Some examples of log Fano structures on blow-ups along subvarieties in products of two projective spaces

Abstract

We give a series of examples of log Fano manifolds in any dimension greater than or equal to three by using successive blow-ups along subvarieties in products of two projective spaces. More precisely, we first blow-up a product of two projective spaces along a smooth hypersurface contained in a fiber of one of the two projections and then along the strict transform of a fiber (intersecting the center of the first blow-up) of the other projection. By computing the nef cone and giving an explicit boundary divisor, we show that the resulting variety is always a log Fano manifold. Moreover, the effective cone and the extremal contractions are described. Note that this variety depends on three integral parameters: the dimensions of the two projective spaces and the degree of the hypersurface, which is the center of the first blow-up. Hence, we obtain a series of examples of log Fano manifolds. We also determine which ones are weak Fano.