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A smooth complex rational affine surface with uncountably many nonisomorphic real forms
Anna Bot (University of Basel, Basel)
Abstract
A real form of a complex algebraic variety $X$ is a real algebraic variety whose complexification is isomorphic to $X$.
Many families of complex varieties have a finite number of nonisomorphic real forms, but up until recently no example with infinitely many had been found.
In 2018, Lesieutre constructed a projective variety of dimension six with infinitely many nonisomorphic real forms, and last year, Dinh, Oguiso and Yu described
projective rational surfaces with infinitely many as well.
In this talk, I’ll present the first example of a rational affine surface having uncountably many nonisomorphic real forms.
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Connected algebraic groups acting on Fano fibrations over $\mathbb{P}^1$
Enrica Floris (Université de Poitiers, Poitiers)
Abstract
Let $G$ be a connected algebraic group and $X$ a variety endowed with a regular action of $G$ and a Mori fibre space $X/\mathbb{P}^1$ whose fibre is
a Fano variety of Picard rank at least 2. In this talk I will explain why there is a proper horizontal subvariety of $X$ which is invariant under
the action of $G$, alongside with some applications of this result to the classification of connected algebraic subgroups of the Cremona group in dimension 4.
This is a joint work with Jérémy Blanc.
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Zariski’s finiteness theorem and properties of some rings of invariants
Buddhadev Hajra (Indian Institute of Technology, Bombay)
Abstract
In this talk I will present a short proof of a special case of O. Zariski’s result about
finite generation in connection with Hilbert’s 14th problem using a new idea. This result
is useful for invariant subrings of unipotent or connected semisimple groups. The next
result I will talk about is a stronger form of one well-known result by A. Tyc. This result
proves that the quotient space under a regular $\mathbb{G}_a$-action on an affine space over the field of
complex numbers has at most rational singularities, under an assumption about the quotient
morphism. I will also sketch the main idea of the proof of a result which is an analogue
of M. Miyanishi’s result for the ring of invariants of a $\mathbb{G}_a$-action on $R[X, Y, Z]$ for an affine
Dedekind domain $R$. This proof involves some classical topological methods. This is a joint
work with R.V. Gurjar and Sudarshan R. Gurjar.
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On the structure of projective manifolds whose tangent bundles are positive
Masataka Iwai (Tohoku University, Sendai)
Abstract
In this talk, I will explain the structure of complex projective manifolds or log
smooth pairs whose tangent bundles are ”positive” (such as ample, nef, big, and pseudo-
effective) and summarize recent studies related to our work. A part of this talk is based on
a joint work with Genki Hosono and Shin-ichi Matsumura
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Projective varieties with nef tangent bundle in positive characteristic
Akihiro Kanemitsu (Saitama University, Saitama)
Abstract
Demailly, Peternell and Schneider proved two structure theorems for complex projective manifold with nef tangent bundle:
(A) The variety X decomposes to the product of a Fano manifold and an ́etale quotient of an abelian variety.
(B) Any extremal contraction of X is smooth.
In this talk, we discuss the positive characteristic version of these theorems. This is a joint work with Kiwamu Watanabe.
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Unipotent group structures on quintic del Pezzo varieties
Pedro Montero (Universidad Técnica Federico Santa María, Valparaiso)
Abstract
Del Pezzo varieties arise as a natural higher-dimensional generalization of the
classical Del Pezzo surfaces. They were extensively studied by T. Fujita in the 1980s, who
classified them according to their degree. In degree 5, it follows from Fujita’s classification
that all of these manifolds are obtained as linear sections of the 6-dimensional Grassmannian
$\mathbb{Gr}(2,5)$ with respect to the Plücker embedding, whose points parametrize 2-dimensional
linear subspaces of a vector space of dimension 5. In this talk, we will discuss the existence
and uniqueness of $\mathbb{G}_a^n$-structures on these varieties, i.e., we will determine when and in how
many ways one can obtain them as equivariant compactifications of the abelian unipotent
group $\mathbb{G}_a^n$. To do so, we study the Hilbert schemes of certain linear subspaces on such varieties
and we analyze some explicit equivariant Sarkisov links. This is a joint work with Adrien
Dubouloz and Takashi Kishimoto.
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Some criteria for a ring to be a unique factorization domain
Takanori Nagamine (National Institute of Technology, Oyama College, Oyama)
Abstract
Let $A$ be a unique factorization domain (UFD). We consider ring extensions of the
following two types:
(i) $A[x]$ where $ax = b$ for relatively prime $a, b \in A \setminus \{0\}$ such that ideals $(a)$ and $(a, b)$ are
prime.
(ii) $A[x]$ where $A$ has a $\mathbb{Z}$-grading, $x^n = F$ for a positive integer $n$ and homogeneous prime
$F \in A$ with $\gcd(n, \deg F ) = 1$.
In 1964, Samuel studied these ring extensions. In case (i), he showed that, if $A$ is noe therian, then $A[x]$ is a UFD.
We show that the noetherian condition can be weakened.
In ase (ii), Samuel showed that, if $A$ is a polynomial ring over a UFD $R$, and either $n \equiv 1$
(mod $\deg F$ ) or every finitely generated projective $R$-module is free, then $A[x]$ is a UFD. We
show, more generally, that $A[x]$ is a UFD whenever the conditions of (ii) hold.
This research is joint work with Daniel Daigle and Gene Freudenburg.
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Pathologies on log del Pezzo surfaces in characteristic five
Masaru Nagaoka (Kyushu University, Fukuoka)
Abstract
Log del Pezzo surfaces are 2-dimensional Fano varieties with klt singularities, which
form a building block in the minimal model program. Keel-McKernan essentially classified
log del Pezzo surfaces over the field of complex numbers around 2000, and Lacini showed
that their classification result also holds in characteristic larger than five recently. On the
other hand, there are many log del Pezzo surfaces which appear only in low (and positive)
characteristics. In this talk, I will explain the classification result on log del Pezzo surfaces
which appear only in characteristic five.