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Singularities of 2 Θ -divisors in the jacobian

Christian Pauly, Emma Previato (2001)

Bulletin de la Société Mathématique de France

We consider the linear system | 2 Θ 0 | of second order theta functions over the Jacobian J C of a non-hyperelliptic curve C . A result by J.Fay says that a divisor D ∈ | 2 Θ 0 | contains the origin 𝒪 ∈ J C with multiplicity 4 if and only if D contains the surface C - C = { 𝒪 ( p - q ) ∣ p , q ∈ C } ⊂ J C . In this paper we generalize Fay’s result and some previous work by R.C.Gunning. More precisely, we describe the relationship between divisors containing 𝒪 with multiplicity 6 , divisors containing the fourfold C 2 - C 2 = { 𝒪 ( p + q - r - s ) ∣ p , q , r , s ∈ C } , and divisors singular along C - C , using the third exterior...

Singularities of theta divisors and the geometry of 𝒜 5

Gavril Farkas, Samuele Grushevsky, Salvati R. Manni, Alessandro Verra (2014)

Journal of the European Mathematical Society

We study the codimension two locus H in 𝒜 g consisting of principally polarized abelian varieties whose theta divisor has a singularity that is not an ordinary double point. We compute the class [ H ] ∈ C H 2 ( 𝒜 g ) for every g . For g = 4 , this turns out to be the locus of Jacobians with a vanishing theta-null. For g = 5 , via the Prym map we show that H ⊂ 𝒜 5 has two components, both unirational, which we describe completely. We then determine the slope of the effective cone of 𝒜 5 ¯ and show that the component N 0 ' ¯ of the Andreotti-Mayer...

Solving an indeterminate third degree equation in rational numbers. Sylvester and Lucas

Tatiana Lavrinenko (2002)

Revue d'histoire des mathématiques

This article concerns the problem of solving diophantine equations in rational numbers. It traces the way in which the 19th century broke from the centuries-old tradition of the purely algebraic treatment of this problem. Special attention is paid to Sylvester’s work “On Certain Ternary Cubic-Form Equations” (1879–1880), in which the algebraico-geometrical approach was applied to the study of an indeterminate equation of third degree.

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