A canonical linear system associated to adjoint divisors in characteristic $p > 0$

Details A canonical linear system associated to adjoint divisors in characteristic $p > 0$ A canonical linear system associated to adjoint divisors in characteristic $p > 0$

2011-07-19

arxiv.org/abs/1107.3833v3

Mathematics

Algebraic Geometry

15 pages, improved exposition and additional historical discussion / references added

Scientific paper

Suppose that $X$ is a projective variety over an algebraically closed field of characteristic $p > 0$. Further suppose that $L$ is an ample (or more generally in some sense positive) divisor. We study a natural linear system in $H^0(X, \O_X(K_X + L))$. We further generalize this to incorporate a boundary divisor $\Delta$. We show that these subsystems behave like the global sections associated to multiplier ideals, $H^0(X, \mJ(X, \Delta) \tensor L)$ in characteristic zero. In particular, we show that these systems are in many cases base-point-free. As an application, we prove that a theorem of Esnault-Viehweg also holds in characteristic $p > 0$. While the original proof utilized Kawamata-Viehweg vanishing and variants of multiplier ideals, our proof uses test ideals.

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