# Birational geometry and Gabriel's theorem (Joint with J. Calabrese)

##
Roberto Pirisi
(Università di Napoli Federico II)

created by daniele on 17 Sep 2018

19 sep 2018
-- 15:00

Aula Mancini, SNS, Pisa

**Abstract.**

It is a well-known result by Gabriel that the category of coherent sheaves of a Noetherian scheme X->S contains sufficient information to reconstruct it, or in another words that an isomorphism Coh(Y) -> Coh(X) (as S-linear categories) induces a corresponding isomorphism X -> Y and vice-versa. As a corollary, the group of automorphisms of Coh(X) is a semidirect product of Aut(X) and Pic(X) (note that tensoring by a line bundle induces an isomorphism on coherent sheaves). In a recent joint work with J.Calabrese, we prove a birational extension of this result, showing that when X is of finite type over a field the geometry of X "up to dimension d or lower" is controlled by the category Coh_{{>d}}(X) of coherent sheaves modulo those supported in dimension d or lower, and identify the automorphisms of this category as a semidirect product of the automorphisms of X "up to dimension d or lower" with the group of line bundles on X "up to dimension d or lower".