Gherardelli linkage and complete intersections

Self-duality of the local cohomology of the Jacobian ring and Gherardelli's Theorem

We prove that the $0$-th local cohomology of the jacobian ring of a projective hypersurface with isolated singularities has a nice interpretation it in the context of linkage theory. Roughly speaking, it represents a measure of the failure of Gherardelli's theorem for the corresponding graded modules. This leads us to a different and characteristic free proof of its self-duality, which turns out to be an easy consequence of Grothendieck's local duality theorem.

非交换沙漏 II 1/2：具有良构奇点的双有理联络的导出范畴

Noncommutative Hourglasses II 1/2: Derived Category of Birational Linkage with Well-Formed Singularities

本文针对奇点为环面上奇点内完全交的良构簇，将有界凝聚层导出范畴的经典公式推广到奇异情形，并讨论终复三维簇及其栈间态射的Bondal-Orlov局部化问题。

Linkage on arithmetically Cohen-Macaulay schemes with application to the classification of curves of maximal genus

In this paper the author provides a generalization of classical linkage, i.e. linkage by a complete intersection of dim. 0 or 1 on arithmetically Cohen-Macaulay schemes of any dimension. Namely she looks at residuals in the scheme theoretic intersection of an aCM scheme of dim. r (resp. r+1) with r hypersurfaces of degree a_1,...,a_r. When the aCM scheme is singular a complete intersection on it may not be Gorenstein, in this case classical linkage, even if suitably generalized, does not apply. The main purpose of this paper is to prove some results related to the invariance of the deficiency …

Linkage on singular rational normal surfaces and three-folds with application to the classification of curves of maximal genus

In this paper the author provides a generalization of classical linkage, i.e. linkage by a complete intersection, in a different context. Namely she looks at residuals in the scheme theoretic intersection of a rational normal surface or 3-fold with two hypersurfaces of degree a and b. When the scroll is singular a complete intersection of type (a, b) on it may not be Gorenstein, in this case classical linkage, even if suitably generalized, does not apply. The main purpose of this article is to establish a framework allowing one to find relations between the dimension of some important cohomolo…

$F$-singularities under generic linkage

Let $R=k[x_1,\dots,x_n]$ be a polynomial ring over a prefect field of positive characteristic. Let $I$ be an unmixed ideal in $R$ and let $J$ be a generic link of $I$ in $S=R[u_{ij}]_{c \times r}$. We describe the parameter test submodule of $S/J$ in terms of the test ideal of the pair $(R, I)$ when a reduction of $I$ is a complete intersection or almost complete intersection. As an application, we deduce a criterion for when $S/J$ has $F$-rational singularities in these cases. We also compare the $F$-pure threshold of $(R, I)$ and $(S, J)$.