A famous theorem of Shokurov states that a general anticanonical divisor of a smooth Fano threefold is a smooth K3 surface. This is quite surprising since there are several examples where the base locus of the anticanonical system has codimension two. In this paper, we show that for four-dimensional Fano manifolds the behaviour is completely opposite: if the base locus is a normal surface, and hence has codimension two, all the anticanonical divisors are singular.
FANO FOURFOLDS WITH LARGE ANTICANONICAL BASE LOCUS / A. Höring, S.A. Secci. - In: JOURNAL OF THE INSTITUTE OF MATHEMATICS OF JUSSIEU. - ISSN 1474-7480. - (2025), pp. 1-31. [Epub ahead of print] [10.1017/s1474748024000604]
FANO FOURFOLDS WITH LARGE ANTICANONICAL BASE LOCUS
S.A. Secci
2025
Abstract
A famous theorem of Shokurov states that a general anticanonical divisor of a smooth Fano threefold is a smooth K3 surface. This is quite surprising since there are several examples where the base locus of the anticanonical system has codimension two. In this paper, we show that for four-dimensional Fano manifolds the behaviour is completely opposite: if the base locus is a normal surface, and hence has codimension two, all the anticanonical divisors are singular.
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