arXiv subjects

Clemens Nollau

Publications and source records attributed to Clemens Nollau.

3 recordsLinked to original sources

arXiv2026

The restricted Hitchin map of wobbly vector bundles

This article studies the restricted Hitchin map $h_V$ of a stable rank 2 vector bundle $V$ on a smooth projective curve $C$. This map associates to a trace-free twisted endomorphism $φ: V \to V \otimes K_C$ its determinant, which is a quadratic differential. We show that if $V$ is a general wobbly vector bundle, then it has a single nilpotent twisted endomorphism, up to scalars. As a consequence we show that $h_V$ is generically finite and we compute its degree. Furthermore, we show that if $C$ is not hyperelliptic, then the image of $h_V$ contains a quadratic differential with simple zeros. This is equivalent to saying that there is a smooth spectral curve associated to a twisted endomorphism of $V$.

math.AG↗

arXiv2025

Towards Brill-Noether Theory for Spectral Curves

We study Brill-Noether loci of three kinds of spectral curves: classical spectral curves as introduced by Hitchin, spectral curves over the projective line and double covers whose branch locus is a canonical divisor. Our techniques are based on the Beauville-Narasimhan-Ramanan correspondence: We push down line bundles on the spectral curve to the base curve and then we study the Higgs bundles obtained in this way. For the first kind we study the spaces of pencils in the Picard variety of a classical spectral curve in detail. In the case of spectral curves over the projective line we deal with their splitting loci which refine the Brill-Noether loci in the Picard variety. We compute their dimensions and investigate whether they are smooth. For the third kind we determine the gonality sequence when the rank of the linear system is much smaller than the genus. For this the base curve and the branch divisor are assumed to be general.

math.AG↗

arXiv2023

Recovery of Plane Curves from Branch Points

We recover plane curves from their branch points under projection onto a line. Our focus lies on cubics and quartics. These have 6 and 12 branch points respectively. The plane Hurwitz numbers 40 and 120 count the orbits of solutions. We determine the numbers of real solutions, and we present exact algorithms for recovery. Our approach relies on 150 years of beautiful algebraic geometry, from Clebsch to Vakil and beyond.

math.AG↗