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.

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 Stability conditions and birational geometry of projective surfaces

2014 ◽  
Vol 150 (10) ◽  
pp. 1755-1788 ◽  
Author(s):  
Yukinobu Toda
Keyword(s):  
Moduli Spaces ◽  
Minimal Model ◽  
Derived Category ◽  
Stability Conditions ◽  
Birational Geometry ◽  
Projective Surface ◽  
Model Program ◽  
Minimal Model Program ◽  
Coherent Sheaves ◽  
Projective Surfaces

AbstractWe show that the minimal model program on any smooth projective surface is realized as a variation of the moduli spaces of Bridgeland stable objects in the derived category of coherent sheaves.

scholarly journals ON THE FIRST STEPS OF THE MINIMAL MODEL PROGRAM FOR THE MODULI SPACE OF STABLE POINTED CURVES

2021 ◽  
pp. 1-67
Author(s):  
Giulio Codogni ◽  
Luca Tasin ◽  
Filippo Viviani
Keyword(s):  
Moduli Space ◽  
Minimal Model ◽  
Model Program ◽  
Minimal Model Program ◽  
Geometric Properties ◽  
Canonical Models ◽  
Log Canonical

Abstract The aim of this paper is to study all the natural first steps of the minimal model program for the moduli space of stable pointed curves. We prove that they admit a modular interpretation, and we study their geometric properties. As a particular case, we recover the first few Hassett–Keel log canonical models. As a by-product, we produce many birational morphisms from the moduli space of stable pointed curves to alternative modular projective compactifications of the moduli space of pointed curves.

scholarly journals FAMILIES OF AFFINE RULED SURFACES: EXISTENCE OF CYLINDERS

2016 ◽  
Vol 223 (1) ◽  
pp. 1-20 ◽  
Author(s):  
ADRIEN DUBOULOZ ◽  
TAKASHI KISHIMOTO
Keyword(s):  
Minimal Model ◽  
Finite Extension ◽  
Smooth Curve ◽  
Ruled Surfaces ◽  
Model Program ◽  
Minimal Model Program ◽  
Generic Fiber ◽  
Projective Model ◽  
The Family

We show that the generic fiber of a family $f:X\rightarrow S$ of smooth $\mathbb{A}^{1}$-ruled affine surfaces always carries an $\mathbb{A}^{1}$-fibration, possibly after a finite extension of the base $S$. In the particular case where the general fibers of the family are irrational surfaces, we establish that up to shrinking $S$, such a family actually factors through an $\mathbb{A}^{1}$-fibration $\unicode[STIX]{x1D70C}:X\rightarrow Y$ over a certain $S$-scheme $Y\rightarrow S$ induced by the MRC-fibration of a relative smooth projective model of $X$ over $S$. For affine threefolds $X$ equipped with a fibration $f:X\rightarrow B$ by irrational $\mathbb{A}^{1}$-ruled surfaces over a smooth curve $B$, the induced $\mathbb{A}^{1}$-fibration $\unicode[STIX]{x1D70C}:X\rightarrow Y$ can also be recovered from a relative minimal model program applied to a smooth projective model of $X$ over $B$.

scholarly journals Relations in the Sarkisov program

2013 ◽  
Vol 149 (10) ◽  
pp. 1685-1709 ◽  
Author(s):  
Anne-Sophie Kaloghiros
Keyword(s):  
Finite Number ◽  
Minimal Model ◽  
Model Program ◽  
Minimal Model Program ◽  
Fibre Space ◽  
Projective Varieties ◽  
Birational Maps ◽  
Birational Map ◽  
End Products

AbstractThe Sarkisov program studies birational maps between varieties that are end products of the Minimal Model Program (MMP) on nonsingular uniruled varieties. If $X$ and $Y$ are terminal $ \mathbb{Q} $-factorial projective varieties endowed with a structure of Mori fibre space, a birational map $f: X\dashrightarrow Y$ is the composition of a finite number of elementary Sarkisov links. This decomposition is in general not unique: two such define a relation in the Sarkisov program. I define elementary relations, and show they generate relations in the Sarkisov program. Roughly speaking, elementary relations are the relations among the end products of suitable relative MMPs of $Z$ over $W$ with $\rho (Z/ W)= 3$.

Sign in / Sign up

Export Citation Format

Share Document