiresolution and Stratification of Hilbert Schemes

Given an algorithm of resolution of singularities satisfying certain conditions (``good algorithms''), natural notions of simultaneous algorithmic resolution, or equiresolution, for families of embedded schemes (parametrized by a reduced scheme $T$) are proposed. It is proved that these conditions are equivalent. Something similar is done for families of sheaves of ideals, here the goal is algorithmic simultaneous principalization. A consequence is that given a family of embedded schemes over a reduced $T$, this parameter scheme can be naturally expressed as a disjoint union of locally closed …

高阶除子的对角线性质与弱点性质及某些Hilbert概型

Diagonal property and weak point property of higher rank divisors and certain Hilbert schemes

本文为ind-簇定义了对角线性质与弱点性质，证明光滑射影曲线上整斜率高阶除子的ind-簇具有弱点性质，$(1,n)$-除子具有对角线性质，并给出相关Hilbert概型同构类数量的精确上界。

Families of divisors

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 $(X,Δ)$.

The geometry of degenerations of Hilbert schemes of points

Given a strict simple degeneration $f \colon X\to C$ the first three authors previously constructed a degeneration $I^n_{X/C} \to C$ of the relative degree $n$ Hilbert scheme of $0$-dimensional subschemes. In this paper we investigate the geometry of this degeneration, in particular when the fibre dimension of $f$ is at most $2$. In this case we show that $I^n_{X/C} \to C$ is a dlt model. This is even a good minimal dlt model if $f \colon X \to C$ has this property. We compute the dual complex of the central fibre $(I^n_{X/C})_0$ and relate this to the essential skeleton of the generic fibre. …

高秩除子与某些希尔伯特概型的对角性质和弱点性质

Diagonal property and weak point property of higher rank divisors and certain Hilbert schemes

该研究引入ind-概型的对角与弱点性质，证明高秩除子ind-概型的相关性质，计算(1,n)-除子ind-概型的皮卡群，并确定一类曲线希尔伯特概型的同构类数量。

An efficient implementation of the algorithm computing the Borel-fixed points of a Hilbert scheme

Borel-fixed ideals play a key role in the study of Hilbert schemes. Indeed each component and each intersection of components of a Hilbert scheme contains at least one Borel-fixed point, i.e. a point corresponding to a subscheme defined by a Borel-fixed ideal. Moreover Borel-fixed ideals have good combinatorial properties, which make them very interesting in an algorithmic perspective. In this paper, we propose an implementation of the algorithm computing all the saturated Borel-fixed ideals with number of variables and Hilbert polynomial assigned, introduced from a theoretical point of view i…

Sklyanin algebras and Hilbert schemes of points

We construct projective moduli spaces for torsion-free sheaves on noncommutative projective planes. These moduli spaces vary smoothly in the parameters describing the noncommutative plane and have good properties analogous to those of moduli spaces of sheaves over the usual (commutative) projective plane P^2. The generic noncommutative plane corresponds to the Sklyanin algebra S constructed from an automorphism sigma of infinite order on an elliptic curve E < P^2. In this case, the fine moduli space of line bundles over S with first Chern class zero and Euler characteristic (1-n) provides a sy…

完全忠实的 p 进 Riemann--Hilbert 函子：作用于滤过凝聚 D-模

A fully faithful p-adic Riemann--Hilbert functor for filtered coherent D-modules

本文证明光滑刚性解析簇上滤过凝聚 D-模的求解函子完全忠实，恢复 Scholze 水平截面函子，并引入 de Rham Zariski 可构造层以推广经典 de Rham 局部系统。

Computable Hilbert Schemes

In this PhD thesis we propose an algorithmic approach to the study of the Hilbert scheme. Developing algorithmic methods, we also obtain general results about Hilbert schemes. In Chapter 1 we discuss the equations defining the Hilbert scheme as subscheme of a suitable Grassmannian and in Chapter 5 we determine a new set of equations of degree lower than the degree of equations known so far. In Chapter 2 we study the most important objects used to project algorithmic techniques, namely Borel-fixed ideals. We determine an algorithm computing all the saturated Borel-fixed ideals with Hilbert poly…