arXiv subjects

Juan Pablo Zúñiga

Publications and source records attributed to Juan Pablo Zúñiga.

5 recordsLinked to original sources

arXiv2026

Orbifold Degenerations of Hirzebruch Surfaces

We study orbifold degenerations of Hirzebruch surfaces. Our main theorem shows that every such degeneration $X$ arises as a partial smoothing of a toric surface. We then give a combinatorial description of the singularities that arise when $-K_X$ is not nef. This complements previous joint work with G. Urzúa, which treated the case in which $-K_X$ is ample. For Hirzebruch surfaces $\mathbb{F}_k$ with $k\geq 3$, we obtain an explicit description of all possible central fibers. For $k\leq 1$, we use the threefold minimal model program to reduce the problem to the case of central fibers whose anticanonical divisor is nef. The remaining surfaces are toric del Pezzo surfaces of degree $8$ with T-singularities. We classify these by studying the birational geometry arising from mutations of Fano polygons.

math.AG↗

arXiv2026

Computing Cox rings via the cone conjecture

We initiate a program to study the Cox ring of Calabi-Yau varieties, employing the notion of Morrison-Kawamata dream spaces. In this setting, we establish an analogue of the Hu-Keel GIT constructions for Mori dream spaces. More precisely, for a Morrison-Kawamata dream space $X$, we establish a correspondence between the small $\mathbb{Q}$-factorial modifications of $X$ and the GIT quotients of $\operatorname{Spec}\operatorname{Cox}(X)$. We further show that the Cox ring of a Morrison-Kawamata dream space is a filtered direct limit of subalgebras, each of which is an inverse limit of finitely generated $\mathrm{Cl}(X)$-graded $\mathbb{K}$-algebras. As an application, we give an explicit presentation of the Cox ring of a very general hypersurface of multidegree $(2,\dots,2,n+1)$ in $(\mathbb{P}^1)^m\times \mathbb{P}^n$. Furthermore, we prove that the Cox ring of such a hypersurface is of dense $F$-pure type.

math.AG↗

arXiv2026

Degenerations of cluster type varieties

We study degenerations of cluster type varieties and pairs. Our first theorem proves that degenerations of toric pairs are finite quotients of toric pairs. In a similar vein, under some mild conditions, we prove that degenerations of cluster type pairs are finite quotients of cluster type pairs. Then, we focus on degenerations of cluster type surfaces. We give some general criteria for the existence of $1$-complements on degenerations of toric surfaces. We prove that for almost all $(a,b,c)\in \mathbb{Z}_{\geq 1}^3$ the weighted projective plane $\mathbb{P}(a,b,c)$ has no non-trivial degenerations. In particular, for a Markov triple $(a,b,c)\in \mathbb{Z}_{\geq 2}^3$, we prove that $\mathbb{P}(a^2,b^2,c^2)$ admits no non-trivial degenerations. Finally, we give a complete classification of the degenerations of $\mathbb{P}(1,1,n)$ for $n\geq 3$.

math.AG↗

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

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↗