Affine Algebraic Geometry Meeting is traditionally held once a year in March.
Main subject are affine and birational algebraic geometry, commutative algebra, group actions, polynomial rings and related topics.
Researchers and (graduate) students who are interested in all these subjects are all welcome.

After an 19th edition held fully online in 2021 and a 20th edition held in hybrid form in 2022 due international travel restrictions caused be the coronavirus pandemic,
the 21st edition will be held on March 3-5 2023 in a traditional form at Saitama University.

- Kenta Hashizume (Kyoto University, Kyoto)
For a projective klt pair whose log canonical divisor is nef, the abundance conjecture
predicts the semi-ampleness of the log canonical divisor. Assuming the abundance conjecture,
then it is natural to study the integer m by which the multiple of log canonical divisor is base
point free. This is called the effectivity of the base point freeness. In this talk, I will explain
a recent result on this topic.

- Ayako Kubota (Waseda University, Tokyo)
The invariant Hilbert scheme is a moduli space of schemes which are stable under
an action of a reductive algebraic group. By a suitable choice of the parameter, it becomes a
candidate for a resolution of singularities of a quotient singularity. In the first half of this talk,
I will explain two main problems in the study of the invariant Hilbert scheme from the point
of view of birational geometry of singularities. One of them is about the invariant Hilbert
scheme of the Cox realization of a singularity, and we discuss the case where the singularity
is the nilpotent cone of type $A$ in the second half of the talk.

- Kayo Masuda (Kwansei Gakuin University, Sanda)
We study a factorial affine variety with an algebraic action of the additive group
$\mathbb{G}_a$ when its plinth ideal is principal. We analyze the quotient morphism by equivariant affine
modifications.

- Masayoshi Miyanishi (Kwansei Gakuin University, Sanda)
I had chances to look back my past results and find them still having rooms for
improvement. My interest lies in Abhyankar’s probelm which asks:

Let $k$ be an algebraically closed field of characteristic $p > 0$. Let $C$ be an affine plane curve
defined by a polynomial $f(x, y) \in k[x, y]$. Is the curve $C_\alpha$ defined by $f = \alpha$ isomorphic to $\mathbb{A}^1$
for every $\alpha \in k$ ?

In the case of characteristic zero, this is the case. Furthermore, $f$ is a ring generator of
$k[x, y]$. But in the case $p > 0$, the last assertion does not hold by an example of Nagata. The
problem remains unsolved more than fifty years. I will speak on the case $C$ is liftable to the
characteristic zero. More precisely, we have

Theorem. Let $(R, \mathfrak{m})$ be a DVR. Let $k = R/\mathfrak{m}$ and $K = Q(R)$. Let $f \in R[x, y]$ be an element
such that $\overline{C}$ and $C_K$ define respectively the curves isomorphic to $\mathbb{A}^1$ in $\mathbb{A}^2_k$
and $\mathbb{A}^2_K$, where $\overline{C}$ and $C_K$ are defined by equating to the zero the following elements $\overline{f} := f (mod \mathfrak{m} ) \in k[x, y]$
and $f_K := f \otimes 1 \in K[x, y]$. Then $f$ is a ring generator over $R$. Namely, there exists $g \in R[x, y]$
such that $R[x, y] = R[f, g]$.

This result implies that Abhyankar’s Problem is solved positively if the given $f$ is liftable
to a ring generator in characteristic zero.

- Masaru Nagaoka (Gakushuin University, Tokyo)
Log del Pezzo surfaces are 2-dimensional Fano varieties with klt singularities, which
form a building block in the minimal model program. Keel-McKernan and Lacini classified
log del Pezzo surfaces of Picard number one in characteristic different from two and three.
Their strategy is based on whether or not there are special divisors called tigers over such
surfaces. In this talk, I will explain the classification result on log del Pezzo surfaces of Picard
number one without tigers in characteristic two and three.

- Hirokazu Nasu (Tokai University, Hiratsuka)
Computing obstructions is useful for determining the dimension and the singularity
of a Hilbert scheme at a given point, but it is a hard task if the obstruction space is nonzero.
Generalizing a technique developed in a joint research with S.Mukai, I have recently obtained
a new sufficient condition for a first order deformation of a curve on a projective threefold to
be primarily obstructed. In 1980s, J.O. Kleppe gave a conjecture concerning the dimension
of the Hilbert scheme of space curves (lying on a smooth cubic surface). In this talk, as an
application, I will talk about a recent progress on the conjecture.

- Takuzo Okada (Saga University, Saga)
It can happen that a special member of a family of Fano 3-folds fails to be bi-
rationally (super)rigid while its general members are birationally (super)rigid. A typical
situation is when general members are smooth and some special member posses a singular
point from which one can construct a birational map (Sarkisov link) to another Mori fiber
space. The aim of this talk is to explain this phenomena, and also explain birational rigidity
of certain Fano 3-folds with only compound singularities of type A. This talk is based on a
joint work with Krylov, Paemurru and Park.

- Karol Palka (Polish Academy of Sciences, Warsaw)
A minimal model of a quasi-projective surface and of a log surface with nonzero boundary can be singular.
To avoid singularities, for log smooth surfaces with reduced boundary Miyanishi developed the notion of an almost minimal model.
It is related to a minimal model by a well described morphism - a peeling - contracting only some curves supported in the boundary.
We show that the idea of almost minimalization can be used more widely. Given a generalized log canonical or a $\mathbb{Q}$-factorial log surface $(X,D)$
defined over an algebraically closed field of arbitrary characteristic we define its almost minimal model,
whose underlying surface has singularities not worse than those of $X$ and which differs from a minimal model by a peeling.
We discuss properties of almost minimalization for boundaries of type $rD$, where $D$ is reduced and r is a positive rational number.

- Andriy Regeta (Universitat Jena, Jena)
I will show that the automorphism group of an affine variety is essentially never isomorphic
to a linear algebraic group as an abstract group. Further, I will show that the commutative connected subgroup of the automorphism group $\mathrm{Aut}(X)$
of an affine variety $X$ is the union of algebraic subgroups of $\mathrm{Aut}(X)$. Using this as a tool I will show that an affine toric variety is determined by its automorphism
group in the category of connected affine algebraic varieties over algebraically closed uncountable fields of any characteristic.

- Masatomo Sawahara (Saitama University, Saitama)
Polarized cylinders in normal projective varieties receive a lot of attention from the
viewpoint of connecting unipotent group actions on affine algebraic varieties. Hence, we shall
focus on the configuration of cylindrical ample cones of normal projective varieties. Cheltsov,
Park and Won studied cylindrical ample cones of smooth del Pezzo surfaces. In this talk, we
will discuss the configuration of cylindrical ample cones of Du Val del Pezzo surfaces. As a
result, if $S$ is a Du Val del Pezzo surface of degree $\geq 3$ such that $\mathrm{Sing}(S)\neq \emptyset$, then $S$ contains
an $H$-polar cylinder for every ample $\mathbb{Q}$-divisor $H$ on $S$.

- Ryuji Tanimoto (Shizuoka University, Shizuoka)
Let $k$ be a field of positive characteristic $p$. A finite group action on the affine
$n$-space $\mathbb{A}^n_k$ over $k$ is said to be wild if $p$ divides the order of the finite group. Few examples of
wild finite group actions on $\mathbb{A}^n_k$ are known. We know a linear action of a $p$-cyclic group $\mathbb{Z}/p\mathbb{Z}$
on $\mathbb{A}^n_k$. Except for this linear example, we hardly know examples of wild $\mathbb{Z}/p\mathbb{Z}$-actions on $\mathbb{A}^n_k$.
So, we started to study constructing examples of wild $\mathbb{Z}/p\mathbb{Z}$--actions on $\mathbb{A}^n_k$. In this talk,
we will describe triangular $\mathbb{Z}/3\mathbb{Z}$-actions on $\mathbb{A}^4_k$
in characteristic three, up to conjugation of
automorphisms of $\mathbb{A}^4_k$.

- Sho Tanimoto (Nagoya University, Nagoya)
Manin’s Conjecture predicts the asymptotic formula for the counting function
of rational points over number fields or global function fields. In the late 80’s, Batyrev
developed a heuristic argument for Manin’s Conjecture over global function fields, and the
assumptions underlying Batyrev’s heuristics are refined and formulated as Geometric Manin’s
Conjecture. Geometric Manin’s Conjecture is a set of conjectures regarding properties of the
space of sections of Fano fibrations, and it consists of three conjectures: (i) Pathological
components are controlled by Fujita invariants; (ii) For each nef algebraic class, a non-
pathological component which should be counted in Manin’s Conjecture is unique (This
component is called as Manin component); (iii) Manin components exhibit homological or
motivic stability. In this talk we discuss our proofs of GMC (i) over complex numbers using
theory of foliations and the minimal model program. Using this result, we prove that these
pathological components are coming from a bounded family of accumulating maps. This is
joint work with Brian Lehmann and Eric Riedl.