On the smoothness of lexicographic points on Hilbert schemes

Minimal Castelnuovo-Mumford regularity for a given Hilbert polynomial

Let $K$ be an algebraically closed field of null characteristic and $p(z)$ a Hilbert polynomial. We look for the minimal Castelnuovo-Mumford regularity $m_{p(z)}$ of closed subschemes of projective spaces over $K$ with Hilbert polynomial $p(z)$. Experimental evidences led us to consider the idea that $m_{p(z)}$ could be achieved by schemes having a suitable minimal Hilbert function. We give a constructive proof of this fact. Moreover, we are able to compute the minimal Castelnuovo-Mumford regularity $m_p(z)^{\varrho}$ of schemes with Hilbert polynomial $p(z)$ and given regularity $\varrho$ of …

不变的希尔伯特方案和卢娜的微分切片定理

Invariant Hilbert Schemes and Luna's Etale Slice Theorem

本文利用卢娜的微分切片定理研究不变希尔伯特方案的局部结构，探讨其在闭轨道处的光滑性问题。

三维空间上的希尔伯特方案：断裂格罗滕迪耶结构与链接

The Hilbert scheme of points on a threefold: broken Gorenstein structures and linkage

研究三维光滑空间的希尔伯特方案，引入断裂格罗滕迪耶结构概念，证明其存在性保证希尔伯特方案的光滑性，并推测其完备性。分析单调理想对应的光滑点，探讨奇点性质，验证Hu的猜想。

An example of an Sl_2-Hilbert scheme with multiplicities

We determine the invariant Hilbert scheme of the zero fibre of the moment map of an action of Sl_2 on (C^2)^6 as one of the first examples of invariant Hilbert schemes with multiplicities. While doing this, we present a general procedure how to realise the calculation of invariant Hilbert schemes, which have been introduced by Alexeev and Brion. We also consider questions of smoothness and connectedness and thereby show that our Hilbert scheme gives a resolution of singularities of the symplectic reduction of the action.

Irreducibility of the Gorenstein loci of Hilbert schemes via ray families

We analyse the Gorenstein locus of the Hilbert scheme of $d$ points on $\mathbb{P}^n$ i.e. the open subscheme parameterising zero-dimensional Gorenstein subschemes of $\mathbb{P}^n$ of degree $d$. We give new sufficient criteria for smoothability and smoothness of points of the Gorenstein locus. In particular we prove that this locus is irreducible when $d\leq 13$ and find its components when $d = 14$. The proof is relatively self-contained and it does not rely on a computer algebra system. As a by--product, we give equations of the fourth secant variety to the $d$-th Veronese reembedding of $…

光滑Quot概型的分类

Classifying Smooth Quot Schemes

本研究针对n维射影空间上的$\text{Quot}$概型$\text{Quot}^{q}(\text{O}_{\text{P}^n}^{r})$，确定多项式q的数值条件以判定其光滑性与不可约性，同时揭示了该概型字典序点的光滑性等几何特征。

Detaching embedded points

We show that if $D \subset \mathbb P^N$ is obtained from a codimension two local complete intersection $C$ by adding embedded points of multiplicity $\leq 3$, then $D$ is a flat limit of $C$ and isolated points. As applications, we determine the irreducible components of Hilbert schemes of space curves with high arithmetic genus, show the smoothness of the Hilbert component whose general member is a plane curve union a point in $\mathbb P^3$, and construct a Hilbert component whose general member has an embedded point.

伴随簇中圆锥的希尔伯特概形的球面几何

Spherical Geometry of Hilbert Schemes of Conics in Adjoint Varieties

本文利用切触几何证明非A、C型伴随簇中光滑圆锥的希尔伯特概形不可约分支的正规化是球面簇，计算其着色扇，并由此描述圆锥的共轭类及正规化的光滑性，同时给出周概形的类似结果。

Consecutive cancellations in Betti numbers of local rings

Let I be a homogeneous ideal in a polynomial ring P over a field. By Macaulay's Theorem, there exists a lexicographic ideal L=Lex(I) with the same Hilbert function as I. Peeva has proved that the Betti numbers of P/I can be obtained from the graded Betti numbers of P/L by a suitable sequence of consecutive cancellations. We extend this result to any ideal I in a regular local ring (R,m) by passing through the associated graded ring. To this purpose it will be necessary to enlarge the list of the allowed cancellations. Taking advantage of Eliahou-Kervaire's construction, several applications ar…

Sur la lissité du schéma Quot ponctuel emboîté

In this paper we characterise the smoothness of the nested Quot scheme of points of a smooth variety, namely the moduli space parametrising flags of $0$-dimensional quotients of a fixed locally free sheaf. Our results extend Cheah's classification of smooth nested Hilbert schemes. -- Dans cet article on caractérise la lissité du schéma Quot ponctuel emboîté d'une variété lisse, c'est-à-dire l'espace de modules paramétrant les drapeaux de quotients de dimension $0$ d'un faisceau localement libre fixé. Nos résultats étendent la classification de Cheah concernant les schémas de Hilbert ponctuels …

Obstructions to deforming curves on a prime Fano 3-fold

We prove that for every smooth prime Fano $3$-fold $V$, the Hilbert scheme $\operatorname{Hilb}^{sc} V$ of smooth connected curves on $V$ contains a generically non-reduced irreducible component of Mumford type. We also study the deformations of degenerate curves $C$ in $V$, i.e., curves $C$ contained in a smooth anti-canonical member $S \in |-K_V|$ of $V$. We give a sufficient condition for $C$ to be stably degenerate, i.e., every small (and global) deformation of $C$ in $V$ is contained in a deformation of $S$ in $V$. As a result, by using the Hilbert-flag scheme of $V$, we determine the dim…