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Register a new userA Minimal Model Program for $mathbb{Q}$-Gorenstein varieties
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Boris Pasquier
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The main results of this paper are already known (V.V. Shokurov, the non-vanishing theorem, 1985). Moreover, the non-$mathbb{Q}$-factorial MMP was more recently considered by O~Fujino, in the case of toric varieties (Equivariant completions of toric contraction morphisms, 2006), for klt pairs (Special termination and reduction to pl flips, 2007) and more generally for log-canonical pairs (Foundation of the minimal model program, 2014). Here we rewrite the proofs of some of these results, by following the proofs given by Y. Kawamata, K. Matsuda, and K. Matsuki (Introduction to the minimal model problem, 1985) of the same results in $mathbb{Q}$-factorial MMP. And, in the family of $mathbb{Q}$-Gorenstein spherical varieties, we answer positively to the questions of existence of flips and of finiteness of sequences of flips. I apologize for the first version of this paper, which I wrote without knowing that these results already exist.
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In a previous work, we described the Minimal Model Program in the family of $Qbb$-Gorenstein projective horospherical varieties, by studying certain continuous changes of moment polytopes of polarized horospherical varieties. Here, we summarize the results of the previous work and we explain how to generalize them in order to describe the Log Minimal Model Program for pairs $(X,D)$ when $X$ is a projective horospherical variety.
We describe the Minimal Model Program in the family of $mathbb{Q}$-Gorenstein projective horospherical varieties, by studying a family of polytopes defined from the moment polytope of a Cartier divisor of the variety we begin with. In particular, we generalize the results on MMP in toric varieties due to M. Reid, and we complete the results on MMP in spherical varieties due to M. Brion in the case of horospherical varieties.
We show that the reduction to positive characteristic of the multiplier ideal in the sense of de Fernex and Hacon agrees with the test ideal for infinitely many primes, assuming that the variety is numerically Q-Gorenstein. It follows, in particular, that this reduction property holds in dimension 2 for all normal surfaces.
We analyze and present an effective solution to the minimal Gorenstein cover problem: given a local Artin k-algebra $A = k[[x 1 ,. .. x n ]]/I$, compute an Artin Gorenstein $k$-algebra $G = k[[x 1 ,. .. x n ]]/J$ such that $ell(G)--ell(A)$ is minimal. We approach the problem by using Macaulays inverse systems and a modification of the integration method for inverse systems to compute Gorenstein covers. We propose new characterizations of the minimal Gorenstein cover and present a new algorithm for the effective computation of the variety of all minimal Gorenstein covers of A for low Gorenstein colength. Experimentation illustrates the practical behavior of the method.
We prove the Conjecture of Catenese--Chen--Zhang: the inequality $K_X^3geq frac{4}{3}p_g(X)-frac{10}{3}$ holds for all projective Gorenstein minimal 3-folds $X$ of general type.
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