Curves Of Low Genus Research Articles

The dimension of the moduli space of pointed algebraic curves of low genus

We explicitly compute the moduli space pointed algebraic curves with a given numerical semigroup as Weierstrass semigroup for many cases of genus at most seven and when possible describe the structure as cone over an explicitly given variety. We determine the dimension of the moduli space for all semigroups of genus seven.

Triangular modular curves of small genus

Triangular modular curves are a generalization of modular curves that arise from quotients of the upper half-plane by congruence subgroups of hyperbolic triangle groups. These curves also arise naturally as a source of Belyi maps with monodromy text {PGL}_2(mathbb {F}_q) or text {PSL}_2(mathbb {F}_q). We present a computational approach to enumerate Borel-type triangular modular curves of low genus, and we carry out this enumeration for prime level and small genus.

Machine-learning the Sato–Tate conjecture

We apply some of the latest techniques from machine-learning to the arithmetic of hyperelliptic curves. More precisely we show that, with impressive accuracy and confidence (between 99 and 100 percent precision), and in very short time (matter of seconds on an ordinary laptop), a Bayesian classifier can distinguish between Sato–Tate groups given a small number of Euler factors for the L-function. Our observations are in keeping with the Sato-Tate conjecture for curves of low genus. For elliptic curves, this amounts to distinguishing generic curves (with Sato–Tate group SU(2)) from those with complex multiplication. In genus 2, a principal component analysis is observed to separate the generic Sato–Tate group USp(4) from the non-generic groups. Furthermore in this case, for which there are many more non-generic possibilities than in the case of elliptic curves, we demonstrate an accurate characterisation of several Sato–Tate groups with the same identity component. Throughout, our observations are verified using known results from the literature and the data available in the LMFDB. The results in this paper suggest that a machine can be trained to learn the Sato–Tate distributions and may be able to classify curves efficiently.

Algebraic geometry codes over abelian surfaces containing no absolutely irreducible curves of low genus

We provide a theoretical study of Algebraic Geometry codes constructed from abelian surfaces defined over finite fields. We give a general bound on their minimum distance and we investigate how this estimation can be sharpened under the assumption that the abelian surface does not contain low genus curves. This approach naturally leads us to consider Weil restrictions of elliptic curves and abelian surfaces which do not admit a principal polarization.

Picard group of moduli of curves of low genus in positive characteristic

We compute the Picard group of the moduli stack of smooth curves of genus g for \(3\le g\le 5\), using methods of equivariant intersection theory. We base our proof on the computation of some relations in the integral Chow ring of certain moduli stacks of smooth complete intersections. As a byproduct, we compute the cycle classes of some divisors on \(\mathcal {M}_g\).

Computing unit groups of curves

The group of units modulo constants of an affine variety over an algebraically closed field is free abelian of finite rank. Computing this group is difficult but of fundamental importance in tropical geometry, where it is necessary in order to realize intrinsic tropicalizations. We present practical algorithms for computing unit groups of smooth curves of low genus. Our approach is rooted in divisor theory, based on interpolation in the case of rational curves and on methods from algebraic number theory in the case of elliptic curves.

Lower degree curves in X3,3 ⊂ ℙ2 × ℙ2

In this paper we study low-degree low-genus curves in a generic hypersurface X of degree (3, 3) in ℙ2 × ℙ2. We prove that the genus 0 and genus 1 curves of degree up to (2, 2) are smooth and rigid. We then use the multiple cover formula to compute the Gromov-Witten invariants of X of degree up to (2, 2) and genus up to 2. This provides some initial conditions to determine the full genus 1 and genus 2 Gromov-Witten invariants via Bershadsky-Cecotti-Ooguri-Vafa’s Feynman rule, which is expected to be proved in the near future.

Logarithmic Co-Higgs Bundles

In this article, we introduce a notion of logarithmic co-Higgs sheaves associated with a simple normal crossing divisor on a projective manifold and show their existence with nilpotent co-Higgs fields for fixed ranks and second Chern classes. Then, we deal with various moduli problems involving logarithmic co-Higgs sheaves, such as coherent systems and holomorphic triples, specially over algebraic curves of low genus.

Twists of Mukai bundles and the geometry of the level $3$ modular variety over $\overline {\mathcal {M}}_{8}$

For a curve $C$ of genus $6$ or $8$ and a torsion bundle $\eta$ of order $\ell$ we study the vanishing of the space of global sections of the twist $E_C \otimes \eta$ of the rank two Mukai bundle $E_C$ of $C$. The bundle $E_C$ was used in a well-known construction of Mukai which exhibits general canonical curves of low genus as sections of Grassmannians in the Plucker embedding. Globalizing the vanishing condition, we obtain divisors on the moduli spaces $\overline{\mathcal{R}}_{6,\ell}$ and $\overline{\mathcal{R}}_{8,\ell}$ of pairs $[C, \eta]$. First we characterize these divisors by different conditions on linear series on the level curves, afterwards we calculate the divisor classes. As an application, we are able to prove that $\overline{\mathcal{R}}_{8,3}$ is of general type.

Quadratic sequences of powers and Mohanty’s conjecture

We prove under the Bombieri–Lang conjecture for surfaces that there is an absolute bound on the length of sequences of integer squares with constant second differences, for sequences which are not formed by the squares of integers in arithmetic progression. This answers a question proposed in 2010 by Browkin and Brzezinski, and independently by Gonzalez-Jimenez and Xarles. We also show that under the Bombieri–Lang conjecture for surfaces, for every [Formula: see text] there is an absolute bound on the length of sequences formed by [Formula: see text]th powers with constant second differences. This gives a conditional result on one of Mohanty’s conjectures on arithmetic progressions in Mordell’s elliptic curves [Formula: see text]. Moreover, we obtain an unconditional result regarding infinite families of such arithmetic progressions. We also study the case of hyperelliptic curves of the form [Formula: see text]. These results are proved by unconditionally finding all curves of genus zero or one on certain surfaces of general type. Moreover, we prove the unconditional analogues of these arithmetic results for function fields by finding all the curves of low genus on these surfaces.

On the Moduli Space of Pointed Algebraic Curves of Low Genus III ---Positive Characteristic---

In his classical work, Pinkham discovered a beautiful theorem on the moduli space of pointed algebraic curves with a fixed Weierstrass gap sequence at the marked point. Namely, the complement of a Weierstrass gap sequence in the set of non-negative integers is a numerical semigroup, and he described such a moduli space in terms of the negative part of the miniversal deformation space of the monomial curve of this semigroup. Unfortunately, his theorem holds only in characteristic 0 and does not hold in positive characteristic in general. In this paper, we will study his theorem in positive characteristic, and give a fairly sharp condition for his theorem to hold in positive characteristic up to genus 4. As an application, we present a complete analysis of his theorem in positive characteristic in the low genus case.

Rational self-maps of moduli spaces

We discuss some examples of geometrically meaningful rational self-maps of moduli space of curves of low genus and homogeneous forms.

Optimal curves of low genus over finite fields

We investigate maximal and minimal curves of genus 4 and 5 over finite fields with discriminant −11 and −19. As a result the Hasse–Weil–Serre bound is improved.

Counting curves over finite fields

This is a survey on recent results on counting of curves over finite fields. It reviews various results on the maximum number of points on a curve of genus g over a finite field of cardinality q, but the main emphasis is on results on the Euler characteristic of the cohomology of local systems on moduli spaces of curves of low genus and its implications for modular forms.

An improvement of the Hasse–Weil–Serre bound for curves over some finite fields

The Hasse–Weil–Serre bound is improved for low genus curves over finite fields with discriminant from {−3,−4,−7,−8} by studying maximal and minimal curves.

On the Koszul cohomology of canonical and Prym-canonical binary curves

In this paper we study Koszul cohomology and the Green and Prym–Green conjectures for canonical and Prym-canonical binary curves. We prove that, if property Np holds for a canonical or a Prym-canonical binary curve of genus g, then it holds for a generic canonical or Prym-canonical binary curve of genus g + 1. We also verify the Green and Prym–Green conjectures for generic canonical and Prym-canonical binary curves of low genus (6 ⩽ g ⩽ 15, g ≠ 8 for Prym-canonical and 3 ⩽ g ⩽ 12 for canonical).

First steps towards radical parametrization of algebraic surfaces

We introduce the notion of radical parametrization of a surface, and we provide algorithms to compute such type of parametrizations for families of surfaces, like: Fermat surfaces, surfaces with a high multiplicity (at least the degree minus 4) singularity, all irreducible surfaces of degree at most 5, all irreducible singular surfaces of degree 6, and surfaces containing a pencil of low-genus curves. In addition, we prove that radical parametrizations are preserved under certain type of geometric constructions that include offset and conchoids.

Some remarks on superspecial and ordinary curves of low genus

AbstractIn the first half of this paper, we investigate a superspecialty criterion with respect to $\lambda$ for the curve which is a generalization of a theorem for an elliptic curve by Max Deuring (1941). In the last half, we study a criterion with respect to the characteristic $p>0$ of the ground field such that the Picard curve is superspecial or ordinary. In the final section (appendix), we consider another viewpoint of the first result by using a fact that the first curve has a completely decomposable Jacobian.

Fano surfaces with 12 or 30 elliptic curves

Curves of low genus on a surface carry important informations on that surface. We study the Fano surfaces of lines of cubic threefolds that contain 12 or 30 elliptic curves. We determine their Picard number and compute a basis of the N\'eron-Severi group of the Fano surface of the Fermat cubic threefold.

Addition formulae for Abelian functions associated with specialized curves

We discuss a family of multi-term addition formulae for Weierstrass functions on specialized curves of low genus with many automorphisms, concentrating mostly on the case of genus 1 and 2. In the genus 1 case, we give addition formulae for the equianharmonic and lemniscate cases, and in genus 2 we find some new addition formulae for a number of curves.