# On the numerical dimension of pseudo-effective divisors in positive characteristic

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Karl Schwede , Penn State University
Fine Hall 322

Suppose that X is a smooth algebraic variety over an algebraically closed field and that D is a pseudo effective R-divisor on X. In characteristic zero, by utilizing vanishing theorems, Nakayama proved that if D is not numerically equivalent to the negative part of its Zariski decomposition, then D is pseudo-effective. We prove the same result in characteristic p > 0 using the Frobenius morphism as a replacement for vanishing theorems. This is joint work with Paolo Cascini, Christopher Hacon and Mircea Mustata.