scholarly journals An approach of the minimal model program for horospherical varieties via moment polytopes

2015 ◽  
Vol 2015 (708) ◽  
Author(s):  
Boris Pasquier
Keyword(s):  
Minimal Model ◽  
Toric Varieties ◽  
Cartier Divisor ◽  
Model Program ◽  
Minimal Model Program ◽  
Spherical Varieties ◽  
The Family ◽  
Moment Polytope ◽  
The Moment ◽  
Moment Polytopes

AbstractWe describe the minimal model program in the family of 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 for toric varieties due to M. Reid, and we complete the results on MMP for spherical varieties due to M. Brion in the case of horospherical varieties.

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 Brick Manifolds and Toric Varieties of Brick Polytopes

10.37236/5038 ◽  
2016 ◽  
Vol 23 (2) ◽  
Author(s):  
Laura Escobar
Keyword(s):  
Toric Variety ◽  
Toric Varieties ◽  
Moment Map ◽  
Twisted Product ◽  
General Fiber ◽  
Moment Polytope ◽  
The Moment ◽  
Subword Complex ◽  
Do So

Bott-Samelson varieties are a twisted product of $\mathbb{C}\mathbb{P}^1$'s with a map into $G/B$. These varieties are mostly studied in the case in which the map into $G/B$ is birational to the image; however in this paper we study a fiber of this map when it is not birational. We prove that in some cases the general fiber, which we christen a brick manifold, is a toric variety. In order to do so we use the moment map of a Bott-Samelson variety to translate this problem into one in terms of the "subword complexes" of Knutson and Miller. Pilaud and Stump realized certain subword complexes as the dual of the boundary of a polytope which generalizes the brick polytope defined by Pilaud and Santos. For a nice family of words, the brick polytope is the generalized associahedron realized by Hohlweg, Lange and Thomas. These stories connect in a nice way: we show that the moment polytope of the brick manifold is the brick polytope. In particular, we give a nice description of the toric variety of the associahedron. We give each brick manifold a stratification dual to the subword complex. In addition, we relate brick manifolds to Brion's resolutions of Richardon varieties.

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 $\rho (Z/ W)= 3$.

scholarly journals Cuspidal curves, minimal models and Zaidenberg’s finiteness conjecture

2019 ◽  
Vol 2019 (747) ◽  
pp. 147-174 ◽  
Author(s):  
Karol Palka
Keyword(s):  
Minimal Model ◽  
Resolution Of Singularities ◽  
Model Program ◽  
Minimal Models ◽  
Minimal Model Program ◽  
Effective Version

Abstract Let {E\subseteq\mathbb{P}^{2}} be a complex rational cuspidal curve and let {(X,D)\to(\mathbb{P}^{2},E)} be the minimal log resolution of singularities. We prove that E has at most six cusps and we establish an effective version of the Zaidenberg finiteness conjecture (1994) concerning Eisenbud–Neumann diagrams of E. This is done by analyzing the Minimal Model Program run for the pair {(X,\frac{1}{2}D)} . Namely, we show that {\mathbb{P}^{2}\setminus E} is {\mathbb{C}^{**}} -fibred or for the log resolution of the minimal model the Picard rank, the number of boundary components and their self-intersections are bounded.

scholarly journals Families of canonically polarized manifolds over log Fano varieties

2013 ◽  
Vol 149 (6) ◽  
pp. 1019-1040
Author(s):  
Daniel Lohmann
Keyword(s):  
Minimal Model ◽  
Projective Variety ◽  
Fano Varieties ◽  
Model Program ◽  
Minimal Model Program ◽  
Smooth Family ◽  
Family Of Curves ◽  
Moving Cone ◽  
Extremal Ray

AbstractLet $(X,D)$ be a dlt pair, where $X$ is a normal projective variety. We show that any smooth family of canonically polarized varieties over $X\setminus \,{\rm Supp}\lfloor D \rfloor $ is isotrivial if the divisor $-(K_X+D)$ is ample. This result extends results of Viehweg–Zuo and Kebekus–Kovács. To prove this result we show that any extremal ray of the moving cone is generated by a family of curves, and these curves are contracted after a certain run of the minimal model program. In the log Fano case, this generalizes a theorem by Araujo from the klt to the dlt case. In order to run the minimal model program, we have to switch to a $\mathbb Q$-factorialization of $X$. As $\mathbb Q$-factorializations are generally not unique, we use flops to pass from one $\mathbb Q$-factorialization to another, proving the existence of a $\mathbb Q$-factorialization suitable for our purposes.

scholarly journals MINIMAL MODEL PROGRAM WITH SCALING AND ADJUNCTION THEORY

2013 ◽  
Vol 24 (02) ◽  
pp. 1350007 ◽  
Author(s):  
MARCO ANDREATTA
Keyword(s):  
Minimal Model ◽  
Projective Variety ◽  
Model Program ◽  
Minimal Model Program ◽  
Projective Varieties ◽  
Complex Projective Variety ◽  
Adjunction Theory ◽  
Birational Equivalence ◽  
Big Line Bundle ◽  
Complex Projective

Let (X, L) be a quasi-polarized pair, i.e. X is a normal complex projective variety and L is a nef and big line bundle on it. We study, up to birational equivalence, the positivity (nefness) of the adjoint bundles KX + rL for high rational numbers r. For this we run a Minimal Model Program with scaling relative to the divisor KX + rL. We give then some applications, namely the classification up to birational equivalence of quasi-polarized pairs with sectional genus 0, 1 and of embedded projective varieties X ⊂ ℙN with degree smaller than 2 codim ℙN(X) + 2.

scholarly journals Reduction maps and minimal model theory

2012 ◽  
Vol 149 (2) ◽  
pp. 295-308 ◽  
Author(s):  
Yoshinori Gongyo ◽  
Brian Lehmann
Keyword(s):  
Model Theory ◽  
Minimal Model ◽  
Model Program ◽  
Minimal Model Program ◽  
Birational Model ◽  
Terminal Pair

AbstractWe use reduction maps to study the minimal model program. Our main result is that the existence of a good minimal model for a Kawamata log terminal pair (X,Δ) can be detected on a birational model of the base of the (KX+Δ)-trivial reduction map. We then interpret the main conjectures of the minimal model program as a natural statement about the existence of curves on X.

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 $\overline {M}_{1,n}$.

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