On the birationality of the adjunction mapping of projective varieties

TL;DR

This paper establishes criteria for when the adjunction mappings of certain smooth projective varieties are birational, extending known results and confirming parts of a conjecture for Calabi-Yau varieties.

Contribution

It provides necessary and sufficient conditions for the birationality of adjoint systems on projective varieties, generalizing previous conjectures especially for Calabi-Yau manifolds.

Findings

Criteria for birationality of $|K_X+kL|$ for $k \,\geq\, n-1$

Extension of results to Calabi-Yau $n$-folds

Partial proof of a conjecture by Gallego and Purnaprajna

Abstract

Let $X$ be a smooth projective $n$-fold such that $q(X)=0$ and $L$ a globally generated, big line bundle on $X$ such that $h^0(K_X+(n-2)L) >0$. We give necessary and sufficient conditions for the adjoint systems $|K_X+kL|$ to be birational for $k \geq n-1$. In particular, for Calabi-Yau $n$-folds we generalize and prove parts of a conjecture of Gallego and Purnaprajna.