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Algebraic Geometry Seminar
Seminar information archive ～05/22｜Next seminar｜Future seminars 05/23～
| Date, time & place | Friday 13:30 - 15:00 118Room #118 (Graduate School of Math. Sci. Bldg.) |
|---|---|
| Organizer(s) | GONGYO Yoshinori, KAWAKAMI Tatsuro, ENOKIZONO Makoto |
Seminar information archive
2012/06/04
15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)
Kiwamu Watanabe (Saitama University)
Smooth P1-fibrations and Campana-Peternell conjecture (ENGLISH)
Kiwamu Watanabe (Saitama University)
Smooth P1-fibrations and Campana-Peternell conjecture (ENGLISH)
[ Abstract ]
We give a complete classification of smooth P1-fibrations
over projective manifolds of Picard number 1 each of which admit another
smooth morphism of relative dimension one.
Furthermore, we consider relations of the result with Campana-Peternell conjecture
on Fano manifolds with nef tangent bundle.
We give a complete classification of smooth P1-fibrations
over projective manifolds of Picard number 1 each of which admit another
smooth morphism of relative dimension one.
Furthermore, we consider relations of the result with Campana-Peternell conjecture
on Fano manifolds with nef tangent bundle.
2012/05/28
15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)
Mihnea Popa (University of Illinois at Chicago)
Generic vanishing and linearity via Hodge modules (ENGLISH)
Mihnea Popa (University of Illinois at Chicago)
Generic vanishing and linearity via Hodge modules (ENGLISH)
[ Abstract ]
I will explain joint work with Christian Schnell, in which we extend the fundamental results of generic vanishing theory (for instance for the canonical bundle of a smooth projective variety) to bundles of holomorphic forms and to rank one local systems, where parts of the theory have eluded previous efforts. To achiever this, we bring all of the old and new results under the same roof by enlarging the scope of generic vanishing theory to the study of filtered D-modules associated to mixed Hodge modules. Besides Saito's vanishing and direct image theorems for Hodge modules, an important input is the Laumon-Rothstein Fourier transform for bundles with integrable connection.
I will explain joint work with Christian Schnell, in which we extend the fundamental results of generic vanishing theory (for instance for the canonical bundle of a smooth projective variety) to bundles of holomorphic forms and to rank one local systems, where parts of the theory have eluded previous efforts. To achiever this, we bring all of the old and new results under the same roof by enlarging the scope of generic vanishing theory to the study of filtered D-modules associated to mixed Hodge modules. Besides Saito's vanishing and direct image theorems for Hodge modules, an important input is the Laumon-Rothstein Fourier transform for bundles with integrable connection.
2012/05/21
15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)
Taku Suzuki (Waseda University)
Characterizations of projective spaces and hyperquadrics
(JAPANESE)
Taku Suzuki (Waseda University)
Characterizations of projective spaces and hyperquadrics
(JAPANESE)
[ Abstract ]
After Mori's works on Hartshorne's conjecture, many results to
characterize projective spaces and hyperquadrics in terms of
positivity properties of the tangent bundle have been provided.
Kov\\'acs' conjecture states that smooth complex projective
varieties are projective spaces or hyperquadrics if the $p$-th
exterior product of their tangent bundle contains the $p$-th
exterior product of an ample vector bundle. This conjecture is
the generalization of many preceding results. In this talk, I will
explain the idea of the proof of Kov\\'acs' conjecture for varieties
with Picard number one by using a method of slope-stabilities
of sheaves.
After Mori's works on Hartshorne's conjecture, many results to
characterize projective spaces and hyperquadrics in terms of
positivity properties of the tangent bundle have been provided.
Kov\\'acs' conjecture states that smooth complex projective
varieties are projective spaces or hyperquadrics if the $p$-th
exterior product of their tangent bundle contains the $p$-th
exterior product of an ample vector bundle. This conjecture is
the generalization of many preceding results. In this talk, I will
explain the idea of the proof of Kov\\'acs' conjecture for varieties
with Picard number one by using a method of slope-stabilities
of sheaves.
2012/05/07
15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)
Atsushi Ito (University of Tokyo)
Algebro-geometric characterization of Cayley polytopes (JAPANESE)
Atsushi Ito (University of Tokyo)
Algebro-geometric characterization of Cayley polytopes (JAPANESE)
[ Abstract ]
A lattice polytope is called a Cayley polytope if it is "small" in some
sense.
In this talk, I will explain an algebro-geometric characterization of
Cayley polytopes
by considering whether or not the corresponding polarized toric
varieties are covered by lines, planes, etc.
We can apply this characterization to the study of Seshadri constants,
which are invariants measuring the positivity of ample line bundles.
That is, we can obtain an explicit description of a polarized toric
variety whose Seshadri constant is one.
A lattice polytope is called a Cayley polytope if it is "small" in some
sense.
In this talk, I will explain an algebro-geometric characterization of
Cayley polytopes
by considering whether or not the corresponding polarized toric
varieties are covered by lines, planes, etc.
We can apply this characterization to the study of Seshadri constants,
which are invariants measuring the positivity of ample line bundles.
That is, we can obtain an explicit description of a polarized toric
variety whose Seshadri constant is one.
2012/04/23
17:10-18:40 Room #122 (Graduate School of Math. Sci. Bldg.)
Takehiko Yasuda (Osaka University)
Motivic integration and wild group actions (JAPANESE)
Takehiko Yasuda (Osaka University)
Motivic integration and wild group actions (JAPANESE)
[ Abstract ]
The cohomological McKay correspondence proved by Batyrev is the equality of an orbifold invariant
and a stringy invariant. The former is an invariant of a smooth variety with a finite group action and the latter is
an invariant of its quotient variety. Denef and Loeser gave an alternative proof of it which uses the motivic integration theory developped by themselves.
Then I pushed forward with their study by generalizing the motivic integration to
Deligne-Mumford stacks and reformulating the cohomological McKay correspondence from the viewpoint of
the birational geometry of stacks.
However all of these are about tame group actions (the order of a group is not divisible by the characteristic of the base field),
and the wild (= not tame) case has remained unexplored.
In this talk, I will explain my attempt to examine the simplest situation of the wild case. Namely linear actions of a cyclic group
of order equal to the characteristic of the base field are treated. A remarkable new phenomenon is that the space of generalized
arcs is a fibration over an infinite dimensional space with infinite dimensional fibers, where the base space is the space of
Artin-Schreier extensions of $k((t))$, the field of Laurent series.
The cohomological McKay correspondence proved by Batyrev is the equality of an orbifold invariant
and a stringy invariant. The former is an invariant of a smooth variety with a finite group action and the latter is
an invariant of its quotient variety. Denef and Loeser gave an alternative proof of it which uses the motivic integration theory developped by themselves.
Then I pushed forward with their study by generalizing the motivic integration to
Deligne-Mumford stacks and reformulating the cohomological McKay correspondence from the viewpoint of
the birational geometry of stacks.
However all of these are about tame group actions (the order of a group is not divisible by the characteristic of the base field),
and the wild (= not tame) case has remained unexplored.
In this talk, I will explain my attempt to examine the simplest situation of the wild case. Namely linear actions of a cyclic group
of order equal to the characteristic of the base field are treated. A remarkable new phenomenon is that the space of generalized
arcs is a fibration over an infinite dimensional space with infinite dimensional fibers, where the base space is the space of
Artin-Schreier extensions of $k((t))$, the field of Laurent series.
2012/04/16
15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)
Makoto Miura (University of Tokyo)
Toric degenerations of minuscule Schubert varieties and mirror symmetry (JAPANESE)
Makoto Miura (University of Tokyo)
Toric degenerations of minuscule Schubert varieties and mirror symmetry (JAPANESE)
[ Abstract ]
Minuscule Schubert varieties admit the flat degenerations to projective
Hibi toric varieties, whose combinatorial structure is explicitly
described by finite posets. In this talk, I will explain these toric
degenerations and discuss the mirror symmetry for complete intersection
Calabi-Yau varieties in Gorenstein minuscule Schubert varieties.
Minuscule Schubert varieties admit the flat degenerations to projective
Hibi toric varieties, whose combinatorial structure is explicitly
described by finite posets. In this talk, I will explain these toric
degenerations and discuss the mirror symmetry for complete intersection
Calabi-Yau varieties in Gorenstein minuscule Schubert varieties.
2012/04/09
15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)
Kazushi Ueda (Osaka University)
On mirror symmetry for weighted Calabi-Yau hypersurfaces (JAPANESE)
Kazushi Ueda (Osaka University)
On mirror symmetry for weighted Calabi-Yau hypersurfaces (JAPANESE)
[ Abstract ]
In the talk, I will discuss relation between homological mirror symmetry for weighted projective spaces, their Calabi-Yau hypersurfaces, and weighted homogeneous singularities.
If the time permits, I will also discuss an application to monodromy of hypergeometric functions.
In the talk, I will discuss relation between homological mirror symmetry for weighted projective spaces, their Calabi-Yau hypersurfaces, and weighted homogeneous singularities.
If the time permits, I will also discuss an application to monodromy of hypergeometric functions.
2012/01/30
15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)
Yoshinori Gongyo (University of Tokyo)
On varieties of globally F-regular type (JAPANESE)
Yoshinori Gongyo (University of Tokyo)
On varieties of globally F-regular type (JAPANESE)
[ Abstract ]
I will talk about recent topics on varieties of globally F-regular type.
I will talk about recent topics on varieties of globally F-regular type.
2012/01/23
15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)
Kimiko Yamada (Okayama university)
Sigularities and Kodaira dimension of the moduli of stable sheaves on Enriques surfaces (JAPANESE)
Kimiko Yamada (Okayama university)
Sigularities and Kodaira dimension of the moduli of stable sheaves on Enriques surfaces (JAPANESE)
[ Abstract ]
We shall estimate singularities of moduli of stable sheaves on Enriques/hyper-elliptic surfaces via the Kuranishi theory, consider when its singularities are canonical, and calculate its Kodaira dimension.
We shall estimate singularities of moduli of stable sheaves on Enriques/hyper-elliptic surfaces via the Kuranishi theory, consider when its singularities are canonical, and calculate its Kodaira dimension.
2012/01/16
15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)
Mihai Paun (Institut Élie Cartan and KIAS)
TBA(Cancelled) (JAPANESE)
Mihai Paun (Institut Élie Cartan and KIAS)
TBA(Cancelled) (JAPANESE)
2011/12/26
15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)
Takuzo Okada (Kyoto University)
Birational unboundedness of Q-Fano varieties and rationally connected strict Mori fiber spaces (JAPANESE)
Takuzo Okada (Kyoto University)
Birational unboundedness of Q-Fano varieties and rationally connected strict Mori fiber spaces (JAPANESE)
[ Abstract ]
It has been known that suitably restricted classes of Q-Fano varieties are bounded. I will talk about the birational unboundedness of (log terminal) Q-Fano varieties with Picard number one and of rationally connected strict Mori fiber spaces in each dimension $¥geq 3$. I will explain the idea of the proof which will be done after passing to a positive characteristic.
It has been known that suitably restricted classes of Q-Fano varieties are bounded. I will talk about the birational unboundedness of (log terminal) Q-Fano varieties with Picard number one and of rationally connected strict Mori fiber spaces in each dimension $¥geq 3$. I will explain the idea of the proof which will be done after passing to a positive characteristic.
2011/12/12
15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)
Robert Laterveer (CNRS, IRMA, Université de Strasbourg)
A version of Barth's theorem for singular varieties (cancelled) (JAPANESE)
Robert Laterveer (CNRS, IRMA, Université de Strasbourg)
A version of Barth's theorem for singular varieties (cancelled) (JAPANESE)
2011/12/05
15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)
Hirokazu Nasu (Tokai University)
Obstructions to deforming curves on a uniruled 3-fold (JAPANESE)
Hirokazu Nasu (Tokai University)
Obstructions to deforming curves on a uniruled 3-fold (JAPANESE)
[ Abstract ]
In this talk, I review some results from a joint work with Mukai:
1. a generalization of Mumford's example of a non-reduced component of the Hilbert scheme, and
2. a sufficient condition for a first order deformation of a curve on a uniruled 3-fold to be obstructed.
As a sequel of the study, we will discuss some obstructed deformations of degenerate curves on a higher dimensional scroll.
In this talk, I review some results from a joint work with Mukai:
1. a generalization of Mumford's example of a non-reduced component of the Hilbert scheme, and
2. a sufficient condition for a first order deformation of a curve on a uniruled 3-fold to be obstructed.
As a sequel of the study, we will discuss some obstructed deformations of degenerate curves on a higher dimensional scroll.
2011/12/01
16:30-18:00 Room #002 (Graduate School of Math. Sci. Bldg.)
Dmitry Kaledin (Steklov Mathematics Institute/ KIAS)
Cyclic K-theory (ENGLISH)
Dmitry Kaledin (Steklov Mathematics Institute/ KIAS)
Cyclic K-theory (ENGLISH)
[ Abstract ]
Cyclic K-theory is a variant of algebraic K-theory introduced by Goodwillie 25 years ago and more-or-less forgotten by now. I will try to convince the audience that cyclic K-theory is actually quite useful -- in particular, it can be effectively computed for varieties over a finite field.
Cyclic K-theory is a variant of algebraic K-theory introduced by Goodwillie 25 years ago and more-or-less forgotten by now. I will try to convince the audience that cyclic K-theory is actually quite useful -- in particular, it can be effectively computed for varieties over a finite field.
2011/11/28
15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)
Kotaro Kawatani (Kyoto University)
Comparison with Gieseker stability and slope stability via Bridgeland's stability (JAPANESE)
Kotaro Kawatani (Kyoto University)
Comparison with Gieseker stability and slope stability via Bridgeland's stability (JAPANESE)
[ Abstract ]
In this talk we compare two classical notions of stability (Gieseker stability and slope stability) for sheaves on K3 surfaces by using stability conditions which was introduced by Bridgeland. As a consequence of this work, we give a classification of 2 dimensional moduli spaces of sheaves on K3 surface depending on the rank of the sheaves.
In this talk we compare two classical notions of stability (Gieseker stability and slope stability) for sheaves on K3 surfaces by using stability conditions which was introduced by Bridgeland. As a consequence of this work, we give a classification of 2 dimensional moduli spaces of sheaves on K3 surface depending on the rank of the sheaves.
2011/11/14
15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)
Kiwamu Watanabe (University of Tokyo)
On projective manifolds swept out by cubic varieties (JAPANESE)
Kiwamu Watanabe (University of Tokyo)
On projective manifolds swept out by cubic varieties (JAPANESE)
[ Abstract ]
The structures of embedded complex projective manifolds swept out by varieties with preassigned properties have been studied by several authors. In this talk, we study structures of embedded projective manifolds swept out by cubic varieties.
The structures of embedded complex projective manifolds swept out by varieties with preassigned properties have been studied by several authors. In this talk, we study structures of embedded projective manifolds swept out by cubic varieties.
2011/11/07
15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)
Atsushi Ito (University of Tokyo)
Okounkov bodies and Seshadri constants (JAPANESE)
Atsushi Ito (University of Tokyo)
Okounkov bodies and Seshadri constants (JAPANESE)
[ Abstract ]
Okounkov bodies, which are convex bodies associated to big line bundles, have rich information of the line bundles. On the other hand, Seshadri constants are invariants which measure the positivities of line bundles. In this talk, I will explain a relation between Okounkov bodies and Seshadri constants.
Okounkov bodies, which are convex bodies associated to big line bundles, have rich information of the line bundles. On the other hand, Seshadri constants are invariants which measure the positivities of line bundles. In this talk, I will explain a relation between Okounkov bodies and Seshadri constants.
2011/10/31
15:30-17:00 Room #122 (Graduate School of Math. Sci. Bldg.)
Osamu Fujino (Kyoto University)
Minimal model theory for log surfaces (JAPANESE)
Osamu Fujino (Kyoto University)
Minimal model theory for log surfaces (JAPANESE)
[ Abstract ]
We discuss the log minimal model theory for log sur- faces. We show that the log minimal model program, the finite generation of log canonical rings, and the log abundance theorem for log surfaces hold true under assumptions weaker than the usual framework of the log minimal model theory.
We discuss the log minimal model theory for log sur- faces. We show that the log minimal model program, the finite generation of log canonical rings, and the log abundance theorem for log surfaces hold true under assumptions weaker than the usual framework of the log minimal model theory.
2011/07/04
16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Yasunari Nagai (Waseda University)
Birational Geometry of O'Grady's six dimensional example over the Donaldson-Uhlenbeck compactification (JAPANESE)
Yasunari Nagai (Waseda University)
Birational Geometry of O'Grady's six dimensional example over the Donaldson-Uhlenbeck compactification (JAPANESE)
[ Abstract ]
O'Grady constructed two sporadic examples of compact irreducible symplectic Kaehler manifold, by resolving singular moduli spaces of sheaves on a K3 surface or an abelian surface. We will give a full description of the birational geometry of O'Grady's six dimensional example over the corresponding Donaldson-Uhlenbeck compactification, using an explicit calculation of certain kind of GIT quotients.
If time permits, we will also discuss an involution of the example induced by a Fourier-Mukai transformation.
O'Grady constructed two sporadic examples of compact irreducible symplectic Kaehler manifold, by resolving singular moduli spaces of sheaves on a K3 surface or an abelian surface. We will give a full description of the birational geometry of O'Grady's six dimensional example over the corresponding Donaldson-Uhlenbeck compactification, using an explicit calculation of certain kind of GIT quotients.
If time permits, we will also discuss an involution of the example induced by a Fourier-Mukai transformation.
2011/06/27
16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Vladimir Lazić (Imperial College London)
MMP revisited, II (ENGLISH)
Vladimir Lazić (Imperial College London)
MMP revisited, II (ENGLISH)
[ Abstract ]
I will talk about how finite generation of certain adjoint rings implies everything we currently know about the MMP. This is joint work with A. Corti.
I will talk about how finite generation of certain adjoint rings implies everything we currently know about the MMP. This is joint work with A. Corti.
2011/06/07
16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Chenyang Xu (MIT)
Log canonical closure (ENGLISH)
Chenyang Xu (MIT)
Log canonical closure (ENGLISH)
[ Abstract ]
(joint with Christopher Hacon) In this talk, we will address the problem on given a log canonical variety, how we compactify it. Our approach is via MMP. The result has a few applications. Especially I will explain the one on the moduli of stable schemes.
If time permits, I will also talk about how a similar approach can be applied to give a proof of the existence of log canonical flips and a conjecture due to Kollár on the geometry of log centers.
(joint with Christopher Hacon) In this talk, we will address the problem on given a log canonical variety, how we compactify it. Our approach is via MMP. The result has a few applications. Especially I will explain the one on the moduli of stable schemes.
If time permits, I will also talk about how a similar approach can be applied to give a proof of the existence of log canonical flips and a conjecture due to Kollár on the geometry of log centers.
2011/06/06
16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Shihoko Ishii (University of Tokyo)
Multiplier ideals via Mather discrepancies (JAPANESE)
Shihoko Ishii (University of Tokyo)
Multiplier ideals via Mather discrepancies (JAPANESE)
[ Abstract ]
For an arbitrary variety we define a multiplier ideal by using Mather discrepancy.
This ideal coincides with the usual multiplier ideal if the variety is normal and complete intersection.
In the talk I will show a local vanishing theorem for this ideal and as corollaries we obtain restriction theorem, subadditivity theorem, Skoda type theorem, and Briancon-Skoda type theorem.
For an arbitrary variety we define a multiplier ideal by using Mather discrepancy.
This ideal coincides with the usual multiplier ideal if the variety is normal and complete intersection.
In the talk I will show a local vanishing theorem for this ideal and as corollaries we obtain restriction theorem, subadditivity theorem, Skoda type theorem, and Briancon-Skoda type theorem.
2011/05/30
16:30-18:00 Room #126 (Graduate School of Math. Sci. Bldg.)
Jungkai Alfred Chen (National Taiwan University and RIMS)
Kodaira Dimension of Irregular Varieties (ENGLISH)
Jungkai Alfred Chen (National Taiwan University and RIMS)
Kodaira Dimension of Irregular Varieties (ENGLISH)
[ Abstract ]
$f:X\\to Y$ be an algebraic fiber space with generic geometric fiber $F$, $\\dim X=n$ and $\\dim Y=m$. Then Iitaka's $C_{n,m}$ conjecture states $$\\kappa (X)\\geq \\kappa (Y)+\\kappa (F).$$ In particular, if $X$ is a variety with $\\kappa(X)=0$ and $f: X \\to Y$ is the Albanese map, then Ueno conjecture that $\\kappa(F)=0$. One can regard Ueno’s conjecture an important test case of Iitaka’s conjecture in general.
These conjectures are of fundamental importance in the classification of higher dimensional complex projective varieties. In a recent joint work with Hacon, we are able to prove Ueno’s conjecture and $C_{n,m}$ conjecture holds when $Y$ is of maximal Albanese dimension. In this talk, we will introduce some relative results and briefly sketch the proof.
$f:X\\to Y$ be an algebraic fiber space with generic geometric fiber $F$, $\\dim X=n$ and $\\dim Y=m$. Then Iitaka's $C_{n,m}$ conjecture states $$\\kappa (X)\\geq \\kappa (Y)+\\kappa (F).$$ In particular, if $X$ is a variety with $\\kappa(X)=0$ and $f: X \\to Y$ is the Albanese map, then Ueno conjecture that $\\kappa(F)=0$. One can regard Ueno’s conjecture an important test case of Iitaka’s conjecture in general.
These conjectures are of fundamental importance in the classification of higher dimensional complex projective varieties. In a recent joint work with Hacon, we are able to prove Ueno’s conjecture and $C_{n,m}$ conjecture holds when $Y$ is of maximal Albanese dimension. In this talk, we will introduce some relative results and briefly sketch the proof.
2011/05/23
17:00-18:30 Room #126 (Graduate School of Math. Sci. Bldg.)
Yuji Sano (Kumamoto University)
Alpha invariant and K-stability of Fano varieties (JAPANESE)
Yuji Sano (Kumamoto University)
Alpha invariant and K-stability of Fano varieties (JAPANESE)
[ Abstract ]
From the results of Tian, it is proved that the lower bounds of alpha invariant implies K-stability of Fano manifolds via the existence of Kähler-Einstein metrics. In this talk, I will give a direct proof of this relation in algebro-geometric way without using Kähler-Einstein metrics. This is joint work with Yuji Odaka (RIMS).
From the results of Tian, it is proved that the lower bounds of alpha invariant implies K-stability of Fano manifolds via the existence of Kähler-Einstein metrics. In this talk, I will give a direct proof of this relation in algebro-geometric way without using Kähler-Einstein metrics. This is joint work with Yuji Odaka (RIMS).
2011/05/16
17:00-18:30 Room #126 (Graduate School of Math. Sci. Bldg.)
Shinnosuke Okawa (University of Tokyo)
On images of Mori dream spaces (JAPANESE)
Shinnosuke Okawa (University of Tokyo)
On images of Mori dream spaces (JAPANESE)
[ Abstract ]
Mori dream space (MDS), introduced by Y. Hu and S. Keel, is a class of varieties whose geometry can be controlled via the VGIT of the Cox ring. It is a generalization of both toric varieties and log Fano varieties.
The purpose of this talk is to study the image of a morphism from a MDS.
Firstly I prove that such an image again is a MDS.
Secondly I introduce a fan structure on the effective cone of a MDS and show that the fan of the image coincides with the restriction of that of the source.
This fan encodes some information of the Zariski decompositions, which turns out to be equivalent to the information of the GIT equivalence. In toric case, this fan coincides with the so called GKZ decomposition.
The point is that these results can be clearly explained via the VGIT description for MDS.
If I have time, I touch on generalizations and an application to the Shokurov polytopes.
Mori dream space (MDS), introduced by Y. Hu and S. Keel, is a class of varieties whose geometry can be controlled via the VGIT of the Cox ring. It is a generalization of both toric varieties and log Fano varieties.
The purpose of this talk is to study the image of a morphism from a MDS.
Firstly I prove that such an image again is a MDS.
Secondly I introduce a fan structure on the effective cone of a MDS and show that the fan of the image coincides with the restriction of that of the source.
This fan encodes some information of the Zariski decompositions, which turns out to be equivalent to the information of the GIT equivalence. In toric case, this fan coincides with the so called GKZ decomposition.
The point is that these results can be clearly explained via the VGIT description for MDS.
If I have time, I touch on generalizations and an application to the Shokurov polytopes.
