We describe the behavior of a free reduced plane projective curve with respect to the addition, respectively, deletion, of a smooth conic. These results apply in particular to conic-line arrangements. We present some obstructions to the geometry and combinatorics of a free reduced curve, generalizing results known a priori only for free projective line arrangements.

In the 17th century, a letter from Fermat to Mersenne considered the problem of finding the Pythagorean triples in which the hypotenuse and the sum of the other two sides are squares. We call this question the Fermat–Mersenne problem. Number-theoretic solutions of the problem are known, but they are complicated. We solve the Fermat–Mersenne problem by using a geometric approach that provides new insight. The argument involves stereographically projecting points onto an elliptic curve and applying complex multiplication on that curve.

Multiscale differentials arise as limits of holomorphic differentials with prescribed zero orders on nodal curves. In this paper, we address the conjecture concerning Gorenstein contractions of multiscale differentials, originally proposed by Ranganathan and Wise and further developed by Battistella and Bozlee. Specifically, in the case of a one-parameter degeneration, we show that multiscale differentials can be contracted to Gorenstein singularities, level by level, from the top down. At each level, these differentials descend to generators of the dualizing bundle at the resulting singularities. Moreover, the global residue condition, which governs the smoothability of multiscale differentials, appears as a special case of the residue condition for descent differentials.

For $r\geq 3$ and $g= \frac {r(r+1)}{2}$, we study the Prym-Brill-Noether variety $V^r(C,\eta )$ associated to Prym curves $[C,\eta ]$. The locus $\mathcal {R}_g^r$ in $\mathcal {R}_g$ parametrizing Prym curves $(C, \eta )$ with nonempty $V^r(C,\eta )$ is a divisor. We compute some key coefficients of the class $[\overline {\mathcal {R}}_g^r]$ in $\mathrm {Pic}_{\mathbb {Q}}(\overline {\mathcal {R}}_g)$. Furthermore, we examine a strongly Brill-Noether divisor in $\overline {\mathcal {M}}_{g-1,2}$: we show its irreducibility and compute some of its coefficients in $\mathrm {Pic}_{\mathbb {Q}}(\overline {\mathcal {M}}_{g-1,2})$. As a consequence of our results, the moduli space $\mathcal {R}_{14,2}$ is of general type.

Let $f \in \mathbb{Q}[x]$ be a square-free polynomial of degree at least $3$, $m_i$, $i=1,2,3$, odd positive integers, and $a_i$, $i=1,2,3$, non-zero rational numbers. We show the existence of a rational function $D\in \mathbb{Q}(v_1,v_2,v_3,v_4)$ such that the Jacobian of the quadratic twist of $y^2=f(x)$ and the Jacobian of the $m_i$-twist, respectively, $2m_i$-twist, of $y^2=x^{m_i}+a_i^2$, $i=1,2,3$, by $D$ are all of positive Mordell–Weil ranks. As an application, we present families of hyperelliptic curves with large Mordell–Weil rank.

We determine the convergence regions of certain local integrals on the moduli spaces of curves in neighborhoods of fixed stable curves with rational components in terms of the combinatorics of the corresponding graphs.

Under the assumption that the adjusted Brill-Noether number $\widetilde {\rho }$ is at least $-g$, we prove that the Brill-Noether loci in ${\mathcal M}_{g,n}$ of pointed curves carrying pencils with prescribed ramification at the marked points have a component of the expected codimension with pointed curves having Brill-Noether varieties of pencils of the minimal dimension. As an application, the map from the Hurwitz scheme to ${\mathcal M}_g$ is dominant if $n+\widetilde {\rho } \geq 0$ and generically finite otherwise, settling a variation of a classical problem of Zariski.

In the second part of the paper, we study the analogous loci of curves in Severi varieties on $K3$ surfaces, proving existence of curves with nongeneral behaviour from the point of view of Brill-Noether theory. This extends previous results of Ciliberto and the first-named author to the ramified case. We apply these results to study correspondences and cycles on $K3$ surfaces in relation to Beauville-Voisin points and constant cycle curves.

We study the relationship between the enumerative geometry of rational curves in local geometries and various versions of maximal contact logarithmic curve counts. Our approach is via quasimap theory, and we show versions of the [vGGR19] local/logarithmic correspondence for quasimaps, and in particular for normal crossings settings, where the Gromov-Witten theoretic formulation of the correspondence fails. The results suggest a link between different formulations of relative Gromov-Witten theory for simple normal crossings divisors via the mirror map. The main results follow from a rank reduction strategy, together with a new degeneration formula for quasimaps.

Given a holomorphic Lie algebroid on an m-pointed connected Riemann surface, we define parabolic Lie algebroid connections on any parabolic vector bundle equipped with parabolic structure over the marked points. An analog of the Atiyah exact sequence for parabolic Lie algebroids is constructed. For any Lie algebroid whose underlying holomorphic vector bundle is stable, we give a complete characterization of all the parabolic vector bundles that admit a parabolic Lie algebroid connection.

We compute the integral Chow rings of $\overline {\mathcal {M}}_{1,n}$ for $n=3,4$. The alternative compactifications introduced by Smyth – and studied further by Lekili and Polishchuk – present each of these stacks as a sequence of weighted blow-ups and blow-downs from a weighted projective space. We compute all the integral Chow rings by repeated application of the blow-up formula.

We obtain explicit expressions for the class in the Grothendieck group of varieties of the moduli space $\overline {\mathcal{M}}_{0,n}$. This information is equivalent to the Poincaré polynomial and yields explicit expressions for the Betti numbers of $\overline {\mathcal{M}}_{0,n}$ in terms of Stirling or Bernoulli numbers. The expressions are obtained by solving a differential equation characterizing the generating function for the Poincaré polynomials, determined by Manin in the 1990s and equivalent to Keel’s recursion for the Betti numbers of $\overline {\mathcal{M}}_{0,n}$. Our proof reduces the solution to two combinatorial identities, verified by applying Lagrange series. We also study generating functions for the individual Betti numbers. These functions are determined by a set of polynomials $p^{(k)}_m(z)$, $k\geqslant m$. These polynomials are conjecturally log-concave; we verify this conjecture for several infinite families, corresponding to generating functions for $2k$-Betti numbers of $\overline {\mathcal{M}}_{0,n}$ for all $k\leqslant 100$. Further, studying the polynomials $p^{(k)}_m(z)$, we prove that the generating function for the Grothendieck class can be written in terms of a series of rational functions in the principal branch of the Lambert W-function. We include an interpretation of the main result in terms of Stirling matrices and a discussion of the Euler characteristic of $\overline {\mathcal{M}}_{0,n}$.

We define the type of a plane curve as the initial degree of the corresponding Bourbaki ideal. Then, we show that this invariant behaves well with respect to the union of curves. Curves of type $0$ are precisely the free curves, while curves of type $1$ are the plus-one generated curves. In this article, we first show that line arrangements and conic-line arrangements can exhibit all the theoretically possible types. In the second part, we study the properties of the curves of type $2$ and construct families of line arrangements and conic-line arrangements of this type.

We pose some conjectures about the existence of certain complete, one-dimensional families of degree $d$, genus $g$ branched covers of an elliptic curve. The conjectures would imply that the slope of the corresponding Hurwitz space is precisely $5+6/d$, and that the slope of the moduli space of stable genus $g$ curves is bounded below by $5$. We provide evidence for the conjectures when $g=2$ or when $d \leq 5$ and $g \gg 0$.

We prove cohomological vanishing criteria for the Ceresa cycle of a curve C embedded in its Jacobian J: (A) if $\mathrm{H}^3(J)^{\mathrm{Aut}(C)} = 0$, then the Ceresa cycle is torsion modulo rational equivalence; (B) if $\mathrm{H}^0(J, \Omega_J^3)^{\mathrm{Aut}(C)} = 0$, then the Ceresa cycle is torsion modulo algebraic equivalence, with criterion (B) conditional on the Hodge conjecture. We then use these criteria to study the simplest family of curves where (B) holds but (A) does not, namely the family of Picard curves $C \colon y^3 = x^4 + ax^2 + bx + c$. Criterion (B) and work of Schoen combine to show that the Ceresa cycle of a Picard curve is torsion in the Griffiths group. We furthermore determine exactly when it is torsion in the Chow group. As a byproduct, we deduce that there exist one-parameter families of plane quartic curves with torsion Ceresa Chow class; that the torsion locus in $\mathcal{M}_3$ of the Ceresa Chow class contains infinitely many components; and that the order of a torsion Ceresa Chow class of a Picard curve over a number field K is bounded, with the bound depending only on $[K\colon \mathbb{Q}]$. Finally, we determine which automorphism group strata are contained in the vanishing locus of the universal Ceresa cycle over $\mathcal{M}_3$.

We consider the question of when a Jacobian of a curve of genus $2g$ admits a $(2,2)$-isogeny to a product of two polarized dimension g abelian varieties. We find that one of them must be a Jacobian itself and, if the associated curve is hyperelliptic, so is the other.

For $g=2,$ this allows us to describe $(2,2)$-decomposable genus $4$ Jacobians in terms of Prym varieties. We describe the locus of such genus $4$ curves in terms of the geometry of the Igusa quartic threefold. We also explain how our characterization relates to Prym varieties of unramified double covers of plane quartic curves, and we describe this Prym map in terms of $6$ and $7$ points in $\mathbb {P}^3$.

We also investigate which genus $4$ Jacobians admit a $2$-isogeny to the square of a genus $2$ Jacobian and give a full description of the hyperelliptic ones. While most of the families we find are of the expected dimension $1$, we also find a family of unexpectedly high dimension $2$.

In this paper we show that the rank of the normal function function of the genus $g$ Ceresa cycle over the moduli space of curves has the maximal rank possible, $3g-3$ , provided that $g\ge 3$. In genus 3 we show that the Green–Griffiths invariant of this normal function is a Teichmüller modular form of weight $(4,0,-1)$ and use this to show that the rank of the Ceresa normal function is exactly 1 along the hyperelliptic locus. We also introduce new techniques and tools for studying the behaviour of normal functions along and transverse to boundary divisors. These include the introduction of residual normal functions and the use of global monodromy arguments to compute them.

We study universal-existential fragments of first-order theories of fields, in particular of function fields and of equicharacteristic henselian valued fields. For example, we discuss to what extent the theory of a field k determines the universal-existential theories of the rational function field over k and of the field of Laurent series over k, and we find various many-one reductions between such fragments.

Dedicated to the memory of Alexander Prestel (1941–2024)

We show that the cohomological Brauer groups of the moduli stacks of stable genus g curves over the integers and an algebraic closure of the rational numbers vanish for any $g\geq 2$. For the n marked version, we show the same vanishing result in the range $(g,n)=(1,n)$ with $1\leq n \leq 6$ and all $(g,n)$ with $g\geq 4.$ We also discuss several finiteness results on cohomological Brauer groups of proper and smooth Deligne-Mumford stacks over the integers.

A Pell–Abel equation is a functional equation of the form $P^{2}-DQ^{2} = 1$, with a given polynomial $D$ free of squares and unknown polynomials $P$ and $Q$. We show that the space of Pell–Abel equations with the degrees of $D$ and of the primitive solution $P$ fixed is a complex manifold. We describe its connected components by an efficiently computable invariant. Moreover, we give various applications of this result, including to torsion pairs on hyperelliptic curves and to Hurwitz spaces, and a description of the connected components of the space of primitive $k$-differentials with a unique zero on genus $2$ Riemann surfaces.

We compute the $\ell$-primary torsion of the Brauer group of the moduli stack of smooth curves of genus three over any field of characteristic different from two and the Brauer group of the moduli stacks of smooth plane curves of degree d over any algebraically closed field of characteristic different from two, three and coprime to d. We achieve this result by computing the low-degree cohomological invariants of these stacks. As a corollary, we are additionally able to compute the $\ell$-primary torsion of the Brauer group of the moduli stack of principally polarized abelian varieties of dimension three over any field of characteristic different from two.