Abstract. Recently Johnson and Kollár determined the complete list of anticanonically embedded quasi-smooth log del Pezzo surfaces in weighted projective 3-spaces. They also proved that many of those surfaces admit a Kähler-Einstein metric, and that some of them do not have tigers.The aim of this paper is to settle the question of the existence of Kähler-Einstein metrics and tigers for those surfaces for which the question was left open. In order to do so, we will use techniques developed earlier by Nadel, Demailly and Kollár.
In this paper we determine which blow-ups X of P n at general points are log Fano, that is, when there exists an effective Q-divisor ∆ such that −(KX + ∆) is ample and the pair (X, ∆) is klt. For these blow-ups, we produce explicit boundary divisors ∆ making X log Fano. and suppose that d ≤ n − s. Thenwhere the union is taken over all subsets J ⊂ I 1 ∪ I 2 satisfying d(I i , J) = s for i = 1, 2. Moreover, for a general point in any irreducible component of the above intersections, the intersection is transverse.Proof. We note that the assumptions of the theorem imply that d = k 1 + k 2 + s − 1 and m 1 − m 2 = 2(k 2 − k 1 ).Let J ⊂ I 1 ∪I 2 be such that d(I i , J) = s for i = 1, 2. We shall prove that Y d−s J ⊂ Y d I 1 ∩Y d I 2 . Write J = J 1 ∪ J 2 , where J i ⊂ I i , i = 1, 2, set ℓ i := |J i | , i = 1, 2, and ℓ = |J| = ℓ 1 + ℓ 2 .
In this paper we investigate complex uniruled varieties X whose rational curves of minimal degree satisfy a special property. Namely, we assume that the tangent directions to such curves at a general point x ∈ X form a linear subspace of T x X. As a first application of our main result, we give a unified geometric proof of Mori's, Wahl's, Campana-Peternell's and Andreatta-Wiśniewski's characterizations of P n . We also give a characterization of products of projective spaces in terms of the geometry of their families of rational curves of minimal degree.
In this paper we describe the geometry of the 2m-dimensional Fano manifold G parametrizing (m − 1)-planes in a smooth complete intersection Z of two quadric hypersurfaces in the complex projective space P 2m+2 , for m ≥ 1. We show that there are exactly 2 2m+2 distinct isomorphisms in codimension one between G and the blow-up of P 2m at 2m + 3 general points, parametrized by the 2 2m+2 distinct m-planes contained in Z, and describe these rational maps explicitly. We also describe the cones of nef, movable and effective divisors of G, as well as their dual cones of curves. Finally, we determine the automorphism group of G.These results generalize to arbitrary even dimension the classical description of quartic del Pezzo surfaces (m = 1).
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