arXiv subjects
Publications and source records attributed to Isabel Vogt.
The first goal of this article is to survey recent progress in Brill--Noether theory, including both the study of the moduli space of maps from a curve to projective space and the geometry of the resulting curves in projective space. The second goal is to introduce newcomers to some of the important techniques that have been introduced or developed in the last decade that made these advances possible.
In this paper, we survey recent developments concerning the stability of naturally defined bundles on curves that play a central role in the deformation theory of the curve.
Zariski dense collections of quadratic points on curves $X$ are well-understood by results of Harris--Silverman and Vojta, but when $\dim X \geq 2$ there is not an analogous geometric characterization, even conjecturally. In this note we consider the case of a double cover $\pi \colon X \to \mathbb{P}^r$, where Hilbert's Irreducibility Theorem implies that the quadratic points in the fibers of $\pi$ are dense. We show that Vojta's Conjecture implies that, once the canonical bundle of $X$ is sufficiently positive, there are no other sources of Zariski dense quadratic points. This is complemented by several examples of surfaces $X \to \mathbb{P}^2$ with an additional source of dense quadratic points.
In this paper, we initiate the systematic study of density of algebraic points on surfaces. We give an effective asymptotic range in which the density degree set has regular behavior dictated by the index. By contrast, in small degree, the question of density is subtle and depends on the arithmetic of the curves. We give several explicit examples displaying these different behaviors, including products of genus $2$ curves with and without dense quadratic points. These results for products of curves have applications to questions about algebraic points on closely related surfaces, such as rank growth on abelian surfaces and bielliptic surfaces.
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 show that parameterized points of very low degree arise for a unique geometric reason. The paper includes several examples that illustrate the possible behaviors of degree d points.
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; however, over nonclosed fields, including $\mathbb{R}$, 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 encodes the obstruction to the existence of a section of the first projection $Y\to\mathbb{P}^1$.
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.
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.
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.
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 $g(Y) \geq 2$.
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 $k$.
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.
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.