scholarly journals Morphisms of projective varieties from the viewpoint of minimal model theory

2003 ◽  
Vol 413 ◽  
pp. 1-72 ◽  
Author(s):  
Marco Andreatta ◽  
Massimiliano Mella
Keyword(s):  
Model Theory ◽  
Minimal Model ◽  
Projective 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 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 On minimal model theory for log abundant lc pairs

2020 ◽  
Vol 2020 (767) ◽  
pp. 109-159
Author(s):  
Kenta Hashizume ◽  
Zheng-Yu Hu
Keyword(s):  
Model Theory ◽  
Minimal Model ◽  
Canonical Divisor ◽  
Detailed Proof ◽  
Log Canonical

AbstractUnder the assumption of the minimal model theory for projective klt pairs of dimension n, we establish the minimal model theory for lc pairs {(X/Z,\Delta)} such that the log canonical divisor is relatively log abundant and its restriction to any lc center has relative numerical dimension at most n. We also give another detailed proof of results by the second author, and study termination of log MMP with scaling.

scholarly journals MINIMAL MODEL THEORY FOR LOG SURFACES IN FUJIKI’S CLASS

2020 ◽  
pp. 1-27
Author(s):  
OSAMU FUJINO
Keyword(s):  
Model Theory ◽  
Minimal Model ◽  
Log Canonical

We establish the minimal model theory for $\mathbb{Q}$ -factorial log surfaces and log canonical surfaces in Fujiki’s class ${\mathcal{C}}$ .

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