Journal of the Ramanujan Mathematical Society

Andreas Leopold Knutsen ¹, Angelo Felice Lopez ², Roberto Munoz ³

Affiliations
1 Andreas Leopold Knutsen, Department of Mathematics, University of Bergen, Johannes Brunsgate 12, 5008 Bergen, NO
2 Angelo Felice Lopez, Dipartimento di Matematica, Universita di Roma Tre, Largo San Leonardo Murialdo 1, 00146, Roma, IT
3 Roberto Munoz, ESCET, Departamento de Matematica Aplicada, Universidad Rey Juan Carlos, 28933 Mostoles (Madrid), ES

Abstract

Given an Enriques surface S embedded in 𝕡^r with a certain linear system, we show that S is not hyperplane section of any threefold X ⊂ 𝕡^r+1 that is not a cone over S. This special case completes the proof of the genus bound for Enriques–Fano threefolds [11, Thm. 1.5].

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Andreas Leopold Knutsen ¹

Affiliations
1 Department of Mathematics, University of Bergen, Johannes Brunsgate 12, 5008 Bergen, NO

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 h⁰(K_X +(n−2)L) > 0. We give necessary and sufficient conditions for the adjoint systems |K_X + kL| to be birational for k ≥ n − 1. In particular, for Calabi-Yau n-folds we generalize and prove parts of a conjecture of Gallego and Purnaprajna.

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