Zoom link:

https://us02web.zoom.us/j/83962386996?pwd=UmFGOGpYRGpkdlBTVUtadEpFbDJwUT09

Zoom Meeting ID: 839 6238 6996

Passcode: 870348

Venue: Rm. 519, Astro-Math Building (NTU Campus)

Title and Abstract

Speaker: Yusuke Nakamura (UTokyo)

Title: Inversion of adjunction for quotient singularities

Abstract:

In this talk, we will discuss the minimal log discrepancies of quotient singularities. The minimal log discrepancy is an invariant of singularities defined in birational geometry, and it is related to the conjecture of termination of flips. I will show that the PIA (precise inversion of adjunction) conjecture holds for quotient singularities. The main tool of this talk involves the theory of the arc space of a quotient sin-gularity established by Denef and Loeser. I will also explain some technical difficulties when dealing with non-linear group actions. This is joint work with Kohsuke Shibata.

Speaker: Yen-An Chen (NCTS)

Title: Log canonical foliation singularities on surfaces

Abstract:

In recent years, algebraic geometers studied foliations from the viewpoints of the Minimal Model Program in which singularity plays a vital role. In this talk, I will show the classification of log canonical foliation singularities on surfaces. As an application, the set of minimal log discrepancies on foliated surfaces satisfies the ascending chain condition.

Speaker: Hiromu Tanaka (UTokyo)

Title: Pathological examples in minimal model program of positive characteristic

Abstract:

We first overview the current status of the minimal model program.

I then exhibit several examples which illustrate typical phenomena that occur only in positive characteristic.

Speaker: Tatsuro Kawakami (UTokyo)

Title: Quasi F-splitting and del Pezzo surfaces

Abstract:

Frobenius split (F-split) varieties are an important class of algebraic varieties in positive characteristic. Yobuko introduced the notion of “quasi” F-splitting, a generalization of F-splitting, and proved this behaves very well for K3 surfaces because of its good corresponding to Artin-Mazur height.

In this talk, we focus on the quasi F-splitting of del Pezzo surfaces. We prove a klt del Pezzo surface in characteristic at least seven is quasi F-split, and see that quasi F-splitting of del Pezzo surfaces defines a nice class in a view of pathological phenomena in positive characteristic.

This talk is based on joint work with Teppei Takamatsu and Shou Yoshikawa.

Speaker: Jeng-Daw Yu (NTU)

Title: Hodge aspects of moments of Bessel and Airy connections

Abstract:

Many classical ordinary differential equations, e.g., Gauss hypergeometric equations, are the connections of certain variations of Hodge structure and often have arithmetic counterparts. However in such circumstances, only regular singularities are allowed to occur. Working in the framework of the irregular Hodge theory, one extends the scopes of the Hodge structures into certain equations with irregular singularities. We illustrate such theory in the cases of moments of classical Bessel and Airy differential equations. In the former the moments turn out to be classical Hodge structures, while in the later genuine irregular Hodge structures occur but not that far from classical ones in a precise sense.

Speaker: Chi-Kang Chang (NTU)

Title: Positivity of anticanonical divisors in algebraic fibre spaces

Abstract.

It is known that the positivity of the anti-canonical divisor is an important property that is closely related to the geometric structure of a variety. Given an algebraic fibre space from X to Y between normal projective varieties with mild singularities, and let F be its general fibre. In this talk, we will introduce results relating the positivity of the anticanonical divisors of X and of Y under some conditions on the asymptotic base loci of -K_X. In particular, we will obtain an inequality between the Iitaka dimension of anti-canonical divisors of X, Y and F under the assumption that the stable base locus of -K_X does not dominant over Y.

Speaker: Shou Yoshikawa (UTokyo)

Title: Fedder's type criterion for quasi-Frobenius splitting

Abstract:

In characteristic zero, the analytic structure on the complex varieties plays the essential role for studying the geometry. In positive characteristic, we sometimes use the Frobenius morphism and its properties instead of analytic structure. Mehta and Ramanathan introduced the notion of Frobenius splitting and proved the Kodaira vanishing for Frobenius splitting varieties. After that, Yobuko introduced the weaker notion, called quasi-Frobenius splitting, and he proved that quasi-Frobenius splitting also gives the Kodaira vanishing.

In this talk, I will introduce a criterion for quasi-Frobenius splitting of hypersurfaces in a projective space or a weighted projective space. It is a generalization of the criterion for Frobenius splitting proved by Fedder. Moreover, I will introduce the interesting examples and applications of the criterion. This talk is based on joint work with Tasturo Kawakami and Teppei Takamastu.

Speaker: Hsueh-Yung Lin (NTU)

Title: Growths of twisted homogeneous coordinate rings

Abstract:

Let X be a projective variety over a field k, endowed with an automorphism f. Fix a polarization L, Artin and Van den Bergh introduced the twisted homogeneous coordinate ring B(X,L,f) associated to the above data. This is a graded associative ring which specializes to the homogeneous coordinate ring when f is the identity. We study the polynomial growth (i.e. the Gelfand-Kirillov dimension) and the exponential growth of the algebra B(X,L,f) from a dynamical viewpoint of f, and obtain new constraints on B(X,L,f). This allows us to show, for instance, that when X is a smooth projective threefold, the Gelfand-Kirillov dimension of B(X,L,f) is either 3, 5, or 9, and all three numbers are realizable. Joint work in progress with Keiji Oguiso and De-Qi Zhang.