arXiv subjects

Carlos A. Serván

Publications and source records attributed to Carlos A. Serván.

6 recordsLinked to original sources

arXiv2026

Local rigidity of covering constructions and Weil--Petersson subvarieties of the moduli space of curves

We show that totally geodesic subvarieties of the moduli space $\mathcal M_{g,n}$ of genus $g$ curves with $n$ marked points, endowed with the Weil--Petersson metric, are locally rigid. This implies that covering constructions -- examples of totally geodesic subvarieties of $\mathcal M_{g,n}$ endowed with the Teichmüller metric -- are locally rigid. We deduce the local rigidity statement from a more general rigidity result for a class of orbifold maps to $\mathcal M_{g,n}$.

math.GT↗

arXiv2026

Infinitely many Lefschetz pencils on ruled surfaces

We show that any ruled surface $X$ with $χ(X) < 0$ admits infinitely many inequivalent Lefschetz pencils of fixed genus and number of base points. Our proof proceeds by building infinitely many inequivalent Lefschetz fibrations on a blow-up $X \# 4 \overline{\mathbb{CP}^2}$ of $X$ with constant fiber class, via a mechanism known as partial conjugation. Furthermore, there exists a symplectic form on $X$ compatible with all such pencils, and similarly for the fibrations in $X\#4\overline{\mathbb{CP}^2}$. This provides the first example of this phenomenon and makes progress on Problem 4.98 of the K3 list of problems in low-dimensional topology in the case of ruled surfaces.

math.GT↗

arXiv2025

On the uniqueness of the Prym map

The classical Prym construction associates to a smooth, genus $g$ complex curve $X$ equipped with a nonzero cohomology class $θ\in H^1(X,\mathbb{Z}/2\mathbb{Z})$, a principally polarized abelian variety (PPAV) $\mbox{Prym}(X,θ)$. Denote the moduli space of pairs $(X,θ)$ by $\mathcal{R}_g$, and let $\mathcal{A}_h$ be the moduli space of PPAVs of dimension $h$. The Prym construction globalizes to a holomorphic map of complex orbifolds $\mbox{Prym}: \mathcal{R}_g \to \mathcal{A}_{g-1}$. For $g\geq 4$ and $h \leq g-1$, we show that $\mbox{Prym}$ is the unique nonconstant holomorphic map of complex orbifolds $F:\mathcal{R}_g \to \mathcal{A}_h$. This solves a conjecture of Farb. A main component in our proof is a classification of homomorphisms $π_1^{\mbox{orb}}(\mathcal{R}_g) \to \mbox{Sp}(2h,\mathbb{Z})$ for $h \leq g-1$. This is achieved using arguments from geometric group theory and low-dimensional topology.

math.AG↗

arXiv2024

Lefschetz fibrations with infinitely many sections

The Arakelov--Parshin rigidity theorem implies that a holomorphic Lefschetz fibration $π: M \to S^2$ of genus $g \geq 2$ admits only finitely many holomorphic sections $σ:S^2 \to M$. We show that an analogous finiteness theorem does not hold for smooth or for symplectic Lefschetz fibrations. We prove a general criterion for a symplectic Lefschetz fibration to admit infinitely many homologically distinct sections and give many examples satisfying such assumptions. Furthermore, we provide examples that show that finiteness is not necessarily recovered by considering a coarser count of sections up to the action of the (smooth) automorphism group of a Lefschetz fibration.

math.GT↗

arXiv2023

Isometric embeddings of Teichmüller spaces are covering constructions

Pulling back complex structures along a branched covering induces a holomorphic isometric embedding of Teichmüller spaces. We show that for dimension at least $2$, all isometric embeddings arise from branched coverings. This generalizes a theorem of Royden. As a consequence we obtain that totally geodesic submanifolds of Teichmüller space, which are isometric to some Teichmüller space, are covering constructions. Another consequence is the classification of locally isometric embeddings of moduli spaces of Riemann surfaces.

math.GT↗

arXiv2020

Thresholding normally distributed data creates complex networks

Network data sets are often constructed by some kind of thresholding procedure. The resulting networks frequently possess properties such as heavy-tailed degree distributions, clustering, large connected components and short average shortest path lengths. These properties are considered typical of complex networks and appear in many contexts, prompting consideration of their universality. Here we introduce a simple model for correlated relational data and study the network ensemble obtained by thresholding it. We find that some, but not all, of the properties associated with complex networks can be seen after thresholding the correlated data, even though the underlying data are not "complex". In particular, we observe heavy-tailed degree distributions, a large numbers of triangles, and short path lengths, while we do not observe non-vanishing clustering or community structure.

cs.SI↗