Let X be a Fano variety of index k such that the non-klt locus Nklt(X) is not empty. We prove that dimNklt(X) >= k-1 and equality holds if and only if Nklt(X) is a linear Pk-1. In this case, X has lc singularities and is a generalized cone with Nklt(X) as vertex. If X has lc singularities and dim Nklt(X)= k we describe the non-klt locus Nklt(X) and the global geometry of X. Moreover, we construct examples to show that all the classification results are effective.
Fano varieties with small non-klt locus
NOVELLI, CARLA
2015
Abstract
Let X be a Fano variety of index k such that the non-klt locus Nklt(X) is not empty. We prove that dimNklt(X) >= k-1 and equality holds if and only if Nklt(X) is a linear Pk-1. In this case, X has lc singularities and is a generalized cone with Nklt(X) as vertex. If X has lc singularities and dim Nklt(X)= k we describe the non-klt locus Nklt(X) and the global geometry of X. Moreover, we construct examples to show that all the classification results are effective.
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