arXiv subjects

J. Spandaw

Publications and source records attributed to J. Spandaw.

3 recordsLinked to original sources

arXiv2000

The modularity of the Barth-Nieto quintic and its relatives

The moduli space of (1,3)-polarized abelian surfaces with full level-2 structure is birational to a double cover of the Barth-Nieto quintic. Barth and Nieto have shown that these varieties have Calabi-Yau models Z and Y, respectively. In this paper we apply the Weil conjectures to show that Y and Z are rigid and we prove that the L-function of their common third étale cohomology group is modular, as predicted by a conjecture of Fontaine and Mazur. The corresponding modular form is the unique normalized cusp form of weight 4 for the group Γ_1(6). By Tate's conjecture, this should imply that Y, the fibred square of the universal elliptic curve S_1(6), and Verrill's rigid Calabi-Yau Z_{A_3}, which all have the same L-function, are in correspondence over Q. We show that this is indeed the case by giving explicit maps.

math.AG↗

arXiv1994

Degenerations of abelian surfaces and Hodge structures

In this paper the authors consider a certain toroidal compactification of the moduli space of degenerations of (1,p)-polarized abelian surfaces with (canonical) level structure. Using Hodge theory we give a proof that a degenerate abelian surface associated to a corank 1 boundary point is (almost) completely determined by the boundary point. The crucial ingredient in the proof is the Local Invariant Cycle theorem which relates the variation of Hodge structure to the mixed Hodge structure on the singular surface.

alg-geom↗

arXiv1993

Very ample linear systems on abelian varieties

Let $(X,L)$ be a polarized complex abelian variety of dimension $g$ where $L$ is a polarization of type $(1,...,1,d)$. For $(X,L)$ genberic we prove the following: (1) If $d \ge g+2$, then $ϕ_L\colon X \to {\bf P}^{d-1}$ defines a birational morphism onto its image. (2) If $d > 2^g$, then $L$ is very ample. We show the latter by checking it on a suitable rank-$(g-1)$-degeneration.

alg-geom↗