arXiv subjects

Robert Auffarth

Publications and source records attributed to Robert Auffarth.

At least 19 recordsLinked to original sources

arXiv2026

Soohak: A Mathematician-Curated Benchmark for Evaluating Research-level Math Capabilities of LLMs

Following the recent achievement of gold-medal performance on the IMO by frontier LLMs, the community is searching for the next meaningful and challenging target for measuring LLM reasoning. Whereas olympiad-style problems measure step-by-step reasoning alone, research-level problems use such reasoning to advance the frontier of mathematical knowledge itself, emerging as a compelling alternative. Yet research-level math benchmarks remain scarce because such problems are difficult to source (e.g., Riemann Bench and FrontierMath-Tier 4 contain 25 and 50 problems, respectively). To support reliable evaluation of next-generation frontier models, we introduce Soohak, a 439-problem benchmark newly authored from scratch by 64 mathematicians. Soohak comprises two subsets. On the Challenge subset, frontier models including Gemini-3-Pro, GPT-5, and Claude-Opus-4.5 reach 30.4%, 26.4%, and 10.4% respectively, leaving substantial headroom, while leading open-weight models such as Qwen3-235B, GPT-OSS-120B, and Kimi-2.5 remain below 15%. Notably, beyond standard problem solving, Soohak introduces a refusal subset that probes a capability intrinsic to research mathematics: recognizing ill-posed problems and pausing rather than producing confident but unjustified answers. On this subset, no model exceeds 50%, identifying refusal as a new optimization target that current models do not directly address. To prevent contamination, the dataset will be publicly released in late 2026, with model evaluations available upon request in the interim.

cs.CL↗

arXiv2025

On the Picard number and the extension degree of period matrices of complex tori

The rank $\rho$ of the N\'eron-Severi group of a complex torus $X$ of dimension $g$ satisfies $0\leq\rho\leq g^2=h^{1,1}.$ The degree $\mathfrak{d}$ of the extension field generated over $\mathbb{Q}$ by the entries of a period matrix of $X$ imposes constraints on its Picard number $\rho$ and, consequently, on the structure of $X$. In this paper, we show that when $\mathfrak{d}$ is $2$, $3$, or $4$, the Picard number $\rho$ is necessarily large. Moreover, for an abelian variety $X$ of dimension $g$ with $\mathfrak{d}=3,$ we establish a structure-type result: $X$ must be isogenous to $E^g$, where $E$ is an elliptic curve without complex multiplication. In this case, the Picard number satisfies $\rho(X)=\frac{g(g+1)}{2}.$ As a byproduct, we obtain that if $\mathfrak{d}$ is odd, then $\rho(X)\leq\frac{g(g+1)}{2}.$

math.AG↗

arXiv2025

A note on multisecants of the Kummer variety of a Jacobian

We show that if $C$ is a smooth projective curve and $\mathfrak{d}$ is a $\mathfrak{g}^{n}_{2n}$ on $C$, then we obtain a rational map $\mathrm{Sym}^{n}(C)\dashrightarrow\mathfrak{d}$ whose fibers can be related in an interesting way to Gunning multisecants of the Kummer variety of $JC$. This generalizes previous work done by the first author with Codogni and Salvati Manni.

math.AG↗

arXiv2024

Pseudoreflections on Prym Varieties

We show that for every g greater or equal than 5, the locus of Prym varieties in the moduli space of principally polarized abelian varieties of dimension g-1 that possess a pseudoreflection of geometric origin is the union of three different non-empty explicit irreducible families. This is in stark contrast to the loci of Jacobian varieties that possess a pseudoreflection of geometric origin, which is empty for any genus greater than 3. In g=6, a distinguished example of Prym varieties with a pseudoreflection is given by intermediate Jacobians of cubic threefolds that possess an Eckardt point.

math.AG↗

arXiv2024

Counting polarizations on abelian varieties with group action

Let $\mathcal{A}_g$ be the moduli space of principally polarized abelian varieties. We study the problem of counting the number of principal polarizations modulo the natural action of the automorphism group of the abelian variety on a very general element of a positive dimensional component of $\mathrm{Sing}(\mathcal{A}_g)$, and show that this number is not always 1.

math.AG↗

arXiv2024

On the Jacobian variety of the Accola-Maclachlan curve of genus four

In this short note, we study the Jacobian variety of the Accola-Maclachlan curve of genus four and obtain explicitly its Poincar\'e isogeny decomposition. More precisely, we show that its Jacobian variety is isomorphic to the product of two abelian surfaces that are simple, and provide explicitly a Riemann matrix for each one of the involved abelian surfaces

math.AG↗

arXiv2023

Galois subspaces for projective varieties

Given an embedding of a projective variety into projective space, we study the structure of the space of all linear projections that, when composed with the embedding, give a Galois morphism from the variety to a projective space of the same dimension.

math.AG↗

arXiv2022

Non-simple polarised abelian surfaces and genus 3 curves with completely decomposable Jacobians

We study the space of non-simple polarised abelian surfaces. Specifically, we describe for which pairs $(m,n)$ the locus of polarised abelian surfaces of type $(1,d)$ that contain two complementary elliptic curve of exponents $m,n$, denoted $\mathcal{E}_d(m,n)$ is non-empty. We show that if $d$ is square-free, the locus $\mathcal{E}_d(m,n)$ is an irreducible surface (if non-empty). We also show that the loci $\mathcal{E}_d(d,d)$ can have many components if $d$ is an odd square. As an application, we show that for a genus $3$ curve with a completely decomposable Jacobian (i.e. isogenous to a product of 3 elliptic curves) the degrees of complementary coverings $f_i:C\rightarrow E_i,\ i=1,2,3$ satisfy $lcm(deg(f_1),deg(f_2))=lcm(deg(f_1),deg(f_3))=lcm(deg(f_2),deg(f_3))$.

math.AG↗

arXiv2021

Smooth quotients of abelian surfaces by finite groups that fix the origin

Let $A$ be an abelian surface and let $G$ be a finite group of automorphisms of $A$ fixing the origin. Assume that the analytic representation of $G$ is irreducible. We give a classification of the pairs $(A,G)$ such that the quotient $A/G$ is smooth. In particular, we prove that $A=E^2$ with $E$ an elliptic curve and that $A/G\simeq\mathbb{P}^2$ in all cases. Moreover, for fixed $E$, there are only finitely many pairs $(E^2,G)$ up to isomorphism. This fills a small gap in the literature and completes the classification of smooth quotients of abelian varieties by finite groups fixing the origin started by the first two authors.

math.AG↗

arXiv2021

Smooth quotients of principally polarized abelian varieties

We give an explicit characterization of all principally polarized abelian varieties $(A,Θ)$ such that there is a finite subgroup of automorphisms $G$ of $A$ that preserve the numerical class of $Θ$, and such that the quotient variety $A/G$ is smooth. We also give a complete classification of smooth quotients of Jacobians of curves.

math.AG↗

arXiv2021

Smooth quotients of complex tori by finite groups (with an appendix by Stephen Griffeth)

Let $A$ be a complex torus and $G$ a finite group acting on $A$ without translations such that $A/G$ is smooth. Consider the subgroup $F\leq G$ generated by elements that have at least one fixed point. We prove that there exists a point $x\in A$ fixed by the whole group $F$ and that the quotient $A/G$ is a fibration of products of projective spaces over an étale quotient of a complex torus (the étale quotient being Galois with group $G/F$). In particular, when $G=F$, we may assume that $G$ fixes the origin. This is related to previous work by the authors, where the case of actions on abelian varieties fixing the origin was treated. Here, we generalize these results to complex tori and use them to reduce the problem of classifying smooth quotients of complex tori to the case of étale quotients. An ingredient of the proof of our fixed-point theorem is a result proving that in every irreducible complex reflection group there is an element which is not contained in any proper reflection subgroup and that Coxeter elements have this property for well-generated groups. This result is proved by Stephen Griffeth in an appendix.

math.AG↗

arXiv2020

Galois subspaces for smooth projective curves

Given an embedding of a smooth projective curve $X$ of genus $g\geq1$ into $\mathbb{P}^N$, we study the locus of linear subspaces of $\mathbb{P}^N$ of codimension 2 such that projection from said subspace, composed with the embedding, gives a Galois morphism $X\to\mathbb{P}^1$. For genus $g\geq2$ we prove that this locus is a smooth projective variety with components isomorphic to projective spaces. If $g=1$ and the embedding is given by a complete linear system, we prove that this locus is also a smooth projective variety whose positive-dimensional components are isomorphic to projective bundles over étale quotients of the elliptic curve, and we describe these components explicitly.

math.AG↗

arXiv2020

A decomposition of the Jacobian of a Humbert-Edge curve

A \textit{Humbert-Edge curve of type} $n$ is a non-degenerate smooth complete intersection of $n-1$ diagonal quadrics. Such a curve has an interesting geometry since it has a natural action of the group $(\mathbb{Z}/2\mathbb{Z})^n$. We present here a decomposition of its Jacobian variety as a product of Prym-Tyurin varieties, and we compute the kernel of the corresponding isogeny.

math.AG↗

arXiv2020

Theta divisors whose Gauss map has a fiber of positive dimension

We construct families of principally polarized abelian varieties whose theta divisor is irreducible and contains an abelian subvariety. These families are used to construct examples when the Gauss map of the theta divisor is only generically finite and not finite. That is, the Gauss map in these cases has at least one positive-dimensional fiber. We also obtain lower-bounds on the dimension of Andreotti-Mayer loci.

math.AG↗

arXiv2019

The Gauss map and secants of the Kummer variety

Fay's trisecant formula shows that the Kummer variety of the Jacobian of a smooth projective curve has a four dimensional family of trisecant lines. We study when these lines intersect the theta divisor of the Jacobian, and prove that the Gauss map of the theta divisor is constant on these points of intersection, when defined. We investigate the relation between the Gauss map and multisecant planes of the Kummer variety as well.

math.AG↗

arXiv2018

Smooth quotients of abelian varieties by finite groups

We give a complete classification of smooth quotients of abelian varieties by finite groups that fix the origin. In the particular case where the action of the group $G$ on the tangent space at the origin of the abelian variety $A$ is irreducible, we prove that $A$ is isomorphic to the self-product of an elliptic curve and $A/G\simeq \mathbb P^n$. In the general case, assuming $\dim(A^G)=0$, we prove that $A/G$ is isomorphic to a direct product of projective spaces.

math.AG↗

arXiv2018

Galois subspaces for the rational normal curve

We characterize all $(n-2)$-dimensional linear subspaces of $\mathbb{P}^{n}$ such that the induced linear projection, when restricted to the rational normal curve, gives a Galois morphism. We give an explicit description of these spaces as a disjoint union of locally closed subvarieties in the Grassmannian $\mathbb{G}(n-2,n)$.

math.AG↗

arXiv2017

Fixed points of endomorphisms of complex tori

We study the asymptotic behavior of the cardinality of the fixed point set of iterates of an endomorphism of a complex torus. We show that there are precisely three types of behavior of this function: it is either an exponentially growing function, a periodic function, or a product of both.

math.AG↗