Families of reduced zero-dimensional schemes

The Geometry of Border Bases

The main topic of the paper is the construction of various explicit flat families of border bases. To begin with, we cover the punctual Hilbert scheme Hilb^μ(A^n) by border basis schemes and work out the base changes. This enables us to control flat families obtained by linear changes of coordinates. Next we provide an explicit construction of the principal component of the border basis scheme, and we use it to find flat families of maximal dimension at each radical point. Finally, we connect radical points to each other and to the monomial point via explicit flat families on the principal com…

关于理想的反向族的约化模与协约化模

Reduced and coreduced modules with respect to inverse families of ideals

本文针对交换环上理想的反向族，引入 Phi 约化模与协约化模，证明广义挠与完备化函子由最大真成员决定，并建立刻画、闭包性质及 Greenlees-May 型伴随，为理想族上的广义挠理论提供模论框架。

收缩族的代数方法

Algebraic approach to contraction families

本文提出了一种纯代数方法处理收缩群概型，并比较了其商概型与其他相关概型在对称和θ-稳定抛物子群情形下的群作用。

Groebner bases for families of affine or projective schemes

Let $I$ be an ideal of the polynomial ring $A[x]=A[x_1,...,x_n]$ over the commutative, noetherian ring $A$. Geometrically $I$ defines a family of affine schemes over $\Spec(A)$: For $\p\in\Spec(A)$, the fibre over $\p$ is the closed subscheme of affine space over the residue field $k(\p)$, which is determined by the extension of $I$ under the canonical map $σ_\p:A[x]\to k(\p)[x]$. If $I$ is homogeneous there is an analogous projective setting, but again the ideal defining the fibre is $\sigI$. For a chosen term order this ideal has a unique reduced Gröbner basis which is known to contain consi…

Deformations of zero-dimensional schemes and applications

In this thesis we consider the geometry of the Hilbert scheme of points in P^n, concentrating on the locus of points corresponding to the Gorenstein subschemes of P^n. New results are given, most importantly we provide tools for constructing flat families and analysis of finite Gorenstein algebras and expose their efficiency by proving smoothability of certain families of algebras. Much of the existing theory and folklore is reviewed, providing a micro-encyclopaedic reference.

Hilbert-Burch矩阵与有限子方案的显式托尔稳定族

Hilbert-Burch matrices and explicit torus-stable families of finite subschemes of $\mathbb A ^2$

本文利用Hilbert-Burch矩阵，研究了Hilbert方案上的Białynicki-Birula细胞，并给出了固定点位留正维时的有理映射构造，同时探讨了形式变形的显式描述。

曲线的完全交差异 II：曲线族

The Complete Intersection Discrepancy of a Curve II: Families of Curves

本文研究了曲线族的等奇异性和完全交差异，证明了魏特奇等奇异性和强同时解析性的等价性，并展示了米尔诺数的上半连续性及其在拓扑等奇异中的应用。

Multiplicity of jet schemes of monomial schemes

This article studies jet schemes of monomial schemes. They are known to be equidimensional but usually are not reduced. We thus investigate their structure further, giving a formula for the multiplicity along every component of the jet schemes of a general reduced monomial hypersurface (that is, the case of a simple normal crossing divisor).

Families of artinian and low dimensional determinantal rings

Let GradAlg(H) be the scheme parameterizing graded quotients of R=k[x_0,...,x_n] with Hilbert function H (it is a subscheme of the Hilbert scheme of P^n if we restrict to quotients of positive dimension, see definition below). A graded quotient A=R/I of codimension c is called standard determinantal if the ideal I can be generated by the t by t minors of a homogeneous t by (t+c-1) matrix (f_{ij}). Given integers a_0\le a_1\le ...\le a_{t+c-2} and b_1\le ...\le b_t, we denote by W_s(\underline{b};\underline{a}) the stratum of GradAlg(H) of determinantal rings where f_{ij} \in R are homogeneous …

Gluing compactly generated t-structures over stalks of affine schemes

We show that compactly generated t-structures in the derived category of a commutative ring $R$ are in a bijection with certain families of compactly generated t-structures over the local rings $R_\mathfrak{m}$ where $\mathfrak{m}$ runs through the maximal ideals in the Zariski spectrum $\mathrm{Spec}(R)$. The families are precisely those satisfying a gluing condition for the associated sequence of Thomason subsets of $\mathrm{Spec}(R)$. As one application, we show that the compact generation of a homotopically smashing t-structure can be checked locally over localizations at maximal ideals. I…

Reduced group schemes as iterative differential Galois groups

This article is on the inverse Galois problem in Galois theory of linear iterative differential equations in positive characteristic. We show that it has an affirmative answer for reduced algebraic group schemes over any iterative differential field which is finitely generated over its algebraically closed field of constants. We also introduce the notion of equivalence of iterative derivations on a given field - a condition which implies that the inverse Galois problem over equivalent iterative derivations are equivalent.

Blocks in flat families of finite-dimensional algebras

We study the behavior of blocks in flat families of finite-dimensional algebras. In a general setting we construct a finite directed graph encoding a stratification of the base scheme according to the block structures of the fibers. This graph can be explicitly obtained when the central characters of simple modules of the generic fiber are known. We show that the block structure of an arbitrary fiber is completely determined by "atomic" block structures living on the components of a Weil divisor. As a byproduct, we deduce that the number of blocks of fibers defines a lower semicontinuous funct…

Hilbert functions of schemes of double and reduced points

It remains an open problem to classify the Hilbert functions of double points in $\mathbb{P}^2$. Given a valid Hilbert function $H$ of a zero-dimensional scheme in $\mathbb{P}^2$, we show how to construct a set of fat points $Z \subseteq \mathbb{P}^2$ of double and reduced points such that $H_Z$, the Hilbert function of $Z$, is the same as $H$. In other words, we show that any valid Hilbert function $H$ of a zero-dimensional scheme is the Hilbert function of a set of a positive number of double points and some reduced points. For some families of valid Hilbert functions, we are also able to sh…