Algebraic Surfaces
and Their Singularities

The workshop is devoted to algebraic surfaces and singularity theory. Its purpose is to facilitate scholarly exchange and collaboration among experts in these areas. 

Titles and Abstracts

• Marcos Canedo _ Pontificia Universidad Católica de Chile

  • • Classification of Small surfaces with only WQHD singularities
      A $\mathbb{Q}$HD singularity is a normal two-dimensional singularity that admits a smoothing with Milnor number equal to zero. Wahl conjectured that every $\mathbb{Q}$HD singularity is weighted homogeneous, and their classification was done by Bhupal and Stipsicz. We refer to them as W$\mathbb{Q}$HD singularities. (They are typically non-log-canonical; the log-canonical ones are Wahl singularities and three quotients of certain elliptic singularities.) On the other hand, small surfaces arise naturally in the classification of W$\mathbb{Q}$HD surfaces on the Noether line $K^2 = 2p_g - 4$. In this talk, we will introduce small surfaces and establish an inequality that allows us to show that W$\mathbb{Q}$HD surfaces on the Noether line must be small. If time permits, we will also discuss progress toward a classification of small surfaces with only W$\mathbb{Q}$HD singularities. This is a joint work with Giancarlo Urzúa.

• DongSeon Hwang _ IBS  Center for Complex Geometry

  • • Canonical Kähler metrics on toric del Pezzo surfaces and beyond
      After a brief review of the known results on canonical Kähler metrics on toric Fano varieties, we will focus on the surface case. More precisely, we show that every symmetric Fano polytope admits a Kähler–Einstein metric, generalizing the work of Batyrev and Selivanova. We also study in detail the automorphism groups of symmetric and Kähler–Einstein Fano polygons. In particular, we prove that every finite subgroup of  GL_2(Z) arises as the automorphism group of some Kähler–Einstein Fano polygon. If time permits, we will also discuss a related notion called a B-transformation.

• Jaekwan Jeon _ IBS Center for Geometry and Physics

  • • Deformations of weighted homogeneous surface singularities via the anti-MMP
      J. Kollár conjectured that for a rational surface singularity $X$, the irreducible components of $Def(X)$ correspond to certain partial resolutions of $X$ which are called P-resolutions. Since every component of $Def(X)$ is a smoothing component and every P-resolution induces a smoothing of $X$, roughly speaking, the problem is how we get a P-resolution from a smoothing. The authors proved the conjecture for weighted homogeneous surface singularities with big central node by using the deformation theory of sandwiched singularities.

• Giancarlo Urzúa _ Pontificia Universidad Católica de Chile

  • • Classification of T-surfaces with $K^2 \le 2p_g-3$
      I will explain the techniques and results of a recent and ongoing work together with Vicente Monreal and Jaime Negrete which classifies surfaces with only T-singularities and $K^2 \le 2p_g-3$. This includes non-smoothable surfaces with $K^2 < 2p_g-4$, Horikawa surfaces $K^2=2p_g-4$, and Horikawa surfaces on the Noether line plus 1, this is, $K^2=2p_g-3$. The latter includes I-surfaces and quintics.

• Yoonjeong Yang _ Chungnam National University

  • • Numerical Godeaux surfaces with $H_1=\mathbb{Z}/4\mathbb{Z}$
      We construct minimal complex surfaces of general type with invariants $K^2=1$ and $p_g=q=0$ as double covers of Enriques surfaces. The main example comes from the seven--nodal Enriques surface studied by Mukai and by Ito--Ohashi; a second example uses Kondo's fifth Enriques surface. In particular,  $H_1(S,\mathbb{Z})=\mathbb{Z}/4\mathbb{Z}$ for the first example $S$.

Timetable

2025-10-31 (Friday)

• 10:00--11:00: DongSeon Hwang

• 11:30--12:30: Giancarlo Urzúa

• 12:30--14:00: Lunch

• 14:00--15:00:  Jaekwan Jeon

• 15:30--16:30: Marcos Canedo

• 17:00--18:00: Yoonjeong Yang

• 18:00--20:00: Banquet