Singularities of 2 Θ -divisors in the jacobian

Christian Pauly; Emma Previato

Bulletin de la Société Mathématique de France (2001)

  • Volume: 129, Issue: 3, page 449-485
  • ISSN: 0037-9484

Abstract

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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 product of the canonical space and the space of quadrics containing the canonical curve. Moreover we show that some of these spaces are equal to the linear span of Brill-Noether loci in the moduli space of semi-stable rank 2 vector bundles with canonical determinant over C , which can be embedded in | 2 Θ 0 | .

How to cite

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Pauly, Christian, and Previato, Emma. "Singularities of $2\Theta $-divisors in the jacobian." Bulletin de la Société Mathématique de France 129.3 (2001): 449-485. <http://eudml.org/doc/272462>.

@article{Pauly2001,
abstract = {We consider the linear system $|2\Theta _0|$ of second order theta functions over the Jacobian $JC$ of a non-hyperelliptic curve $C$. A result by J.Fay says that a divisor $D \in |2\Theta _0|$ contains the origin $\mathcal \{O\} \in JC$ with multiplicity $4$ if and only if $D$ contains the surface $C-C = \lbrace \mathcal \{O\}(p-q) \mid p,q \in C \rbrace \subset JC$. 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 $\mathcal \{O\}$ with multiplicity $6$, divisors containing the fourfold $C_2 - C_2 = \lbrace \mathcal \{O\}(p+q-r-s) \mid p,q,r,s \in C \rbrace $, and divisors singular along $C-C$, using the third exterior product of the canonical space and the space of quadrics containing the canonical curve. Moreover we show that some of these spaces are equal to the linear span of Brill-Noether loci in the moduli space of semi-stable rank $2$ vector bundles with canonical determinant over $C$, which can be embedded in $|2\Theta _0|$.},
author = {Pauly, Christian, Previato, Emma},
journal = {Bulletin de la Société Mathématique de France},
keywords = {theta functions; jacobian; canonical curve; vector bundle},
language = {eng},
number = {3},
pages = {449-485},
publisher = {Société mathématique de France},
title = {Singularities of $2\Theta $-divisors in the jacobian},
url = {http://eudml.org/doc/272462},
volume = {129},
year = {2001},
}

TY - JOUR
AU - Pauly, Christian
AU - Previato, Emma
TI - Singularities of $2\Theta $-divisors in the jacobian
JO - Bulletin de la Société Mathématique de France
PY - 2001
PB - Société mathématique de France
VL - 129
IS - 3
SP - 449
EP - 485
AB - We consider the linear system $|2\Theta _0|$ of second order theta functions over the Jacobian $JC$ of a non-hyperelliptic curve $C$. A result by J.Fay says that a divisor $D \in |2\Theta _0|$ contains the origin $\mathcal {O} \in JC$ with multiplicity $4$ if and only if $D$ contains the surface $C-C = \lbrace \mathcal {O}(p-q) \mid p,q \in C \rbrace \subset JC$. 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 $\mathcal {O}$ with multiplicity $6$, divisors containing the fourfold $C_2 - C_2 = \lbrace \mathcal {O}(p+q-r-s) \mid p,q,r,s \in C \rbrace $, and divisors singular along $C-C$, using the third exterior product of the canonical space and the space of quadrics containing the canonical curve. Moreover we show that some of these spaces are equal to the linear span of Brill-Noether loci in the moduli space of semi-stable rank $2$ vector bundles with canonical determinant over $C$, which can be embedded in $|2\Theta _0|$.
LA - eng
KW - theta functions; jacobian; canonical curve; vector bundle
UR - http://eudml.org/doc/272462
ER -

References

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