THE 16TH AFFINE ALGEBRAIC GEOMETRY MEETING

ABSTRACTS of TALKS⋆ 8th March (Thursday)• Takeshi TAKAHASHI (Niigata University):
Title: Number of weak Galois Weierstraß points with semigroup < a, b >Abstract: Let C be a nonsingular projective curve of genus > 1 over an algebraically closed
field of characteristic 0. For a point P in C, the Weierstraß semigroup H(P ) is defined as
the set of non-negative integers n for which there exists a rational function f on C such thatthe order of the pole of f at P is equal to n, and f is regular away from P . A point P in C
is referred to as a weak Galois Weierstraß point if P is a Weierstraß point and there exists
a Galois morphism from C to the projective line such that P is a total ramification point ofthe morphism. Prof. Komeda and I are investigating weak Galois Weierstraß points whose
Weierstraß semigroups are generated by two positive integers, as a generalization of studies
on Galois points for plane curves defined by Prof. Yoshihara. In my talk, I will introduce
some of these results.
• Yuta KAMBE (Saitama University):
Title: On the moduli space of reduced Gröbner bases
Abstract: For a given monomial ideal J and a given monomial order < on a polynomial ring
S, the moduli space of reduced Gröbner bases in S whose initial ideal is J is determined. I
will talk about a construction and an application of such moduli spaces.
• Yoshinori GONGYO (The University of Tokyo):
Title: Nef anti-canonical divisors and rationally connected fibrations
Abstract: We study the Iitaka–Kodaira dimension of nef relative anti-canonical divisors byusing semi-positivity theorems. Thus we give aﬃrmative answers for some two questions.
This is a joint work with Sho Ejiri.
• Kei MIURA (National Institute of Technology, Ube College):
Title: Birational transformations belonging to Galois points
Abstract: In this talk, we study birational transformations belonging to Galois points. Fora certain plane quartic curve, we determine the number of Galois points. Then, we obtain
birational transformations belonging to these points. In fact, we see that they can be extended
to Cremona transformations. In particular, we determine their conjugacy class and show thatthey are all conjugate to linear transformations.
12ABSTRACT⋆ 9th March (Friday)• Kiwamu WATANABE (Saitama University):
Title: Varieties with nef diagonal
Abstract: For a smooth projective variety X, we consider when the diagonal ∆X is nef as acycle on X × X. In particular, we give a classification of complete intersections and smoothdel Pezzo varieties where the diagonal is nef. We also study the nefness of the diagonal for
spherical varieties. This is a joint work with Taku Suzuki.
• Masayoshi MIYANISHI (Kwansei Gakuin University):
Title: Geometry of Artin-Schreier coverings
Abstract: In positive characteristic, a finite morphism q : Y → X of algebraic varieties iscalled an Artin-Schreier covering if the extension of function fields k(X) ⊂ k(Y ) is an ArtinSchreier covering, i.e., a Galois covering of group Z/pZ. We consider the case where X andY are smooth projective surfaces. There is a pioneering work by Y. Takeda [J. Math. Soc.
Japan, vol 41, No.3, 1989], where basic results and Enriques classification of surfaces obtainedas Artin-Schreier coverings of simple type are outlined. But it seems that the details have tobe exploited and can lead to an interesting geometry. This is a report of a review that thespeaker has made recently.
• Adrien DUBOULOZ (Université de Bourgogne):
Title: Algebraic models of the real aﬃne plane
Abstract: An algebraic model of the plane R2 is a smooth complex surface S defined over
the real numbers, with trivial rational singular homology and real locus homeomorphic toR2 . Two such models S and S ′ are considered to be ”the same” if they are ”birationally
diﬀeomorphic”, that is, there exists a birational map S 99K S ′ defined over the real numbers
which induces a diﬀeomorphism between the real loci of S and S ′ . In this talk, I will explain
how to construct a variant of the logarithmic Kodaira dimension which is an invariant ofsuch surfaces up to birational diﬀeomorphisms. An application is the existence of infinitely
many pairwise non birationally diﬀeomorphic algebraic models of R2 , a situation which contrasts with the analogue problem for rational algebraic models of compact diﬀerentiable realsurfaces. If time permits, I will also indicate some relation between the classification of suchmodels and a birational variant of the classical Abhyankar-Moh embedding problem.
• Masayuki KAWAKITA (RIMS):
Title: Minimal log discrepancies on smooth threefolds
Abstract: The minimal log discrepancy is an important invariant of singularities in the minimal model program. We will discuss several equivalent conjectures on the minimal log
discrepancies on smooth threefolds.

ABSTRACT3⋆ 10th March (Saturday)• Pierre-Marie POLONI (The University of Bern):
Title: Aﬃne plane bundles over the punctured aﬃne plane
Abstract: An A2 -fibration is a flat morphism between complex algebraic varieties whose
fibers are isomorphic to the complex aﬃne plane A2 . In this talk, we study explicit families
f : A4 → A2 of A2 -fibrations over A2 . The famous Dolgachev-Weisfeiler conjecture predicts
that such fibrations are in fact all isomorphic to the trivial bundle over A2 . Our aim is todevelop tools for verifying that this conjecture holds true in some particular examples. For
instance, we will see that the A2 -fibration induced by the second Vénéreau polynomial istrivial. Our strategy is inspired by a previous work of Kaliman and Zaidenberg and consists
in first showing that the considered fibrations f : A4 → A2 have a fiber bundle structure
when restricted over the punctured plane A2 \ {(0, 0)}. This is joint work in progress withJérémy Blanc.
• Ivan CHELTSOV (The University of Edinburgh):
Title: Super-rigid Aﬃne Fano Varieties
Abstract: In my talk I will introduce a wide class of aﬃne varieties (aﬃne Fano varieties). By
analogy with birationally super-rigid Fano varieties, I will define super-rigidity for aﬃne Fanovarieties, and provide many examples and non-examples of super-rigid aﬃne Fano varieties.
The talk is based on a joint work with Adrien Dubouloz (Dijon) and Jihun Park (Pohang).

• Osamu FUJINO (Osaka University):
Title: On semi-log canonical Fano pairs
Abstract: It is well known that every Fano manifold is simply connected and is rationallychain connected. This result was already generalized for Fano log pairs with log canonical singularities. The notion of semi-log canonical singularities was introduced by Kollár–ShepherdBarron in order to investigate deformations of surface singularities and compactifications ofmoduli spaces for surfaces of general type. We have already known that the appropriate singularities to permit on the varieties at the boundaries of moduli spaces of higher-dimensionalvarieties are semi-log canonical. In this talk, I would like to explain the following theorem:
Theorem 0.1. Let (X, ∆) be a connected projective semi-log canonical pair such that −(KX +∆) is ample. Then X is simply connected and is rationally chain connected.
We note that X may be non-normal and reducible. This talk is based on the joint workwith Wenfei Liu.

• Kento FUJITA (RIMS):
Title: K-stability of log Fano hyperplane arrangements
Abstract: We completely determine which log Fano hyperplane arrangements are uniformly
K-stable, K-stable, K-polystable, K-semistable or not.

4ABSTRACT• Takuzo OKADA (Saga University):
Title: On birational rigidity of singular del Pezzo fibrations of degree 1
Abstract: A del Pezzo fibration (or more generally, a Mori fiber space) is said to be birationallyrigid if it cannot be birationally transformed into a Mori fiber space which is not isomorphic
to the original del Pezzo fibration. Birational rigidity of nonsingular del Pezzo fibrations ofdegree 1 is completely understood by Pukhlikov and Grinenko. However almost nothing is
known for singular case. In this talk I will report some results of on-going project studying
birational rigidity of del Pezzo fibrations of degree 1 with terminal quotient singularities.

⋆ 11th March (Sunday)• Isac HEDÉN (The University of Warwick):
Title: Extensions of principal additive bundles over a punctured surface
Abstract: We study complex aﬃne Ga -threefolds X such that the restriction of the quotientmorphism π : X → S to π −1 (S∗ ) is a principal Ga -bundle, where S∗ = S\{o} denotes thecomplement of a closed point o in S and Ga denotes the additive group over the field ofcomplex numbers. Changing the point of view, we look for aﬃne extensions of Ga -principal
bundles over punctured surfaces, i.e., aﬃne varieties that are obtained by adding an extra
fiber to the bundle projection over o. Special attention will be given to the case where X issmooth, the Ga -action on X is proper and π −1 (o) = A2 is the aﬃne plane.
• Takeru FUKUOKA (The University of Tokyo):
Title: Relative linear extensions of sextic del Pezzo fibrations
Abstract: An extremal contraction from a non-singular projective 3-fold onto a smooth curveis called a del Pezzo fibration. It is classically known that every del Pezzo surface S isa (weighted) complete intersection of a certain Fano variety. In order to study del Pezzofibrations, it is important to relativize such descriptions for those. The main result of thistalk shows that the sextic del Pezzo fibrations are relative linear sections of P1 × P1 × P1 fibrations and P2 ×P2 -fibrations, which are constructed as Mori fiber spaces with smooth totalspace. As an application, we will classify the singular fibers of sextic del Pezzo fibrations.