Title and abstract for Mini workshop ``Various aspects of singularities”

Christopher Chiu
Title: On arc fibers of morphisms of schemes
Abstract: In this talk we aim to study the fibers of the map on arc spaces induced by amorphism of algebraic varieties. For finite morphisms, we prove that the arc fibers aretopologically finite in general; and that they are scheme-theoretically finite away from theramification locus. As a byproduct, we obtain a criterion for maps on arc spaces to be locallyof finite type. We will discuss various consequences of these results, including that the local
ring at stable points is topologically Noetherian and semicontinuity properties for embedding(co)dimension in the arc space. This is joint work with Tommaso de Fernex and Roi Docampo.

Makoto Enokizono
Title: Slope inequality of fibered surfaces and moduli of curves
Abstract: The slope inequality of fibered surfaces plays an important role in the classification
problem of algebraic surfaces of general type.
It is known that some positive properties of a linear combination of tautological divisors on
the moduli spaces of stable curves produce many slope inequalities of semi-stable fiberedsurfaces via the moduli maps.
In this talk, I will explain that many slope inequalities of not necessarily semi-stable fiberedsurfaces can be obtained by using the moduli space of curves which admit worse singularities
than nodes.
If time permits, I will also explain related topics and problems.

Herwig Hauser(University of Vienna)
Title : "Algebraicity in Geometry (and Transcendence)"
Abstract
The singular locus of an algebraic variety is a Zariski-closed subvariety. This is not completelyobvious, as regularity is defined pointwise via the local rings, or, alternatively, by requiringthat their completions are formal power series rings.
More generally, one can then ask for the subsets where the completed local rings are
isomorphic to an arbitrarily given complete local ring. Using Artin approximation, one canshow, as one would expect, that these sets are indeed Zariski-locally closed. We further showthat the variety has a local product structure along them (joint with C. Chiu). The proof of bothstatements is, however, a bit involved.
(In a second part, and if time permits, we will address algebraicity criteria for a prioritranscendent formal power series. They are given as solutions of linear differential equations,and yield to very deep and difficult conjectures. Some examples will illustrate the intricacy ofthe problem.)Shinobu Hikami (OIST)
Title: Amida-kuji (阿弥陀籤) and singularities
Abstract: The title looks like for high-school students. It is, however related to the knots andquantum gravity.
We start from random matrix formulation in the replica limit N=0, where N is a size of matrix.
This is equivalent to Seifert surface for 1-knot. The 2-knot is related to time-dependent matrix
model, which is expressed by 2-matrix model. This knot is embedded into four dimensional
sphere S^4 and is related to four dimensional gauge theory. This talk is aimed for theunderstanding of the singularity from matrix models.

Geometric aspects of ideals in a normal surface singularity
by Tomohiro Okuma (Yamagata University)
This is a joint work with Ken-ichi Yoshida and Kei-ichi Watanabe (M. E. Rossi for somepart). Throughout this talk, let (A, m) be an excellent 2-dimensional normal local domain
containing an algebraically closed ﬁeld k, unless otherwise speciﬁed. Let I ⊂ A be an mprimary integrally closed ideal. Then there exist a resolution of singularities X → Spec A
and an anti-nef cycle Z on X such that IOX = OX (−Z), I = H 0 (X, OX (−Z)); we saythat I is represented by Z on X, and write I = IZ . If we have a representation of anideal I = IZ , then the ring-theoretic properties of I are determined by the invertible sheaf
OX (−Z) on the resolution space X. In this talk, we introduce some results obtained fromthis viewpoint. For example, the normal reduction number of I = IZ is given in terms ofcohomology of OX (−nZ), and the characterization and the existence of the pg -ideals aredescribed by using the cohomological cycle (similar results hold for elliptic ideals). We
also have a representation of the core of pg -ideals and certain special elliptic ideals, andthe ﬁniteness of the ideals I with Gorenstein normal tangent cone G(I) for a Gorenstein
elliptic singularity (A, m).
References[OWY1] T. Okuma, K.-i. Watanabe, and K. Yoshida, Good ideals and pg -ideals in two-dimensional normal
singularities, Manuscripta Math. 150 (2016), no. 3-4, 499–520.
[OWY2]
Rees algebras and pg -ideals in a two-dimensional normal local domain, Proc. Amer. Math.
Soc. 145 (2017), no. 1, 39–47.
[OWY3]
A characterization of 2-dimensional rational singularities via core of ideals, J. of Algebra
499 (2018), 450–468.
[OWY4]
Normal reduction numbers for normal surface singularities with application to elliptic
singularities of Brieskorn type, Acta Mathematica Vietnamica 44 (2019), no. 1, 87–100.
[OWY5]The normal reduction number of two-dimensional cone-like singularities, Proc. Amer.
Math. Soc. 149 (2021), no. 11, 4569–4581.
[OWY7]
Gorensteinness for normal tangent cones of elliptic ideals, submitted (available fromarXiv: 2302.07991v1 [math.AC] 15 Feb 2023).
[ORWY] T. Okuma, M. E. Rossi, K.-i. Watanabe, and K. Yoshida, Normal Hilbert coeﬃcients and elliptic
ideals in normal 2-dimensional singularities, Nagoya Math. J. 248 (2022), 779–800.

1Kenta SatoTitle: Geometrically log canonicity of log canonical surfaces in positive characteristic
Abstract:
In this talk, we give a sufficient condition for log canonical surface singularities over a field tobe geometrically log canonical.
As an application, we prove that if a 3-dimensional quasi-projective variety X over analgebraically closed field of characteristic p>3 has only log canonical singularities, then sodoes a general hyperplane section H of X.

Kohsuke ShibataTitle: Shokurov's index conjecture for quotient singularities
Abstract
The minimal log discrepancy is an invariant of singularities defined in birational geometry.
Shokurov conjectured that the Gorenstein index of a Q-Gorenstein germ can be bounded interms of its dimension and minimal log discrepancy. This conjecture is useful to study minimallog discrepancies. In this talk I will explain basic properties for quotient singularities and show
Shokurov's index conjecture for quotient singularities. This is joint work with Yosuke
Nakamura.

Masataka Tomari and Tadashi Tomaru
Title : Comparison of fundamental cycles and maximal ideal cycles for normal surface
double points. - due to Laufer decomposition
Abstract:
For a resolution space of a normal complex surface singularity (X,o), Artin's fundamental
cycle Z (the minimal anti-nef non-zero cycle) and the maximal ideal cycle M (defined as thepull back of the maximal ideal) are defined and important geometric objects to study various
properties of singularity. M. Artin proved that M = Z for all resolutions of all rationalsingularities. However, for non-rational singularities, it is a delicate problem whether M =Z ornot. In this lecture, we will give the criterion for M = Z, under the assumption that themultiplicity is two. Then the singularity defined as z^2=f(x,y), and we prove that M >Z holdson the minimal resolution if and only if f has the Laufer decomposition ;a canonicaldecomposition f=f_{[L]}f_{[c]}f_{[o]}. We characterize these three factors f_{[L]}( Laufer type) ,f_{[c]} (contact type)and f_{[o]} (odd type) in terms of Puiseux pair. Our results explain theLaufer's famous example z^2=y (x^4 + y^6) naturally. Here we would like to present several

related topics and basic techniques for interested beginners.

Shoji Yokura
Title: Ranks of homotopy and homology groups of rationally elliptic spaces and algebraic
varieties
Abstract:
A rationally elliptic space is a simply connected space such that the ranks of its total
homotopy group and (co)homology group are both finite. A conjecture by M.R.Hilali claims
that the homotopy rank does not exceed the homology rank for a rationally elliptic space. In
this talk I will explain about some fundamental properties of a rationally elliptic space, someexamples for which the Hilali conjecture holds, and some comparisons of Poincare
polynomials and mixed Hodge polynomials of homotopy and (co)homology groups ofrationally elliptic complex algebraic varieties. This talk is partly based on joint works with
Anatoly Libgober.

Introduction to ring-theoretic properties of geometric ideals
by Ken-ichi Yoshida (Nihon University)
This is a joint work with Tomohiro Okuma and Kei-ichi Watanabe (M. E. Rossi forsome part). Throughout this talk, let (A, m) be an excellent 2-dimensional normal local
domain containing an algebrically closed field k, unless otherwise specified. In this case,for any m-primary integrally closed ideal I ⊂ A, there exist a resolution of singularities
X → Spec A and an antinef cycle Z on X such that I can be represented as follows:
IOX = OX (−Z), I = H 0 (X, OX (−Z)). Put q(nI) = dimK H 1 (X, OX (−nZ)) for every
integer n ≥ 1. Then the normal reduction number r̄(I) can be desribed in terms of q(nI) :r̄(I) = min{r ∈ Z≥0 | I n+1 = QI n (n ≥ r)}= min{n ∈ Z≥0 | q((n − 1)I) = q(nI)}.
An m-primary integrally closed ideal I is called an elliptic ideal (resp. a pg -ideal) ifr̄(I) = 2 (resp. r̄(I) = 1). These are typical examples of geometric ideals. For any
integrally lclosed ideal I ⊂ A, we consider the following geometric blow-up algebra:⊕G(I) :=I n /I n+1 .
n≥0

We call this algebra the normal tangent cone of I.
In this talk, we introduce characterizations, some ring-theoretic properties (e.g.
Cohen-Macaulayness, Gorensteinness and so on) of G(I) of such an ideal I.
A geometric aspect of this talk will be discussed in Okuma-san’s talk.
References[OWY1] T. Okuma, K.-i. Watanabe, and K. Yoshida, Good ideals and pg -ideals in two-dimensional normal
singularities, Manuscripta Math. 150 (2016), no. 3-4, 499–520.
[OWY2]Rees algebras and pg -ideals in a two-dimensional normal local domain, Proc. Amer. Math.
Soc. 145 (2017), no. 1, 39–47.
[OWY3]
A characterization of 2-dimensional rational singularities via core of ideals, J. of Algebra
499 (2018), 450–468.
[OWY4]
Normal reduction numbers for normal surface singularities with application to elliptic
singularities of Brieskorn type, Acta Mathematica Vietnamica 44 (2019), no. 1, 87–100.
[OWY5]The normal reduction number of two-dimensional cone-like singularities, Proc. Amer.
Math. Soc. 149 (2021), no. 11, 4569–4581.
[OWY7]
Gorensteinness for normal tangent cones of elliptic ideals, submitted (available fromarXiv: 2302.07991v1 [math.AC] 15 Feb 2023).
[ORWY] T. Okuma, M. E. Rossi, K.-i. Watanabe, and K. Yoshida, Normal Hilbert coeﬃcients and elliptic
ideals in normal 2-dimensional singularities, Nagoya Math. J. 248 (2022), 779–800.

1