非奇异三次曲面上的ℓ-away ACM线丛

ACM sheaves on a nonsingular quadric hypersurface in $P^4_k$

We prove that on a nonsingular quadric hypersurface Q in $P^4_k$ even CI liaison classes of ACM curves are in bijective correspondence with the stable equivalence classes (up to shift in degree) of maximal Cohen-Macaulay graded modules over the coordinate ring R of Q, which in turn, are in bijective correspondence with stable equivalence classes (up to shift in degree) of ACM sheaves on Q . In the situation of a nonsingular quadric hypersurface Q in $P^4_k$ work of Knorrer shows that there is a unique nonfree indecomposable MCM module over R. We also describe the ACM sheaf corresponding to thi…

Hypersurface arrangements of aCM type

We investigate the arrangement of hypersurfaces on a nonsingular varieties whose associated logarithmic vector bundle is arithmetically Cohen-Macaulay (for short, aCM), and prove that the projective space is the only smooth complete intersection with Picard rank one that admits an aCM logarithmic vector bundle. We also obtain a number of results on aCM logarithmic vector bundles over several specific varieties. As an opposite situation we investigate the Torelli-type problem that the logarithmic cohomology determines the arrangement.

法诺曲面上的$\boldsymbol{\text{ℓ}}$-分离ACM丛

$\ell$-away ACM Bundles on Fano Surfaces

该研究提出极化簇上ℓ-分离ACM丛的定义，在三类法诺曲面上构造此类丛，完成两类曲面上特定秩或行列式条件的2秩ℓ-分离ACM丛分类，并证明对应分次模的连通性，推广了射影平面上的已有结论。

ACM bundles on cubic surfaces

In this paper we prove that, for every $r \geq 2$, the moduli space $M^s_X(r;c_1,c_2)$ of rank $r$ stable vector bundles with Chern classes $c_1=rH$ and $c_2=(3r^2-r)/2$ on a nonsingular cubic surface $X \subset \mathbb{P}^3$ contains a nonempty smooth open subset formed by ACM bundles, i.e. vector bundles with no intermediate cohomology. The bundles we consider for this study are extremal for the number of generators of the corresponding module (these are known as Ulrich bundles), so we also prove the existence of indecomposable Ulrich bundles of arbitrarily high rank on $X$.

Gorenstein Biliaison and ACM Sheaves

Let $X$ be a normal arithmetically Gorenstein scheme in ${\mathbb P}^n$. We give a criterion for all codimension two ACM subschemes of $X$ to be in the same Gorenstein biliaison class on $X$, in terms of the category of ACM sheaves on $X$. These are sheaves that correspond to the graded maximal Cohen--Macaulay modules on the homogeneous coordinate ring of $X$. Using known results on MCM modules, we are able to determine the Gorenstein biliaison classes of codimension two subschemes of certain varieties, including the nonsingular quadric surface in ${\mathbb P}^3$, and the cone over it in ${\ma…

Stable Ulrich bundles

The existence of stable ACM vector bundles of high rank on algebraic varieties is a challenging problem. In this paper, we study stable Ulrich bundles (that is, stable ACM bundles whose corresponding module has the maximum number of generators) on nonsingular cubic surfaces $X \subset \mathbb{P}^3.$ We give necessary and sufficient conditions on the first Chern class $D$ for the existence of stable Ulrich bundles on $X$ of rank $r$ and $c_1=D$. When such bundles exist, we prove that that the corresponding moduli space of stable bundles is smooth and irreducible of dimension $D^2-2r^2+1$ and co…