Abstract
We show, building on a recent work of Totaro (The failure of Kodaira vanishing for Fano varieties, and terminal singularities that are not Cohen–Macaulay, 2017. arXiv:1710.04364v1), that for every prime number p geqslant 3 there exists a purely log terminal pair (Z, S) of dimension 2p+2 whose plt centre S is not normal.
Highlights
The minimal model program (MMP for short) has become a fundamental tool in the study of the geometry of algebraic varieties
We show, building on a recent work of Totaro (The failure of Kodaira vanishing for Fano varieties, and terminal singularities that are not Cohen–Macaulay, 2017. arXiv:1710.04364v1), that for every prime number p 3 there exists a purely log terminal pair (Z, S) of dimension 2 p + 2 whose plt centre S is not normal
One of the main obstacles in proving the existence of flips lies in the failure of Kodaira type vanishing theorems, which makes the study of singularities in positive characteristic far more complicated than in characteristic zero
Summary
The minimal model program (MMP for short) has become a fundamental tool in the study of the geometry of algebraic varieties. Kodaira vanishing is no longer valid in positive characteristic, in [13, Theorem 3.1.1 and Proposition 4.1] the authors show the normality of plt centres for threefolds over an algebraically closed field of characteristic p > 5 and, using tools from the theory of F-singularities, they succeed in proving the existence of (pl-)flips [13, Theorem 4.12]. This result was the starting point of a series of works (see [3,5,7,15]) which established a large part of the log MMP for threefolds over fields of characteristic p > 5.
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