The Minimal Model Program Revisited
We give a light introduction to some recent developments in Mori theory, and to our recent direct proof of the finite generation of the canonical ring.
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arXiv subjects
Paolo Cascini
Publications and source records attributed to Paolo Cascini.
The augmented base locus in positive characteristic
Let L be a nef line bundle on a projective scheme X in positive characteristic. We prove that the augmented base locus of L is equal to the union of the irreducible closed subsets V of X such that the restriction of L to V is not big. For a smooth variety in characteristic zero, this was proved by Nakamaye using vanishing theorems.
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Symplectic generic complex structures on 4-manifolds with b_+ = 1
We study symplectic structures on Kähler surfaces with p_g = 0. We give an example of a projective surface which admits a symplectic structure which is not compatible with any Kähler metric.
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Existence of minimal models for varieties of log general type
We prove that the canonical ring of a smooth projective variety is finitely generated.
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Kähler-Ricci Flow and the Minimal Model Program for Projective Varieties
In this note we propose to show that the Kähler-Ricci flow fits naturally within the context of the Minimal Model Program for projective varieties. In particular we show that the flow detects, in finite time, the contraction theorem of any extremal ray and we analyze the singularities of the metric in the case of divisorial contractions for varieties of general type. In case one has a smooth minimal model of general type (i.e., the canonical bundle is nef and big), we show infinite time existence and analyze the singularities.
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On the moduli space of the Schwarzenberger bundles
By proving a particular case of a conjecture of Drezet, we show that a component of the Maruyama scheme of the semi-stable sheaves on the projective space $\PP^n$ of rank n and Chern polynomial $(1+t)^{n+2}$ is isomorphic to the Kronecher moduli $N(n+1,2,n+2)$, for any odd n. In particular, such scheme defines a smooth minimal compactification of the moduli space of the rational normal curves in $\PP^n$, that generalizes the construction defined by G. Ellinsgrud, R. Piene and S. Strømme in the case $n=3$.
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On a compactification of the moduli space of the rational normal curves
For any odd $n$, we describe a smooth minimal (i.e. obtained by adding an irreducible hypersurface) compactification $\tilde S_n$ of the quasi-projective homogeneous variety $S_{n}=PGL(n+1)/SL(2)$ that parameterizes the rational normal curves in $P^n$. We show that $\tilde S_{n}$ is isomorphic to a component of the Maruyama scheme of the semi-stable sheaves on $P^n$ of rank $n$ and Chern polynomial $(1+t)^{n+2}$ and we compute its Betti numbers. In particular $\tilde S_{3}$ is isomorphic to the variety of nets of quadrics defining twisted cubics, studied by G. Ellinsgrud, R. Piene and S. Strømme (Space curves, Proc. Conf., LNM 1266).
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Weighted Tango Bundles And Their Moduli Spaces
We define a new class of holomorphic (n-1)-bundles on the n-dimensional complex projective space, that contains the Tango bundles and their stable Horrock's pull-backs; we show that these bundles are invariant under small deformations and that their isomorphism classes correspond to smooth points of moduli spaces.
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