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

### Lei Zhang

Vol. 16 (2022), No. 10, 2339–2384
##### Abstract

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

##### Keywords
effectivity problems, pluricanonical systems, positive characteristic, Frobenius stable sections
##### Mathematical Subject Classification
Primary: 14E05, 14E30
##### Milestones
Received: 27 February 2021
Revised: 5 January 2022
Accepted: 18 February 2022
Published: 28 January 2023
##### Authors
 Lei Zhang School of Mathematical Science University of Science and Technology of China Hefei China