Frobenius stable pluricanonical systems on threefolds of general type in positive characteristic

TL;DR

This paper studies the effectivity of Frobenius stable pluricanonical systems on threefolds of general type in positive characteristic, establishing bounds for non-emptiness and birationality of certain linear systems.

Contribution

It introduces a sub-linear system generated by Frobenius stable sections and proves effective bounds for non-emptiness and birationality on minimal terminal threefolds.

Findings

Non-empty for n ≥ 28

Defines a birational map for n ≥ 42

Applicable to threefolds with q(X)>0 or Gorenstein singularities

Abstract

This paper aims to investigate effectivity problems of pluricanonical systems on varieties of general type in positive characteristic. In practice, we will consider a sub-linear system $|S^0_{-}(X, K_X + nK_X)| \subseteq |H^0(X, K_X +nK_X)|$ generated by certain Frobenius stable sections, and prove that for a minimal terminal threefold $X$ of general type with either $q(X)>0$ or Gorenstein singularities, if $n\geq 28$ then $|S^0_{-}(X, K_X + nK_X)| \neq \emptyset$; if $n\geq 42$ then the linear system $|S^0_{-}(X, K_X + nK_X)|$ defines a birational map.