Workshop
Affine and Birational Geometry
March 5-6 2020, Department of Mathematics, Saitama University
Program
Program and Schedule in pdf
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Generalized Zariski cancellation problem and principal $\mathbb{G}_a$-bundles
Riku KUDO (Waseda University)
Abstract
Generalized Zariski cancellation problem asks whether or not $V\! \times \mathbb{A}^{1}\!\simeq\!W\!\times \mathbb{A}^{1}$ implies $V\!\simeq\!W$ for
varieties $V$ and $W$. Counter examples for this problem have been constructed as principal $\mathbb{G}_{a}$-bundles over prevarieties.
In 2007, {Dry\l o} showed that vector bundles over non $\mathbb{A}^{1}$-uniruled affine varieties have the cancellation property.
In this talk I will explain a slight generalization of {Dry\l o}'s lemma used to show the above theorem and show the following theorem;
if an affine variety $V$ has a principal $\mathbb{G}_{a}$-bundle structure over a non $\mathbb{A}^{1}$-uniruled prevariety $X$,
then for an affine variety $W$, $V\!\times\!\mathbb{A}^{1}\!\simeq\!W\!\times\!\mathbb{A}^{1}$ if and only if $W$ has a principal
$\mathbb{G}_{a}$-bundle structure over $X$.
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$\mathbb{G}_a^3$-structures on del Pezzo fibrations
Masaru NAGAOKA (University of Tokyo)
Abstract
A $\mathbb{G}_{a}^{n}$-structure on a variety $X$ is a $\mathbb{G}_{a}^{n}$-action on $X$ with the dense open orbit
isomorphic to $\mathbb{G}_{a}^{n}$. Projective varieties with $\mathbb{G}_{a}^{n}$-structures are considered as equivariant
compactifications of the affine $n$-space. Hassett-Tschinkel initiated the study of $\mathbb{G}_{a}^{n}$-structures, and they and
Huang-Montero completed the determination of smooth Fano $3$-folds admitting $\mathbb{G}_{a}^{3}$-structures. In this talk, we discuss
the determination of del Pezzo fibrations admitting $\mathbb{G}_{a}^{3}$-structures.
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Equivariant compactifications of the vector group into smooth Fano manifolds
Pedro MONTERO (Universidad Tecnica Santa Maria, Valparaiso)
Abstract
hanks to the recent works of Birkar, we know that there are only finitely many families of midly singular Fano varieties in every fixed dimension.
However, even for smooth Fano varieties, there is no complete classification in dimension greater than or equal to four. Because of that, it is
natural to impose some geometric conditions in order to try to classify those varieties. In this talk we will study the geometry of Fano manifolds
that are obtained as equivariant compactifications of the vector group. Historically, Hassett and Tschinkel iniciated the study of the geometry of
those varieties, which enjoy some nice arithmetic properties such as the Batyrev-Manin principle (concerning the asymptotical distribution of rational points).
After giving some general properties and examples of such varieties, we will discuss how the works of Hassett and Tschinkel, Kishimoto, Arzhantsev et al.
fit together with the classification of Fujita, Mori and Mukai in order to allow us to give a complete classification of “additive Fano manifolds”
when the dimension is 3 (joint work with Zhizhong Huang) and when the Fano index is high (joint work with Baohua Fu).
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Generalized Jacobian Conjecture and deformations
Karol PALKA (Polish Academy of Sciences, Warsaw)
Abstract
We say that the Generalized Jacobian Conjecture holds for a complex variety X if and only every etale endomorphism of X is a
covering (equivalently: is proper). The conjecture is a well-known open problem for affine spaces. Varieties for which the conjecture fails
do exist but a rare and their constructions are ingenious. We discuss the conjecture for smooth affine surfaces and we show that the answer
may change drastically for varieties which are members of the same smooth family of varieties.
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$\mathbb{Q}$-homology planes satisfying the Negativity Conjecture
Tomasz PELKA (University of Bern, Bern)
Abstract
A smooth complex normal algebraic surface $S$ is a $\mathbb{Q}$-homology plane if $H_{i}(S,\Q)=0$ for $i>0$. This holds
for example if $S$ is a complement of a rational cuspidal curve in $\mathbb{P}^{2}$. The Negativity Conjecture of K. Palka asserts
that for a smooth completion $(X,D)$ of $S$, $\kappa(K_{X}+\tfrac{1}{2}D)=-\infty$. Assume that $S$ is of log general type, otherwise
the geometry is well understood. It turns out that all $S$ satisfying the Negativity Conjecture can be arranged in finitely many discrete families,
each obtainable in a uniform way,, as expected by tom Dieck and Petrie, from certain arrangements of lines and conics on $\mathbb{P}^{2}$.
As a consequence, all such $S$ satisfy the Strong Rigidity Conjecture of Flenner and Zaidenberg; and all their automorphism groups are subgroups of $S_{3}$.
To illustrate this rigidity, I will show how to construct all rational cuspidal curves (with complements of log general type, satisfying the Negativity Conjecture)
inductively, by iterating quadratic Cremona maps.
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SNC log symplectic structures on Fano products
Katsuhiko OKUMURA (Waseda University)
Abstract
In this talk, we classify SNC log symplectic structures on the product of Fano varieties with cyclic Picard group.
A log symplectic structure is a Poisson structure with the reduced degeneracy divisor, and SNC means that the degeneracy divisor
has only simple normal crossing singularity. In 2014, Lima and Pereira classified such structures on the
Fano varieties of Picard rank 1. And then Pym gave another proof. I will introduce Pym's method and construction of Poisson
structures on the projective spaces from that on the affine spaces.
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Cylinders in canonical del Pezzo fibrations
Masatomo SAWAHARA (Saitama University)
Abstract
It is known that by the work due to Dubouloz and Kishimoto a del Pezzo fibration $\pi: V \to W$ of degree $d$ contains a
vertical cylinder if and only if $d \geqq 5$ and the generic fiber $V_\eta$, which is a smooth del Pezzo surface of Picard rank
one defined over the field ${\mathbb C}(\eta) ={\mathbb C}(W)$ of functions of the base variety $W$, admits a ${\mathbb C}(W)$-rational point.
Instead, in this talk, we will observe a del Pezzo fibration $\pi: V \to W$ of degree $d$ with {\it canonical} singularities and look for a
criterion for $V$ to contain vertical cylinders with respect to $\pi$. The problem is reduced to the existence of cylinder found on the generic
fiber $V_\eta=\pi^{-1}(\eta )$, which is a normal Gorenstein del Pezzo surface of Picard rank one defined over the field ${\mathbb C}(\eta ) ={\mathbb C}(W)$.
We shall give a complete answer about the existence of vertical cylinder found in $V$ with respect to $\pi$ depending on the degree $d$ and type of singularities.
The case of $d \leqq 2$ is especially complicated, so that we will devote ourselves mainly to the case of $d\leqq 2$ in the talk.
Practical Informations
Contacts
For more information, please contact:
Takashi KISHIMOTO
Venue
The workshop will take place at Kisosuri-ensyushitsu; Falculty of Sciences, 3rd floor of the building no. 14 on
the campus map.
Organizers and supports
Takashi Kishimoto, Hideo Kojima and Adrien Dubouloz
