arXiv subjects

Noah Ruhland

Publications and source records attributed to Noah Ruhland.

3 recordsLinked to original sources

arXiv2026

Genus $2$ pencils on surfaces with $p_g=K^2=1$, envelopes, and conics tangent to plane cubic curves

We consider $(1,1)$-surfaces, namely, minimal compact complex surfaces $S$ with $p_g (S) =K_S^2=1$: for these the bicanonical map is a covering of degree $4$ of the plane $\mathbb{P}^2$. And we answer a question posed by Meng Chen, whether they can contain a genus 2 pencil (this is the standard reason of failure of birationality of the bicanonical map). Our main theorem says that those which admit a genus 2 pencil form an irreducible subvariety of codimension $3$ in their moduli space $\frak M_{[1,1]}$; moreover, the general such surface admits exactly $12$ such pencils. The real fun is to relate this variety to the geometry of pencils of conics in the plane everywhere tangent to a cubic curve and a line. We investigate the corresponding variety $\mathcal{T}$ of triples and provide explicit equations using the classical theory of envelopes: among others, equations given in terms of the Weierstrass normal form of the cubic.

math.AG↗

arXiv2026

Computation of Singular Godeaux Surfaces and a New Explicit Fake Quadric (With an Appendix by Christian Gleissner and Noah Ruhland)

We present a computational method for detecting highly singular members in families of algebraic varieties. Applying this approach to a family of numerical Godeaux surfaces, we obtain explicit examples with many singularities. In particular, we construct a Godeaux surface whose singular locus consists of two $\mathsf A_1$ and two $\mathsf A_3$ singularities. We show that this surface admits a $\mathbb{Z}/2 \times \mathbb{Z}/4$ abelian cover which is a smooth minimal surface of general type with invariants $K^2=8$ and $p_g=0$, i.e. a fake quadric. Together with the result in the Appendix, this provides the first explicit construction of a fake quadric that does not arise as a quotient of a product of curves.

math.AG↗

arXiv2025

On Rigid Varieties Isogenous to a Product of Curves

In this note, we study rigid complex manifolds that are realized as quotients of a product of curves by a free action of a finite group. They serve as higher-dimensional analogues of Beauville surfaces. Using uniformization, we outline the theory to characterize these manifolds through specific combinatorial data associated with the group under the assumption that the action is diagonal and the manifold is of general type. This leads to the notion of a $n$-fold Beauville structure. We define an action on the set of all $n$-fold Beauville structures of a given finite group that allows us to distinguish the biholomorphism classes of the underlying rigid manifolds. As an application, we give a classification of these manifolds with group $\mathbb Z_5^2$ in the three dimensional case and prove that this is the smallest possible group that allows a rigid, free and diagonal action on a product of three curves. In addition, we provide the classification of rigid 3-folds $X$ given by a group acting faithfully on each factor for any value of the holomorphic Euler number $χ(\mathcal O_X) \geq -5$.

math.AG↗