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代数幾何学セミナー
過去の記録 ～05/22｜次回の予定｜今後の予定 05/23～
| 開催情報 | 金曜日 13:30～15:00 数理科学研究科棟(駒場) 118号室 |
|---|---|
| 担当者 | 權業 善範、河上 龍郎 、榎園 誠 |
過去の記録
2012年06月04日(月)
15:30-17:00 数理科学研究科棟(駒場) 122号室
渡辺究 氏 (埼玉大学)
Smooth P1-fibrations and Campana-Peternell conjecture (ENGLISH)
渡辺究 氏 (埼玉大学)
Smooth P1-fibrations and Campana-Peternell conjecture (ENGLISH)
[ 講演概要 ]
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年05月28日(月)
15:30-17:00 数理科学研究科棟(駒場) 122号室
Mihnea Popa 氏 (University of Illinois at Chicago)
Generic vanishing and linearity via Hodge modules (ENGLISH)
Mihnea Popa 氏 (University of Illinois at Chicago)
Generic vanishing and linearity via Hodge modules (ENGLISH)
[ 講演概要 ]
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年05月21日(月)
15:30-17:00 数理科学研究科棟(駒場) 122号室
鈴木拓 氏 (早稲田理工)
Characterizations of projective spaces and hyperquadrics
(JAPANESE)
鈴木拓 氏 (早稲田理工)
Characterizations of projective spaces and hyperquadrics
(JAPANESE)
[ 講演概要 ]
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年05月07日(月)
15:30-17:00 数理科学研究科棟(駒場) 122号室
伊藤 敦 氏 (東京大学大学院数理科学研究科)
Algebro-geometric characterization of Cayley polytopes (JAPANESE)
伊藤 敦 氏 (東京大学大学院数理科学研究科)
Algebro-geometric characterization of Cayley polytopes (JAPANESE)
[ 講演概要 ]
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年04月23日(月)
17:10-18:40 数理科学研究科棟(駒場) 122号室
いつもと時間が違います。
安田 健彦 氏 (大阪大学)
Motivic integration and wild group actions (JAPANESE)
いつもと時間が違います。
安田 健彦 氏 (大阪大学)
Motivic integration and wild group actions (JAPANESE)
[ 講演概要 ]
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年04月16日(月)
15:30-17:00 数理科学研究科棟(駒場) 122号室
三浦 真人 氏 (東大数理)
Toric degenerations of minuscule Schubert varieties and mirror symmetry (JAPANESE)
三浦 真人 氏 (東大数理)
Toric degenerations of minuscule Schubert varieties and mirror symmetry (JAPANESE)
[ 講演概要 ]
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年04月09日(月)
15:30-17:00 数理科学研究科棟(駒場) 122号室
植田 一石 氏 (大阪大学)
On mirror symmetry for weighted Calabi-Yau hypersurfaces (JAPANESE)
植田 一石 氏 (大阪大学)
On mirror symmetry for weighted Calabi-Yau hypersurfaces (JAPANESE)
[ 講演概要 ]
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年01月30日(月)
15:30-17:00 数理科学研究科棟(駒場) 122号室
権業 善範 氏 (東京大学数理科学研究科)
On varieties of globally F-regular type (JAPANESE)
権業 善範 氏 (東京大学数理科学研究科)
On varieties of globally F-regular type (JAPANESE)
[ 講演概要 ]
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年01月23日(月)
15:30-17:00 数理科学研究科棟(駒場) 122号室
山田 紀美子 氏 (岡山理科大学理学部)
Sigularities and Kodaira dimension of the moduli of stable sheaves on Enriques surfaces (JAPANESE)
山田 紀美子 氏 (岡山理科大学理学部)
Sigularities and Kodaira dimension of the moduli of stable sheaves on Enriques surfaces (JAPANESE)
[ 講演概要 ]
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年01月16日(月)
15:30-17:00 数理科学研究科棟(駒場) 122号室
中止になりました
Mihai Paun 氏 (Institut Élie Cartan and KIAS)
TBA(中止になりました) (JAPANESE)
中止になりました
Mihai Paun 氏 (Institut Élie Cartan and KIAS)
TBA(中止になりました) (JAPANESE)
2011年12月26日(月)
15:30-17:00 数理科学研究科棟(駒場) 122号室
岡田拓三 氏 (京都大学大学大学院理学研究科)
Birational unboundedness of Q-Fano varieties and rationally connected strict Mori fiber spaces (JAPANESE)
岡田拓三 氏 (京都大学大学大学院理学研究科)
Birational unboundedness of Q-Fano varieties and rationally connected strict Mori fiber spaces (JAPANESE)
[ 講演概要 ]
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年12月12日(月)
15:30-17:00 数理科学研究科棟(駒場) 122号室
都合により中止となりました。
Robert Laterveer 氏 (CNRS, IRMA, Université de Strasbourg)
A version of Barth's theorem for singular varieties (中止になりました) (JAPANESE)
都合により中止となりました。
Robert Laterveer 氏 (CNRS, IRMA, Université de Strasbourg)
A version of Barth's theorem for singular varieties (中止になりました) (JAPANESE)
2011年12月05日(月)
15:30-17:00 数理科学研究科棟(駒場) 122号室
那須 弘和 氏 (東海大学理学部情報数理学科)
Obstructions to deforming curves on a uniruled 3-fold (JAPANESE)
那須 弘和 氏 (東海大学理学部情報数理学科)
Obstructions to deforming curves on a uniruled 3-fold (JAPANESE)
[ 講演概要 ]
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年12月01日(木)
16:30-18:00 数理科学研究科棟(駒場) 002号室
Dmitry Kaledin 氏 (Steklov Mathematics Institute/ KIAS)
Cyclic K-theory (ENGLISH)
Dmitry Kaledin 氏 (Steklov Mathematics Institute/ KIAS)
Cyclic K-theory (ENGLISH)
[ 講演概要 ]
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年11月28日(月)
15:30-17:00 数理科学研究科棟(駒場) 122号室
川谷康太郎 氏 (京都大学大学院理学研究科)
Comparison with Gieseker stability and slope stability via Bridgeland's stability (JAPANESE)
川谷康太郎 氏 (京都大学大学院理学研究科)
Comparison with Gieseker stability and slope stability via Bridgeland's stability (JAPANESE)
[ 講演概要 ]
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年11月14日(月)
15:30-17:00 数理科学研究科棟(駒場) 122号室
渡辺 究 氏 (東京大学数理科学研究科)
On projective manifolds swept out by cubic varieties (JAPANESE)
渡辺 究 氏 (東京大学数理科学研究科)
On projective manifolds swept out by cubic varieties (JAPANESE)
[ 講演概要 ]
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年11月07日(月)
15:30-17:00 数理科学研究科棟(駒場) 122号室
伊藤 敦 氏 (東京大学数理科学研究科)
Okounkov bodies and Seshadri constants (JAPANESE)
伊藤 敦 氏 (東京大学数理科学研究科)
Okounkov bodies and Seshadri constants (JAPANESE)
[ 講演概要 ]
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年10月31日(月)
15:30-17:00 数理科学研究科棟(駒場) 122号室
藤野 修 氏 (京都大学理学系研究科)
Minimal model theory for log surfaces (JAPANESE)
藤野 修 氏 (京都大学理学系研究科)
Minimal model theory for log surfaces (JAPANESE)
[ 講演概要 ]
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年07月04日(月)
16:30-18:00 数理科学研究科棟(駒場) 126号室
永井 保成 氏 (早稲田大学理工学術院基幹理工学部数学科)
Birational Geometry of O'Grady's six dimensional example over the Donaldson-Uhlenbeck compactification (JAPANESE)
永井 保成 氏 (早稲田大学理工学術院基幹理工学部数学科)
Birational Geometry of O'Grady's six dimensional example over the Donaldson-Uhlenbeck compactification (JAPANESE)
[ 講演概要 ]
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年06月27日(月)
16:30-18:00 数理科学研究科棟(駒場) 126号室
Vladimir Lazić 氏 (Imperial College London)
MMP revisited, II (ENGLISH)
Vladimir Lazić 氏 (Imperial College London)
MMP revisited, II (ENGLISH)
[ 講演概要 ]
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年06月07日(火)
16:30-18:00 数理科学研究科棟(駒場) 126号室
Chenyang Xu 氏 (MIT)
Log canonical closure (ENGLISH)
Chenyang Xu 氏 (MIT)
Log canonical closure (ENGLISH)
[ 講演概要 ]
(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年06月06日(月)
16:30-18:00 数理科学研究科棟(駒場) 126号室
石井 志保子 氏 (東京大学数理科学研究科)
Multiplier ideals via Mather discrepancies (JAPANESE)
石井 志保子 氏 (東京大学数理科学研究科)
Multiplier ideals via Mather discrepancies (JAPANESE)
[ 講演概要 ]
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年05月30日(月)
16:30-18:00 数理科学研究科棟(駒場) 126号室
Jungkai Alfred Chen 氏 (National Taiwan University and RIMS)
Kodaira Dimension of Irregular Varieties (ENGLISH)
Jungkai Alfred Chen 氏 (National Taiwan University and RIMS)
Kodaira Dimension of Irregular Varieties (ENGLISH)
[ 講演概要 ]
$f:X\\to Y$ 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年05月23日(月)
17:00-18:30 数理科学研究科棟(駒場) 126号室
佐野 友二 氏 (熊本大学大学院自然科学研究科)
Alpha invariant and K-stability of Fano varieties (JAPANESE)
佐野 友二 氏 (熊本大学大学院自然科学研究科)
Alpha invariant and K-stability of Fano varieties (JAPANESE)
[ 講演概要 ]
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年05月16日(月)
17:00-18:30 数理科学研究科棟(駒場) 126号室
大川 新之介 氏 (東京大学数理科学研究科)
On images of Mori dream spaces (JAPANESE)
大川 新之介 氏 (東京大学数理科学研究科)
On images of Mori dream spaces (JAPANESE)
[ 講演概要 ]
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.
