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We give a self-contained introduction to isolated points on curves and their counterpoint, parameterized points, that situates these concepts within the study of the arithmetic of curves. In particular, we show how natural geometric constructions of infinitely many degree $d$ points on curves motivate the definitions of $\mathbb{P}^1$- and AV-parameterized points and explain how a result of Faltings implies that there are only finitely many isolated points on any curve. We use parameterized points to deduce properties of the density degree set and review how the minimum density degree relates to the gonality. The paper includes several examples that illustrate the possible behaviors of degree $d$ points., Comment: 26 pages

A double cover $Y$ of $\mathbb{P}^1 \times \mathbb{P}^2$ ramified over a general $(2,2)$-divisor will have the structure of a geometrically standard conic bundle ramified over a smooth plane quartic $\Delta \subset \mathbb{P}^2$ via the second projection. These threefolds are rational over algebraically closed fields, but over nonclosed fields, including over $\mathbb{R}$, their rationality is an open problem. In this paper, we characterize rationality over $\mathbb{R}$ when $\Delta(\mathbb{R})$ has at least two connected components (extending work of M. Ji and the second author) and over local fields when all odd degree fibers of the first projection have nonsquare discriminant. We obtain these applications by proving general results comparing the conic bundle structure on $Y$ with the conic bundle structure on a well-chosen intersection of two quadrics. The difference between these two conic bundles is encoded by a constant Brauer class, and we prove that this class measures a certain failure of Galois descent for the codimension 2 Chow group of $Y$., Comment: 18 pages

The interpolation problem is a natural and fundamental question whose roots trace back to ancient Greece. The story is long and rich, with many chapters, and a complete solution has been obtained only recently. Exploring it leads us on a tour through a number of general themes in geometry. This concrete problem motivates fundamental concepts such as moduli spaces and their properties, deformation theory, normal bundles, and more. Questions about smooth objects lead us to consider singular (non-smooth) objects, and in fact these smooth objects are studied by instead focusing on somehow simpler "non-smooth" objects, and then deforming them., Comment: Expository article, to appear in Bulletin of the AMS

In projective space over fields of characteristic different from 2, the normal bundle of a general nondegenerate rational curve is balanced. The corresponding statement for rational curves in other Grassmannians can fail. Nevertheless, we prove that the normal bundle of a general rational curve in a Grassmannian decomposes into a direct sum of line bundles whose degrees are at most 2 apart.

On a projective variety defined over a global field, any Brauer--Manin obstruction to the existence of rational points is captured by a finite subgroup of the Brauer group. We show that this subgroup can require arbitrarily many generators., Comment: 17 pages

Let $\alpha: X \to Y$ be a finite cover of smooth curves. Beauville conjectured that the pushforward of a general vector bundle under $\alpha$ is semistable if the genus of $Y$ is at least $1$ and stable if the genus of $Y$ is at least $2$. We prove this conjecture if the map $\alpha$ is general in any component of the Hurwitz space of covers of an arbitrary smooth curve $Y$.

We generalize the Embedding Theorem of Eisenbud-Harris from classical Brill-Noether theory to the setting of Hurwitz-Brill-Noether theory. More precisely, in classical Brill-Noether theory, the embedding theorem states that a general linear series of degree d and rank r on a general curve of genus g is an embedding if r is at least 3. If \(f \colon C \to \mathbb{P}^1\) is a general cover of degree k, and L is a line bundle on C, recent work of the authors shows that the splitting type of \(f_* L\) provides the appropriate generalization of the pair (r, d) in classical Brill--Noether theory. In the context of Hurwitz-Brill-Noether theory, the condition that r is at least 3 is no longer sufficient to guarantee that a general such linear series is an embedding. We show that the additional condition needed to guarantee that a general linear series |L| is an embedding is that the splitting type of \(f_* L\) has at least three nonnegative parts. This new extra condition reflects the unique geometry of k-gonal curves, which lie on scrolls in \(\mathbb{P}^r\).

For a genus $2$ curve $C$ over $\mathbb{Q}$ whose Jacobian $A$ admits only trivial geometric endomorphisms, Serre's open image theorem for abelian surfaces asserts that there are only finitely many primes $\ell$ for which the Galois action on $\ell$-torsion points of $A$ is not maximal. Building on work of Dieulefait, we give a practical algorithm to compute this finite set. The key inputs are Mitchell's classification of maximal subgroups of $\mathrm{PSp_4}(\mathbb{F}_\ell)$, sampling of the characteristic polynomials of Frobenius, and the Khare--Wintenberger modularity theorem. The algorithm has been submitted for integration into Sage, executed on all of the genus~$2$ curves with trivial endomorphism ring in the LMFDB, and the results incorporated into the homepage of each such curve.

We define an enrichment of the logarithmic derivative of the zeta function of a variety over a finite field to a power series with coefficients in the Grothendieck--Witt group. We show that this enrichment is related to the topology of the real points of a lift. For cellular schemes over a field, we prove a rationality result for this enriched logarithmic derivative of the zeta function as an analogue of part of the Weil conjectures. We also compute several examples, including toric varieties, and show that the enrichment is a motivic measure., Comment: 41 pages, to appear in Transactions of the AMS

This paper is devoted to understanding curves $X$ over a number field $k$ that possess infinitely many solutions in extensions of $k$ of degree at most $d$; such solutions are the titular low degree points. For $d=2,3$ it is known (by the work of Harris-Silverman and Abramovich-Harris) that such curves, after a base change to $\overline{k},$ admit a map of degree at most $d$ onto $\mathbb{P}^1$ or an elliptic curve. For $d \geqslant 4$ the analogous statement was shown to be false by Debarre and Fahlaoui. We prove that once the genus of $X$ is high enough, the low degree points still have geometric origin: they can be obtained as pullbacks of low degree points from a lower genus curve. We introduce a discrete-geometric invariant attached to such curves: a family of subspace configurations, with many interesting properties. This structure gives a natural alternative construction of curves with many low degree points, that were first discovered by Debarre and Fahlaoui. As an application of our methods, we obtain a classification of such curves over $k$ for $d=2,3$, and a classification over $\overline{k}$ for $d=4,5$.

Let $\alpha : X \to Y$ be a general degree $r$ primitive map of nonsingular, irreducible, projective curves over an algebraically closed field of characteristic zero or larger than $r$. We prove that the Tschirnhausen bundle of $\alpha$ is semistable if $g(Y) \geq 1$ and stable if $g(Y) \geq 2$., Comment: To appear in International Math Research Notices

We undertake a study of conic bundle threefolds $\pi\colon X\to W$ over geometrically rational surfaces whose associated discriminant covers $\tilde{\Delta}\to\Delta\subset W$ are smooth and geometrically irreducible. First, we determine the structure of the group $\mathrm{CH}^2 X_{\overline{k}}$ of rational equivalence classes of curves. Precisely, we construct a Galois-equivariant group homomorphism from $\mathrm{CH}^2X_{\overline{k}}$ to a group scheme associated to the discriminant cover $\tilde{\Delta}\to \Delta$ of $X$. The target group scheme is a generalization of the Prym variety of $\tilde{\Delta}\to\Delta$ and so our result can be viewed as a generalization of Beauville's result that the algebraically trivial curve classes on $X_{\overline{k}}$ are parametrized by the Prym variety. We apply our structural result on curve classes to study the refined intermediate Jacobian torsor (IJT) obstruction to rationality introduced by Hassett--Tschinkel and Benoist--Wittenberg. The first case of interest is $W = \mathbb P^2$ and $\Delta$ is a smooth plane quartic. In this case, we show that the IJT obstruction characterizes rationality when the ground field has less arithmetic complexity (precisely, when the $2$-torsion in the Brauer group of the ground field is trivial). We also show that a hypothesis of this form is necessary by constructing, over any $k \subset\mathbb R$, a conic bundle threefold with $\Delta$ a smooth quartic where the IJT obstruction vanishes, yet $X$ is irrational over $k$., Comment: 39 pages. Comments welcome! v2: Updated introduction. v3: Added Section 3, Subsection 5.5, and Example 8.6

Let $C$ be a general canonical curve of genus $g$ defined over an algebraically closed field of arbitrary characteristic. We prove that if $g \notin \{4,6\}$, then the normal bundle of $C$ is semistable. In particular, if $g \equiv 1$ or $3$ mod $6$, then the normal bundle is stable., Comment: To appear in Journal of the European Math Society

In this paper we determine the number of general points through which a Brill--Noether curve of fixed degree and genus in any projective space can be passed.

In this paper we investigate the $\mathbb{Q}$-rational points of a class of simply connected Calabi-Yau threefolds, which were originally studied by Hosono and Takagi in the context of mirror symmetry. These varieties are defined as a linear section of a double quintic symmetroid; their points correspond to rulings on quadric hypersurfaces. They come equipped with a natural $2$-torsion Brauer class. Our main result shows that under certain conditions, this Brauer class gives rise to a transcendental Brauer-Manin obstruction to weak approximation. Hosono and Takagi showed that over $\mathbb{C}$ each of these Calabi-Yau threefolds $Y$ is derived equivalent to a Reye congruence Calabi-Yau threefold $X$. We show that these derived equivalences may also be constructed over $\mathbb{Q}$, and we give sufficient conditions for $X$ to not satisfy weak approximation. In the appendix, N. Addington exhibits the Brauer groups of each class of Calabi--Yau variety over $\mathbb{C}$., Comment: minor edits

Let $C$ be a curve of genus $g$. A fundamental problem in the theory of algebraic curves is to understand maps $C \to \mathbb{P}^r$ of specified degree $d$. When $C$ is general, the moduli space of such maps is well-understood by the main theorems of Brill--Noether theory. Despite much study over the past three decades, a similarly complete picture has proved elusive for curves of fixed gonality. Here we complete such a picture, by proving analogs of all of the main theorems of Brill--Noether theory in this setting. As a corollary, we prove a conjecture of Eisenbud and Schreyer regarding versal deformation spaces of vector bundles on $\mathbb{P}^1$., Comment: Weakened characteristic hypotheses

In this paper, we prove that the normal bundle of a general Brill-Noether space curve of degree $d$ and genus $g \geq 2$ is stable if and only if $(d,g) \not\in \{ (5,2), (6,4) \}$. When $g\leq1$ and the characteristic of the ground field is zero, it is classical that the normal bundle is strictly semistable. We show that this fails in characteristic $2$ for all rational curves of even degree., Comment: 29 pages

Recent work of Kass--Wickelgren gives an enriched count of the $27$ lines on a smooth cubic surface over arbitrary fields. Their approach using $\mathbb{A}^1$-enumerative geometry suggests that other classical enumerative problems should have similar enrichments, when the answer is computed as the degree of the Euler class of a relatively orientable vector bundle. Here, we consider the closely related problem of the $28$ bitangents to a smooth plane quartic. However, it turns out the relevant vector bundle is not relatively orientable and new ideas are needed to produce enriched counts. We introduce a fixed "line at infinity," which leads to enriched counts of bitangents that depend on their geometry relative to the quartic and this distinguished line., Comment: 17 pages, comments welcome!

In this paper we investigate an arithmetic analogue of the gonality of a smooth projective curve $C$ over a number field $k$: the minimal $e$ such there are infinitely many points $P \in C(\bar{k})$ with $[k(P):k] \leq e$. Developing techniques that make use of an auxiliary smooth surface containing the curve, we show that this invariant can take any value subject to constraints imposed by the gonality. Building on work of Debarre--Klassen, we show that this invariant is equal to the gonality for all sufficiently ample curves on a surface $S$ with trivial irregularity., Comment: 15 pages. Final version, in IMRN (2022)

Let $E$ be an elliptic curve over a number field $K$. If for almost all primes of $K$, the reduction of $E$ modulo that prime has rational cyclic isogeny of fixed degree, we can ask if this forces $E$ to have a cyclic isogeny of that degree over $K$. Building upon the work of Sutherland, Anni, and Banwait-Cremona in the case of prime degree, we consider this question for cyclic isogenies of arbitrary degree., Comment: 30 pages

In this paper, we compute the number of general points through which a general Brill-Noether curve in $\mathbb{P}^4$ passes. We also prove an analogous theorem when some points are constrained to lie in a transverse hyperplane. As explained in arXiv:1809.05980, these results play an essential role in the first author's proof of the Maximal Rank Conjecture., Comment: 25 pages

Given an elliptic curve $E/k$ and a Galois extension $k'/k$, we construct an exact functor from torsion-free modules over the endomorphism ring ${\rm End}(E_{k'})$ with a semilinear ${\rm Gal}(k'/k)$ action to abelian varieties over $k$ that are $k'$-isogenous to a power of $E$. As an application, we show that every elliptic curve with complex multiplication geometrically is isogenous over the ground field to one with complex multiplication by a maximal order., Comment: 6 pages, added references

Inspired by the analogy between the group of units $\mathbb{F}_p^{\times}$ of the finite field with $p$ elements and the group of points $E(\mathbb{F}_p)$ of an elliptic curve $E/\mathbb{F}_p$, E. Kowalski, A. Akbary & D. Ghioca, and T. Freiberg & P. Kurlberg investigated the asymptotic behaviour of elliptic curve sums analogous to the Titchmarsh divisor sum $\sum_{p \leq x} \tau(p + a) \sim C x$. In this paper, we present a comprehensive study of the constants $C(E)$ emerging in the asymptotic study of these elliptic curve divisor sums. Specifically, by analyzing the division fields of an elliptic curve $E/\mathbb{Q}$, we prove upper bounds for the constants $C(E)$ and, in the generic case of a Serre curve, we prove explicit closed formulae for $C(E)$ amenable to concrete computations. Moreover, we compute the moments of the constants $C(E)$ over two-parameter families of elliptic curves $E/\mathbb{Q}$. Our methods and results complement recent studies of average constants occurring in other conjectures about reductions of elliptic curves by addressing not only the average behaviour, but also the individual behaviour of these constants, and by providing explicit tools towards the computational verifications of the expected asymptotics., Comment: 28 pages

We consider the K3 surfaces that arise as double covers of the elliptic modular surface of level 5, $R_{5,5}$. Such surfaces have a natural elliptic fibration induced by the fibration on $R_{5,5}$. Moreover, they admit several other elliptic fibrations. We describe such fibrations in terms of linear systems of curves on $R_{5,5}$. This has a major advantage over other methods of classification of elliptic fibrations, namely, a simple algorithm that has as input equations of linear systems of curves in the projective plane yields a Weierstrass equation for each elliptic fibration. We deal in detail with the cases for which the double cover is branched over the two reducible fibers of type $I_5$ and for which it is branched over two smooth fibers, giving a complete list of elliptic fibrations for these two scenarios., Comment: 32 pages

In this note we compute the number of general points through which a general Brill-Noether space curve passes., Comment: 8 pages; fixed some typos and added a remark on relation with semistability

Let $u_n$ be a nondegenerate Lucas sequence. We generalize the results of Bugeaud, Mignotte, and Siksek, 2006 to give a systematic approach towards the problem of determining all perfect powers in any particular Lucas sequence. We then prove a general bound on admissible prime powers in a Lucas sequence assuming the Frey-Mazur conjecture on isomorphic mod $p$ Galois representations of elliptic curves., Comment: 14 pages