Effective Adjunction Theory

Marco Andreatta; Claudio Fontanari

paper

Effective Adjunction Theory

arXiv:1703.00758

Abstract

Here we investigate the property of effectivity for adjoint divisors. Among others, we prove the following results: (i) A normal projective variety $X$ with at most canonical singularities is uniruled if and only if for each very ample Cartier divisor $H$ on $X$ we have $H^{0} (X, m_{0} K_{X} + H) = 0$ for some $m_{0} = m_{0} (H) > 0$ . (ii) Let $(X, L)$ be a polarized manifold of dimension $4$ and let $t$ be an integer with $t \geq 3$ . If $K_{X} + t L$ is pseudo-effective, then $H^{0} (X, K_{X} + t L) \neq = 0$ .

Minor revision, to appear in "Annali dell'Università di Ferrara"