$n$-blocks collections on Fano manifolds and sheaves with regularity $-\infty$

TL;DR

This paper investigates the regularity of coherent sheaves on Fano manifolds with specific block collections, establishing conditions under which the sheaves have infinite regularity and exploring the implications of geometric collections.

Contribution

It proves that on Fano manifolds with nice $n$-blocks collections, sheaves with finite support have regularity $- obreak ext{ extasciitilde}$, and conversely under mild assumptions, extending results to geometric collections.

Findings

Sheaves with finite support have regularity $- obreak ext{ extasciitilde}$ on Fano manifolds with $n$-blocks collections.

The converse holds under mild assumptions, linking support finiteness to regularity.

Results extend to manifolds with geometric collections.

Abstract

Let $X$ be a smooth Fano manifold equipped with a `` nice '' $n$-blocks collection in the sense of \cite{cm2} and $\mathcal {F}$ a coherent sheaf on $X$. Assume that $X$ is Fano and that all blocks are coherent sheaves. Here we prove that $\mathcal {F}$ has regularity $-\infty$ in the sense of \cite{cm2} if ${Supp}(\mathcal {F})$ is finite, the converse being true under mild assumptions. The corresponding result is also true when $X$ has a geometric collection in the sense of \cite{cm1}.