Nef and pseudoeffective cones of product of projective bundles over a curve

摘要

Let X = P(E-1) x (C) P(E-2), where C is any smooth irreducible curve and E-1, E-2 are vector bundles over C. In this paper, we calculate the nef cone and pseudoeffective cone of X whenever E-1 and E-2 are semistable, and show that they coincide. We also calculate the nef cone and pseudoeffective cone of X in case both E-1 and E-2 have rank 2, and neither E-1 nor E-2 is semistable or at least one of them is semistable. As a result, in the case when both E-1 and E-2 are of rank 2 bundles on C, we get that nef cone and pseudoeffective cone of X coincide iff both E-1 and E-2 are semistable. (C) 2018 Elsevier Masson SAS. All rights reserved.