Subvarieties of SUC and divisors in the Jacobian

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41 pages

English

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Subvarieties of SUC(2) and 2?-divisors in the Jacobian W.M. Oxbury, C. Pauly and E. Previato Let SUC(2, L) denote the projective moduli variety of semistable rank 2 vector bundles with determinant L ? Pic(C) on a smooth curve C of genus g > 2; and suppose that degL is even. It is well-known that, on the one hand, the singular locus of SUC(2, L) is isomorphic to the Kummer variety of the Jacobian; and on the other hand that when C is nonhyperelliptic SUC(2,O) has an injective morphism into the linear series |2?| on the Jacobian Jg?1C which restricts to the Kummer embedding a 7? ?a + ??a on the singular locus. Dually SUC(2, K) injects into the linear series |L| on J0C , where L = O(2??) for any theta characteristic ?, and again this map restricts to the Kummer map Jg?1C ? |2?|? = |L| on the singular locus. This map to projective space (the two cases are of course isomorphic) comes from the complete series on the ample generator of the Picard group, and (at least for a generic curve) is an embedding of the moduli space. Moreover, its image contains much of the geometry studied in connection with the Schottky problem; notably the configuration of Prym-Kummer varieties.

jg?1c ?

moduli space

base locus

kummer variety

maximal degree

wr ?

line bundles

Sujets

Moduli space

Base locus

Kummer variety

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Publié par	pefav
Nombre de lectures	9
Langue	English

Extrait

Subvarieties

SUC(2) and 2-divisors Jacobian

W.M. Oxbury, C. Pauly and E. Previato

the

LetSUC(2 L) denote the projective moduli variety of semistable rank 2 vector bundles with determinantL∈Pic(C) on a smooth curveCof genus g >2; and suppose that degL is well-known that, on the one hand, Itis even. the singular locus ofSUC(2 L) is isomorphic to the Kummer variety of the Jacobian; and on the other hand that whenCis nonhyperellipticSUC(2O) has an injective morphism into the linear series|2|on the JacobianJCg1 which restricts to the Kummer embeddinga7→a+ aon the singular locus. DuallySUC(2 K) injects into the linear series|L|onJ0C, whereL= O(2) for any theta characteristic, and again this map restricts to the Kummer mapJgC1→ |2|∨=|L|on the singular locus. This map to projective space (the two cases are of course isomorphic) comes from the complete series on the ample generator of the Picard group, and (at least for a generic curve) is an embedding of the moduli space. Moreover, its image contains much of the geometry studied in connection with the Schottky problem; notably the conguration of Prym-Kummer varieties. In this paper we explore a little of the interplay, via this embedding, between the geometry of vector bundles and the geometry of 2-divisors. On the vector bundle side we are principally concerned with the Brill-Noether lociWr SUC(2 K) dened by the conditionh0(E)> ron stable bundes E are analogous to the very classical varieties. TheseWgr1JCg1. Unlike the line bundle theory, however, general results—connectedness, dimension, smoothness and so on—are not known for the varietiesWr(see [6]). On the 2side we shall consider the Fay trisecants of the 2-embedded Kummer variety, and the subseriesP00 |L|consisting of divisors having multiplicity This4 at the origin. subseries is known to be important in the study of principally polarised abelian varieties [10]: in the Jacobian of a curve its base locus is the surfaceCCJ0C(plus a pair of isolated points