1,675 research outputs found
Rational curves of minimal degree and characterizations of
In this paper we investigate complex uniruled varieties whose rational
curves of minimal degree satisfy a special property. Namely, we assume that the
tangent directions to such curves at a general point form a linear
subspace of . As an application of our main result, we give a unified
geometric proof of Mori's, Wahl's, Campana-Peternell's and
Andreatta-Wi\'sniewski's characterizations of .Comment: 14 page
K\"ahler-Einstein Metrics for some quasi smooth log del Pezzo Surfaces
Recently Johnson and Koll\'ar 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 K\"ahler-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 K\"ahler-Einstein metrics and tigers for those
surfaces for which the question was still open. In order to do so, we will use
techniques developed earlier by Nadel and Demailly and Koll\'ar.Comment: 10 page
Codimension 1 Mukai foliations on complex projective manifolds
In this paper we classify codimension 1 Mukai foliations on complex
projective manifold
On Fano foliations 2
In this paper we study mildly singular del Pezzo foliations on complex
projective manifolds with Picard number oneComment: 13 pages, several minor clarifications and correction
On Fano foliations
In this paper we address Fano foliations on complex projective varieties.
These are foliations whose anti-canonical class is ample. We focus our
attention on a special class of Fano foliations, namely del Pezzo foliations on
complex projective manifolds. We show that these foliations are algebraically
integrable, with one exceptional case when the ambient space is a projective
space. We also provide a classification of del Pezzo foliations with mild
singularities.Comment: 38 pages, several minor clarifications and correction
On smooth lattice polytopes with small degree
Toric geometry provides a bridge between the theory of polytopes and
algebraic geometry: one can associate to each lattice polytope a polarized
toric variety. In this paper we explore this correspondence to classify smooth
lattice polytopes having small degree, extending a classification provided by
Dickenstein, Di Rocco and Piene. Our approach consists in interpreting the
degree of a polytope as a geometric invariant of the corresponding polarized
variety, and then applying techniques from Adjunction Theory and Mori Theory.Comment: 12 pages. v2: attributions fixed, reference adde
Explicit log Fano structures on blow-ups of projective spaces
In this paper we determine which blow-ups of at general
points are log Fano, that is, when there exists an effective
-divisor such that is ample and the pair
is klt. For these blow-ups, we produce explicit boundary divisors
making log Fano.Comment: 30 page
On the Fano variety of linear spaces contained in two odd-dimensional quadrics
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>0. 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).Comment: 31 pages. Minor revision, to appear in Geometry & Topolog
On codimension 1 del Pezzo foliations on varieties with mild singularities
In this paper we extend to the singular setting the theory of Fano foliations
developed in our previous paper. A Q-Fano foliation on a complex projective
variety X is a foliation F whose anti-canonical class is an ample Q-Cartier
divisor. In the spirit of Kobayashi-Ochiai Theorem, we prove that under some
conditions the index i of a Q-Fano foliation is bounded by the rank r of F, and
classify the cases in which i=r. Next we consider Q-Fano foliations F for which
i=r-1. These are called del Pezzo foliations. We classify codimension 1 del
Pezzo foliations on mildly singular varieties.Comment: 20 pages, to appear in Mathematische Annale
Rational Curves on Varieties
The aim of these notes is to give a introduction to the ideas and techniques
of handling rational curves on varieties. The main emphasis is on varieties
with many rational curves which are the higher dimensional analogs of rational
curves and surfaces.
The first six sections closely follow the lectures of J. Koll\'ar given at
the summer school ``Higher dimensional varieties and rational points''. The
notes were written up by C. Araujo and substantially edited later. Revised
version with 2 new sections
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