scholarly journals Log minimal model program for the moduli space of stable curves: the first flip

2013 ◽  
Vol 177 (3) ◽  
pp. 911-968 ◽  
Author(s):  
Brendan Hassett ◽  
Donghoon Hyeon
Keyword(s):  
Moduli Space ◽  
Minimal Model ◽  
Model Program ◽  
Minimal Model Program ◽  
Stable Curves ◽  
Log Minimal Model Program

scholarly journals Modular compactifications of the space of pointed elliptic curves II

2011 ◽  
Vol 147 (6) ◽  
pp. 1843-1884 ◽  
Author(s):  
David Ishii Smyth
Keyword(s):  
Elliptic Curves ◽  
Moduli Spaces ◽  
Minimal Model ◽  
Model Program ◽  
Minimal Model Program ◽  
Stable Curves ◽  
Log Minimal Model Program

AbstractWe prove that the moduli spaces of n-pointed m-stable curves introduced in our previous paper have projective coarse moduli. We use the resulting spaces to run an analogue of Hassett’s log minimal model program for $\overline {M}_{1,n}$.

scholarly journals On existence of log minimal models

2010 ◽  
Vol 146 (4) ◽  
pp. 919-928 ◽  
Author(s):  
Caucher Birkar
Keyword(s):  
Minimal Model ◽  
Kodaira Dimension ◽  
Reduction Theorem ◽  
Model Program ◽  
Minimal Models ◽  
Minimal Model Program ◽  
Log Minimal Model Program ◽  
Negative Kodaira Dimension

AbstractIn this paper, we prove that the log minimal model program in dimension d−1 implies the existence of log minimal models for effective lc pairs (e.g. of non-negative Kodaira dimension) in dimension d. In fact, we prove that the same conclusion follows from a weaker assumption, namely, the log minimal model program with scaling in dimension d−1. This enables us to prove that effective lc pairs in dimension five have log minimal models. We also give new proofs of the existence of log minimal models for effective lc pairs in dimension four and of the Shokurov reduction theorem.

scholarly journals Smooth projective horospherical varieties of Picard group $\mathbb{Z}^2$

2020 ◽  
Vol Volume 4 ◽  
Author(s):  
Boris Pasquier
Keyword(s):  
Minimal Model ◽  
Picard Group ◽  
Model Program ◽  
Minimal Model Program ◽  
Log Minimal Model Program ◽  
International Audience

International audience We classify all smooth projective horospherical varieties of Picard group $\mathbb{Z}^2$ and we give a first description of their geometry via the Log Minimal Model Program.

Geography of log models via asymptotic base loci

2014 ◽  
Vol 25 (09) ◽  
pp. 1450089
Author(s):  
Sung Rak Choi
Keyword(s):  
Minimal Model ◽  
Partial Answer ◽  
Model Program ◽  
Minimal Model Program ◽  
Structure Theorems ◽  
Log Minimal Model Program ◽  
Base Loci

The geography of log models refers to the decomposition of the set of effective adjoint divisors into the polytopes defined by the resulting models obtained by the log minimal model program. We will describe the geography of log models in terms of the asymptotic base loci and Zariski decompositions of adjoint divisors. As an application, we prove some structure theorems on partially ample cones, thereby giving a partial answer to a question of B. Totaro.

scholarly journals Moduli of products of stable varieties

2013 ◽  
Vol 149 (12) ◽  
pp. 2036-2070 ◽  
Author(s):  
Bhargav Bhatt ◽  
Wei Ho ◽  
Zsolt Patakfalvi ◽  
Christian Schnell
Keyword(s):  
Moduli Space ◽  
Moduli Spaces ◽  
Minimal Model ◽  
Complex Numbers ◽  
Model Program ◽  
Minimal Model Program ◽  
Cotangent Complex ◽  
Geometric Methods ◽  
Local Results

AbstractWe study the moduli space of a product of stable varieties over the field of complex numbers, as defined via the minimal model program. Our main results are: (a) taking products gives a well-defined morphism from the product of moduli spaces of stable varieties to the moduli space of a product of stable varieties; (b) this map is always finite étale; and (c) this map very often is an isomorphism. Our results generalize and complete the work of Van Opstall in dimension$1$. The local results rely on a study of the cotangent complex using some derived algebro-geometric methods, while the global ones use some differential-geometric input.

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