New trends in moduli inspired by K-stability

14:00 -- 15:00 Roktim Mascharak

15:30 -- 16:30 Stefania Vassiliadis

19:00 Dinner

10:00 -- 11:00 Kenta Hashizume

11:30 -- 12:30 Kristin DeVleming

14:00 -- 15:30 Makoto Enokizono

16:00 -- 17:00 Heath Pearson

9:30 -- 10:30 Masafumi Hattori

10:45 -- 11:45 Kento Fujita

KRISTIN DEVLEMING: (TBA)

MAKOTO ENOKIZONO: Normal stable degenerations of Noether-Horikawa surfaces

Abstract: Noether-Horikawa surfaces are surfaces of general type satisfying the equation K^2=2p_g−4, which represents the equality of the Noether inequality K^2≥2p_g−4 for surfaces of general type. In the 1970s, Horikawa conducted a detailed study of smooth Noether-Horikawa surfaces, providing a classification of these surfaces and describing their moduli spaces.

In this talk, I will present an explicit classification of normal stable degenerations of Noether-Horikawa surfaces. Specifically, I will discuss the following results:

(1) Classification of Noether-Horikawa surfaces with Q-Gorenstein smoothable log canonical singularities.

(2) Criterion for determining the (global) Q-Gorenstein smoothability of the surfaces described in (1).

(3) Description of the KSBA moduli spaces for Q-Gorenstein smoothable normal stable Noether-Horikawa surfaces.

This is joint work with Hiroto Akaike, Masafumi Hattori and Yuki Koto.

KENTO FUJITA: Another view on smooth prime Fano threefolds of degree 22 with infinite automorphism groups

Abstract: All smooth Fano threefolds with infinite automorphism groups are understood due to Prokhorov, Kuznetsov and Shramov by use of deep studies of their Hilbert scheme of lines. I will present as our joint work with Adrien Dubouloz and Takashi Kishimoto an alternative and self-contained proof of it, allowing us to use several properties on the smooth quintic del Pezzo threefold. Moreover, I would like to explain an interesting elementary link joining prime Fano threefolds of degree 22 with Fano threefolds of No. 2.21 in Mori-Mukai's list.

KENTA HASHIZUME: On boundedness and moduli space of special klt-trivial fibrations

Abstract: A klt-trivial fibration is a kind of fibration which often appears in birational geometry. In this talk, I will introduce the boundedness result and the existence of the coarse moduli space of special klt-trivial fibrations over curves. I will mainly explain the boundedness of the special klt-trivial fibrations over curves with some fixed invariants. This talk is based on a joint work with Masafumi Hattori.

MASAFUMI HATTORI: Applications of K-moduli of quasimaps to K-moduli conjecture for Calabi-Yau fibrations over curves

Abstract: Odaka proposed the K-moduli conjecture in 2010, predicting the existence of a moduli space of K-polystable objects with an ample CM line bundle.  While this conjecture has been solved in the Fano case, it remains open in general.

Recent developments of Fine, Dervan-Sektnan and Ortu have highlited the relevance of the existence of cscK metrics and K-stability for (X,\epsilon A+L) for sufficiently small \epsilon, where f\colon (X,A)\to (B,L) is a fibration. According to their works, such K-stability is closely related to some K-stability of fibers and the bases. Especially in the Calabi-Yau fibration over curve case, uniform K-stability in this context (uniform adiabatic K-stability) coincides with the log twisted K-stability on the base.

In this talk, we will regard the base curve as a quasimap and introduce the notion of K-moduli of quasimaps. By using this framework, we address the K-moduli conjecture for Calabi-Yau fibrations over curves whose generic fibers are either Abelian varieties or HyperKahler manifolds. This is a joint work arXiv:2504.21519 with Kenta Hashizume.

ROKTIM MASCHARAK: Birational Geometry of Foliated Threefolds

Abstract: Foliation is an ornament of algebraic varieties which provides us more tool to investigate geometry of the underlying variety. In this talk we will take a step towards birational classification of these ornaments through the lens of the Minimal Model Program. I will report some recent development in this direction of research. If time permits, we will see how the birational geometry of this varieties are more “rigid” than that of the varieties.

HEATH PEARSON: The generalised Mukai conjecture for spherical log Fano pairs

Abstract: Given a smooth Fano variety X, the generalised Mukai conjecture is an inequality, which features the dimension, the Picard number, and the pseudoindex -- i.e. the minimum anticanonical degree of a rational curve on X. The boundary case of this inequality characterises powers of projective spaces.

Kento Fujita has studied this conjecture in the more general setting of log Fano pairs (with singularities), and has shown it to hold for the toric log Fano pairs.

Spherical varieties naturally generalise toric varieties, replacing the algebraic torus by a non-abelian reductive group. In this talk, we will sketch a proof of the generalised Mukai conjecture for spherical log Fano pairs. Joint work with Johannes Hofscheier.

STEFANIA VASSILIADIS: (TBA)