arXiv subjects

Giancarlo Urzúa

Publications and source records attributed to Giancarlo Urzúa.

At least 19 recordsLinked to original sources

arXiv2026

Degenerations of the complex projective plane with only rational singularities

Wahl's conjecture states that two-dimensional singularities admitting a rational homology disk smoothing are weighted homogeneous. Assuming the conjecture, we classify all normal degenerations of the complex projective plane with only rational singularities. They are precisely the surfaces classified by Manetti and Hacking--Prokhorov, which are controlled by the Markov equation, together with six new degenerations containing four non-log canonical singularities.

math.AG↗

arXiv2026

Rational homology disk degenerations of elliptic surfaces

In this paper, a $\mathbb{Q}$HD singularity is a weighted homogeneous normal surface singularity admitting a rational homology disk ($\mathbb{Q}$HD) smoothing. These singularities are rational but often not log canonical. We classify all $\mathbb{Q}$HD degenerations of nonsingular projective elliptic surfaces, extending Kawamata's classification of the case with only Wahl singularities (i.e., log terminal $\mathbb{Q}$HD singularities). We also realize all $\mathbb{Q}$HD degenerations of Dolgachev surfaces $D_{a,b}$ with one $\mathbb{Q}$HD singularity, for every pair of integers $a,b$. For each such degeneration, we construct a minimal semi log canonical (slc) birational model via a Seifert partial resolution in the sense of Wahl followed by semistable flips. Finally, we prove that these minimal slc models are unobstructed and deform to the recent degenerations of Dolgachev surfaces constructed by D. Lee and Y. Lee.

math.AG↗

arXiv2026

Tropical methods for stable octic double planes

This paper has been written to illustrate the power of techniques from tropical geometry and mirror symmetry for studying the KSBA moduli space of surfaces on or near the Noether line. We focus on the moduli space of octic double planes ($K^2 = 2$, $p_g = 3$) and use methods from tropical and toric geometry to classify the strata corresponding to normal KSBA-stable surfaces, focusing on the non-Gorenstein case.

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arXiv2025

Wahl singularities in degenerations of del Pezzo surfaces

For any fixed $1 \leq \ell \leq 9$, we characterize all Wahl singularities that appear in degenerations of del Pezzo surfaces of degree $\ell$. This extends the work of Manetti and Hacking-Prokhorov in degree $9$, where Wahl singularities are classified using the Markov equation. To achieve this, we introduce del Pezzo Wahl chains with markings. They define marked del Pezzo surfaces $W_{*m}$ that govern all such degenerations. We also prove that every marked del Pezzo surface degenerates into a canonically defined toric del Pezzo surface with only T-singularities. In addition, we establish a one-to-one correspondence between the $W_{*m}$ surfaces and certain fake weighted projective planes. As applications, we show that every Wahl singularity occurs for del Pezzo surfaces of degree $\leq 4$, but that there are infinitely many Wahl singularities that do not arise for degrees $\geq 5$. We also use Hacking's exceptional collections to provide geometric proofs of recent results by Polishchuk and Rains on exceptional vector bundles on del Pezzo surfaces.

math.AG↗

arXiv2025

Classification of Horikawa surfaces with T-singularities

We classify all projective surfaces with only T-singularities, ample canonical class, and $K^2=2p_g-4$. In this way, we identify all surfaces, smoothable or not, with only T-singularities in the Kollár--Shepherd-Barron--Alexeev (KSBA) moduli space of Horikawa surfaces. We also prove that they are not smoothable when $p_g \geq 10$, except for the Lee-Park (Fintushel-Stern) examples, which we show to have only one deformation type unless $p_g=6$ (in which case they have two). This demonstrates that the challenging Horikawa problem cannot be addressed through complex T-degenerations. We propose new questions regarding diffeomorphism types based on our classification. Furthermore, the techniques developed in this paper enable us to classify all KSBA surfaces with only T-singularities and $K^2\leq 2p_g-3$, for example, quintic surfaces and I-surfaces.

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arXiv2025

The birational geometry of Markov numbers

It is known that all degenerations of the complex projective plane into a surface with only quotient singularities are controlled by the positive integer solutions $(a,b,c)$ of the Markov equation $$x^2+y^2+z^2=3xyz.$$ It turns out that these degenerations are all connected through finite sequences of other simpler degenerations by means of birational geometry. In this paper, we explicitly describe these birational sequences and show how they are bridged among all Markov solutions. For a given Markov triple $(a,b,c)$, the number of birational modifications depends on the number of branches that it needs to cross in the Markov tree to reach the Fibonacci branch. We show that each of these branches corresponds exactly to a Mori train of the flipping universal family of a particular cyclic quotient singularity defined by $(a,b,c)$. As a byproduct, we obtain new numerical/combinatorial data for each Markov number, and new connections with the Markov conjecture (Frobenius Uniqueness Conjecture), which rely on Hirzebruch-Jung continued fractions of Wahl singularities.

math.AG↗

arXiv2024

An exotic 5RP^2 in the 4-sphere

We show an example of an embedded copy of 5RP^2 in the four-sphere which is topologically standard but smoothly knotted, i.e. smoothly not isotopic to the standard embedding.

math.GT↗

arXiv2024

Optimal bounds for many T-singularities in stable surfaces

We effectively bound T-singularities on non-rational projective surfaces with an arbitrary amount of T-singularities and ample canonical class. This fully generalizes the previous work for the case of one singularity, and illustrates the vast increase in combinatorial complexity as the number of singularities grows. We find that certain combinatorial configurations lead to relatively high bounds. We classify all such configurations, and show that their non-existence gives a strong and optimal bound. As an application, we work out in detail the case of two singularities.

math.AG↗

arXiv2023

The Coble-Mukai lattice from $\mathbb Q$-Gorenstein deformations

We show some geometric properties of Enriques surfaces via $\mathbb Q$-Gorenstein smoothings of Coble surfaces. In particular, we explicitly identify the Enriques lattice of the general fiber with the Coble-Mukai lattice. At the end, we discuss applications to Gorenstein $\mathbb Q$-Homology projective planes with trivial canonical class.

math.AG↗

arXiv2022

Categorical aspects of the Kollár--Shepherd-Barron correspondence

It is well known that a $2$-dimensional cyclic quotient singularity $\overline{W}$ has the same singularity category as a finite dimensional associative algebra $\overline{R}$ introduced by Kalck and Karmazyn. We study the deformations of the algebra $\overline{R}$ induced by the deformations of the surface $\overline{W}$ to a smooth surface. We show that they are Morita--equivalent to path algebras $\hat{R}$ of acyclic quivers for general smoothings within each irreducible component of the versal deformation space of $\overline{W}$ (as described by Kollár and Shepherd-Barron). Furthermore, $\hat{R}$ is semi-simple if and only if the smoothing is $\mathbb{Q}$-Gorenstein (one direction is due to Kawamata). We provide many applications. For example, we describe strong exceptional collections of length $10$ on all Dolgachev surfaces and classify admissible embeddings of derived categories of quivers into derived categories of rational surfaces.

math.AG↗

arXiv2022

Exotic surfaces

We construct the first exotic $\mathbb C \mathbb P^2 \# 4 \overline{\mathbb C \mathbb P^2}$ by means of rational blowdowns. Similarly, we construct the first exotic $3\mathbb C \mathbb P^2 \# b^- \overline{\mathbb C \mathbb P^2}$ for $b^-=9,8,7$ . All of them are minimal and symplectic, as they are produced from projective surfaces $W$ with Wahl singularities, and $K_W$ big and nef. In more generality, we elaborate on the problem of finding exotic $(2χ(\mathcal{O}_W)-1) \mathbb C \mathbb P^2 \# (10χ(\mathcal{O}_W)-K^2_W-1) \overline{\mathbb C \mathbb P^2}$ from these Kollár--Shepherd-Barron--Alexeev surfaces $W$, obtaining explicit geometric obstructions.

math.AG↗

arXiv2021

Rational configurations in K3 surfaces and simply-connected $p_g=1$ surfaces for $K^2=1,2,3,4,5,6,7,8,9$

We prove the existence of $(20-2K^2)$-dimensional families of simply-connected surfaces with ample canonical class, $p_g=1$, and $1 \leq K^2 \leq 9$, and we study the relation with configurations of rational curves in K3 surfaces via $\mathbb Q$-Gorenstein smoothings. Our surfaces with $K^2=7$ and $K^2=9$ are the first surfaces known in the literature, together with the existence of a $4$-dimensional family for $K^2=8$.

math.AG↗

arXiv2021

On wormholes in the moduli space of surfaces

We study a certain wormholing phenomenon that takes place in the Kollár--Shepherd-Barron--Alexeev (KSBA) compactification of the moduli space of surfaces of general type. It occurs because of the appearance of particular extremal P-resolutions in surfaces on the KSBA boundary. We state a general wormhole conjecture, and we prove it for a wide range of cases. At the end, we discuss some topological properties and open questions.

math.AG↗

arXiv2020

Transversality of sections on elliptic surfaces with applications to elliptic divisibility sequences and geography of surfaces

We consider elliptic surfaces $\mathcal{E}$ over a field $k$ equipped with zero section $O$ and another section $P$ of infinite order. If $k$ has characteristic zero, we show there are only finitely many points where $O$ is tangent to a multiple of $P$. Equivalently, there is a finite list of integers such that if $n$ is not divisible by any of them, then $nP$ is not tangent to $O$. Such tangencies can be interpreted as unlikely intersections. If $k$ has characteristic zero or $p>3$ and $\mathcal{E}$ is very general, then we show there are no tangencies between $O$ and $nP$. We apply these results to square-freeness of elliptic divisibility sequences and to geography of surfaces. In particular, we construct mildly singular surfaces of arbitrary fixed geometric genus with $K$ ample and $K^2$ unbounded.

math.AG↗

arXiv2020

Chilean configuration of conics, lines and points

Using the theory of rational elliptic fibrations, we construct and discuss a one parameter family of configurations of $12$ conics and $9$ points in the projective plane that realizes an abstract configuration $(12_6,9_8)$. This is analogous to the famous Hesse configuration of $12$ lines and $9$ points forming an abstract configuration $(12_3,9_4)$. We also show that any Halphen elliptic fibration of index $2$ with four triangular singular fibers arises from such configuration of conics.

math.AG↗

arXiv2020

Savage Surfaces

Let $G$ be the topological fundamental group of a given nonsingular complex projective surface. We prove that the Chern slopes $c_1^2(S)/c_2(S)$ of minimal nonsingular projective surfaces of general type $S$ with $π_1(S) \simeq G$ are dense in the interval $[1,3]$.

math.AG↗

arXiv2020

Bounding tangencies of sections on elliptic surfaces

Given an elliptic surface $\mathcal{E}\to\mathcal{C}$ over a field $k$ of characteristic zero equipped with zero section $O$ and another section $P$ of infinite order, we give a simple and explicit upper bound on the number of points where $O$ is tangent to a multiple of $P$.

math.AG↗

arXiv2020

Antiflips, mutations, and unbounded symplectic embeddings of rational homology balls

The Milnor fibre of a $\mathbb{Q}$-Gorenstein smoothing of a Wahl singularity is a rational homology ball $B_{p,q}$. For a canonically polarised surface of general type $X$, it is known that there are bounds on the number $p$ for which $B_{p,q}$ admits a symplectic embedding into $X$. In this paper, we give a recipe to construct unbounded sequences of symplectically embedded $B_{p,q}$ into surfaces of general type equipped with non-canonical symplectic forms. Ultimately, these symplectic embeddings come from Mori's theory of flips, but we give an interpretation in terms of almost toric structures and mutations of polygons. The key point is that a flip of surfaces, as studied by Hacking, Tevelev and Urzúa, can be formulated as a combination of mutations of an almost toric structure and deformation of the symplectic form.

math.SG↗