Characterization of some projective subschemes by locally free resolutions

Liaison addition and the structure of a Gorenstein liaison class

We study the concept of liaison addition for codimension two subschemes of an arithmetically Gorenstein projective scheme. We show how it relates to liaison and biliaison classes of subschemes and use it to investigate the structure of Gorenstein liaison equivalence classes, extending the known theory for complete intersection liaison of codimension two subschemes. In particular, we show that on the non-singular quadric threefold in projective 4-space, every non-licci ACM curve can be obtained from a single line by successive liaison additions with lines and CI-biliaisons.

Linear normality of general linear sections and some graded Betti numbers of 3-regular projective schemes

In this paper we study graded Betti numbers of any nondegenerate 3-regular algebraic set $X$ in a projective space $\mathbb P^{n}$. More concretely, via Generic initial ideals (Gins) method we mainly consider `tailing' Betti numbers, whose homological index is not less than $\mathrm{codim}(X,\mathbb P^{n})$. For this purpose, we first introduce a key definition `$\mathrm{ND(1)}$ property', which provides a suitable ground where one can generalize the concepts such as `being nondegenerate' or `of minimal degree' from the case of varieties to the case of more general closed subschemes and give a…

Smooth Hilbert schemes: their classification and geometry

Closed subschemes in projective space with a fixed Hilbert polynomial are parametrized by a Hilbert scheme. We classify the smooth ones. We identify numerical conditions on a polynomial that completely determine when the Hilbert scheme is smooth. We also reinterpret these smooth Hilbert schemes as generalized partial flag varieties and describe the subschemes being parametrized.

Matrix factorizations over projective schemes

We study matrix factorizations of locally free coherent sheaves on a scheme. For a scheme that is projective over an affine scheme, we show that homomorphisms in the homotopy category of matrix factorizations may be computed as the hypercohomology of a certain mapping complex. Using this explicit description, we give another proof of Orlov's theorem that there is a fully faithful embedding of the homotopy category of matrix factorizations into the singularity category of the corresponding zero subscheme. We also give a complete description of the image of this functor.

On some Gorenstein loci in $\Hilb_{6}(\p 4)$

Let k be an algebraically closed field and let HaG(d) be the open locus inside H(d) (the Hilbert scheme of 0-dimensional length d subschemes of the projective (d-2)-space over k) corresponding to arithmetically Gorenstein subschemes. We prove the irreducibility and characterize the singularities of HaG(6). In order to achieve these results we also classify all Artinian, Gorenstein, not necessarily graded, k-algebras up to degree 6. Moreover we describe the loci in HaG(6) obtained via some geometric construction. Finally we prove the obstructedness of some families of points in HaG(d) for each …

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…

算术 Cohen-Macaulay 簇的 Cohen-Macaulay 表示型

The Cohen-Macaulay representation type of arithmetically Cohen-Macaulay varieties

本文证明射影空间中具有 Cohen-Macaulay 分次坐标环的约化闭子概形几乎都是野 Cohen-Macaulay 型，并对少数例外情形给出完全分类。