arXiv subjects

Herbert Lange

Publications and source records attributed to Herbert Lange.

23 records · Page 2Linked to original sources

arXiv2005

Principal polarizations on products of elliptic curves

An abelian variety admits only a finite number of isomorphism classes of principal polarizations. The paper gives an interpretation of this number in terms of class numbers of definite Hermitian forms in the case of a product of elliptic curves without complex multiplication. In the case of a self-product of an elliptic curve, as well as in the two-dimensional case, classical class number computations can be applied to determine this number.

math.AG↗

arXiv2002

Curves of genus g on an abelian variety of dimension g

In this paper we prove a general theorem concerning the number of translation classes of curves of genus $g$ belonging to a fixed cohomology class in a polarized abelian variety of dimension $g$. For $g = 2$ we recover results of Göttsche and Bryan-Leung. For $g = 3$ we deduce explicit numbers for these classes.

math.AG↗

arXiv2002

On the Hilbert scheme of a Prym variety

To any unramified double cover $π:\tilde C \to C$ of projective irreducible and nonsingular curves one associates the Prym variety $P = P(π)$. For $C$ nonhyperelliptic of genus $g \geq 6$ we consider the natural embedding $\tilde C \subset P$ (defined up to translation) of $\tilde C$ into $P$ and we study the local structure of the Hilbert scheme $Hilb^P$ of $P$ at the point $[\tilde C]$. We show that this structure is related with the local geometry of the Prym map, or more precisely with the validity of the infinitesimal version of Torelli's theorem for Pryms at $[π]$

math.AG↗

arXiv2001

The moduli spaces of hyperelliptic curves and binary forms

There is a canonical isomorphism between the coarse moduli spaces of somooth hyperelliptic curves of genus g and binary forms of degree 2g+2 with nonzero discriminant. In this paper, we study the extension of this isomorphism to the compactification of these moduli spaces. The main result is that the canonical isomorphism above extends to a holomorphic map f_g from the compactication of the moduli space of hyperelliptic curves of genus g to the compactification of the moduli space of binary forms of degree 2g+2. Moreover, we work out how the boundary components are contracted by the map f_g.

math.AG↗

arXiv2001

Curves of genus 2 and Desargues configurations

We study the moduli space M_D of Desargues configurations and how to define a holomorphic map from M_D to M^b_6 the moduli space of stable binary forms of degree 6. The construction of the map goes back to the greek mathematician C. Stephanos who published a paper about this in 1883, using classical geometric invariant theory. We put this in a contemporary context and give a new proof of the properties of the map.We also study degenerations.

math.AG↗