# Prym varieties as Jacobian varieties

A generic abelian variety of dimension 2 or 3 is a jacobian of a curve. Is there a canonical way to determine a curve whose jacobian is a prym variety of a unramified double cover of a curve of genus 3 or 4?

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Yes, assuming you mean a way to go from the double cover to the curve, (rather than how to go from the Prym variety to the curve, which is just a constructive Torelli argument). This is based on the fact that curves of genus 3 and 4 are "trigonal", i.e. admit a degree 3 map to the projective line. The corresponding "trigonal construction" is attributed to S. Recillas who apparently generalized work of Roth, and it may have antecedents in the work of Wirtinger. An excellent reference is Beauville's paper:

http://math.unice.fr/~beauvill/pubs/specprym.pdf

Basically, a double cover C-->D, where D is trigonal of genus g+1, induces a map of degree 8 on the corresponding third symmetric products of these curves. By definition, that of D contains a projective line, whose preimage in the 3rd symmetric product of C, and modulo the induced involution, is a tetragonal curve E of genus g. The Jacobian of E is isomorphic to the Prym variety of C-->D.

In fact the preimage curve in the symmetric cube of C, is a disjoint union of two copies of E, and both embed via the Abel map into copies of the Prym variety of C-->D, realized as the inverse image in Pic^3(C), of the g(1,3) on D. Then Beauville calculates the homology classes of those embedded curves and applies the criterion of Matsusaka. Recillas himself gave an argument based on Hurwitz schemes.

The link with classical projective geometry is via the "complete quadrangle" or "complete quadrilateral". (These objects had apparently been studied by Wirtinger and Roth.) Note that 4 general points determine 3 pairs of distinct diagonals, hence three points of intersection of such pairs, together with a "double cover" of the triple consisting of the 6 diagonals. On the canonical model of a tetragonal curve E there is a pencil of coplanar 4-tuples of points, hence also another trigonal curve D swept out by the associated pencil of coplanar triples of intersection points. This curve D is presumably the Prym canonical model of the curve D associated to the double cover C-->D by the curve C of diagonals.

Here is a relevant classical reference: Roth P., U ̈ber Beziehungen zwischen algebraischen Gebilden vom Geschlecht drei und vier, Monatsh. fu ̈r Math. und Phys., 22, (1911), 64–88.

And some more recent ones: http://www.math.lsa.umich.edu/~idolga/rat3.pdf