Explicit \(\log\) Fano structures on blow-ups of projective spaces (Q2834911)

Let \(X_k^n\) be the blow-up of the projective space \(\mathbb P^n\) along \(k\) points in general position; for \(n=2\) and \(k \leq 8\) these varieties are the well-known del Pezzo surfaces, i.e. their anticanonical bundle is ample. For \(n \geq 3\), as soon as \(k \geq 2\), the line bundle \(-K_X\) is no longer ample (ampleness fails for instance on strict transforms of lines joining two of the blown-up points), but, for small values of \(k\), the geometry of \(X_{k}^n\) is very similar to the one of a Fano manifold (for instance \(X_{k}^n\) is a Mori dream space).NEWLINENEWLINEThe precise notion is that of a log-Fano variety, that is a normal projective \(\mathbb Q\)-factorial variety with an effective \(\mathbb Q\)-divisor \(\Delta\) such that \((-K_X+\Delta)\) is ample and the singularities of \((X,\Delta)\) are klt.NEWLINENEWLINEIn the paper under review the authors first determine for which pairs \((n,k)\) the variety \(X_k^n\) is a log-Fano variety (Theorem 1.3). The proof of this result does not produce a divisor \(\Delta\) which makes \(X_k^n\) log-Fano, so this problem is investigated in the second part of the paper (Theorems 4.3, 4.5, 5.8 and 5.10).NEWLINENEWLINEFinally, the techniques developed in the paper are used to address a question of Hassett about log canonicity of moduli spaces of weighted pointed stable curves.