English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

In this paper we compare a torsion free sheaf $\FF$ on $\PP^N$ and the freevector bundle $\oplus_{i=1}^n\OPN(b_i)$ having same rank and splitting type. Weshow that the first one has always "less" global sections, while it has ahigher second Chern class. In both cases bounds for the difference are found interms of the maximal free subsheaves of $\FF$. As a consequence we obtain adirect, easy and more general proof of the "Horrocks' splitting criterion",also holding for torsion free sheaves, and lower bounds for the Chern classes$c_i(\FF(t))$ of twists of $\FF$, only depending on some numerical invariantsof $\FF$. Especially, we prove for rank $n$ torsion free sheaves on $\PP^N$,whose splitting type has no gap (i.e. $b_i\geq b_{i+1}\geq b_i-1$ for every$i=1, ...,n-1$), the following formula for the discriminant: \[\Delta(\FF):=2nc_2-(n-1)c_1^2\geq -{1/12}n^2(n^2-1)\] Finally in the case ofrank $n$ reflexive sheaves we obtain polynomial upper bounds for the absolutevalue of the higher Chern classes $c_3(\FF(t)), ..., c_n(\FF(t))$, for thedimension of the cohomology modules $H^i\FF(t)$ and for the Castelnuovo-Mumfordregularity of $\FF$; these polynomial bounds only depend only on $c_1(\FF)$,$c_2(\FF)$, the splitting type of $\FF$ and $t$.

SPLITTING TYPE, GLOBAL SECTIONS AND CHERN CLASSES FOR TORSION FREE SHEAVES ON PN