Vanishing thetanull and hyperelliptic curves

Let ??_g,2 be the moduli space of curves of genus g with a level‐2 structure. We prove here that there is always a non hyperelliptic element in the intersection of four thetanull divisors in ??_6,2. We prove also that for all g ≥ 3, each component of the hyperelliptic locus in ??_g,2 is a connected component of the intersection of g – 2 thetanull divisors. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Elliptic subcovers of hyperelliptic curves

The main result of this paper states that if C is a hyperelliptic curve of even genus over an arbitrary field K , then there is a natural bijection between the set of equivalence classes of elliptic subcovers of and the set of elliptic subgroups of its Jacobian .     

Compactifying moduli of hyperelliptic curves

We construct a new compactification of the moduli spaceH _g of smooth hyperelliptic curves of genusg. We compare our compactification with other well-known remarkable compactifications ofH _g. The author was partially supported byCNP _q, Proc. 151610/2005-3, and by Faperj, Proc. E-26/152-629/2005.     

Higher syzygies of hyperelliptic curves

Let X be a hyperelliptic curve of arithmetic genus g and let f:X→P¹ be the hyperelliptic involution map of X. In this paper we study higher syzygies of linearly normal embeddings of X of degree d≤2g. Note that the minimal free resolution of X of degree ≥2g+1 is already completely known. Let A=f^∗O_P¹(1), and let L be a very ample line bundle on X of degree d≤2g. For , we call the pair (m,d−2m)the factorization type ofL. Our main result is that the Hartshorne-Rao module and the graded Betti numbers of the linearly normal curve embedded by |L| are precisely determined by the factorization type of L.     

Asymmetric and pseudo-symmetric hyperelliptic surfaces

 In this paper we examine hyperelliptic Riemann surfaces which possess an anticonformal automorphism but are not symmetric. We determine that all such surfaces must have a full automorphism group which is either cyclic, or an abelian extension of a cyclic group by ℤ₂. We give defining equations for all of these hyperelliptic surfaces and show how they can be constructed by using NEC groups. As a special case, we determine all hyperelliptic surfaces which are pseudo-symmetric but not symmetric. Received: 15 November 2001     

Fundamental S-units in hyperelliptic fields and the torsion problem in Jacobians of hyperelliptic curves

In 2010, Platonov proposed a fundamentally new approach to the torsion problem in Jacobi varieties of hyperelliptic curves over the field of rational numbers. This new approach is based on the calculation of fundamental units in hyperelliptic fields. It was applied to prove the existence of torsion points of new orders. In the paper, the notion of the degree of an S-unit is introduced and a relationship between the degree of an S-unit and the order of the corresponding torsion point of the Jacobian of a hyperelliptic curve is established. A complete exposition of the new method and results obtained on the basis of this method is contained in [2].     

On double coverings of hyperelliptic curves

We prove various properties of varieties of special linear systems on double coverings of hyperelliptic curves. We show and determine the irreducibility, generically reducedness and singular loci of the variety for bi-elliptic curves and double coverings of genus two curves. Similar results for double coverings of hyperelliptic curves of genus h≥3 are also presented.     

2-Weierstrass points of genus 3 hyperelliptic curves with extra involutions

We consider families of curves with extra automorphisms in ?₃, the moduli space of smooth hyperelliptic curves of genus g = 3. Such families of curves are explicitly determined in terms of the absolute invariants of binary octavics. For each family of positive dimension where |Aut (C)|>4, we determine the possible distributions of weights of 2-Weierstrass points.     

Towers of 2-covers of hyperelliptic curves

In this article, we give a way of constructing an unramified Galois-cover of a hyperelliptic curve. The geometric Galois-group is an elementary abelian -group. The construction does not make use of the embedding of the curve in its Jacobian, and it readily displays all subcovers. We show that the cover we construct is isomorphic to the pullback along the multiplication-by- map of an embedding of the curve in its Jacobian.

We show that the constructed cover has an abundance of elliptic and hyperelliptic subcovers. This makes this cover especially suited for covering techniques employed for determining the rational points on curves. In particular the hyperelliptic subcovers give a chance for applying the method iteratively, thus creating towers of elementary abelian 2-covers of hyperelliptic curves.

As an application, we determine the rational points on the genus curve arising from the question of whether the sum of the first fourth powers can ever be a square. For this curve, a simple covering step fails, but a second step succeeds.