Asian Journal of Mathematics
- Asian J. Math.
- Volume 8, Number 1 (2004), 161-172.
CUBIC EQUATIONS FOR THE HYPERELLIPTIC LOCUS
We prove a conjecture from [BK2] that the multi-dimensional vector addition formula for Baker-Akhiezer functions obtained there characterizes Jacobians among principally polarized abelian varieties. We also show that this addition formula is equivalent to Gunning's multisecant formula for the Kummer variety obtained in [Gu2].
We then use this addition formula to obtain cubic relations among theta functions that characterize the locus of hyperelliptic Jacobians among irreducible abelian varieties. In genus 3 our equations are equivalent to the vanishing of one theta-null, and thus are classical (see [M], [P]), but already for genus 4 they appear to be new.
Asian J. Math., Volume 8, Number 1 (2004), 161-172.
First available in Project Euclid: 21 June 2004
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GRUSHEVSKY, SAMUEL. CUBIC EQUATIONS FOR THE HYPERELLIPTIC LOCUS. Asian J. Math. 8 (2004), no. 1, 161--172. https://projecteuclid.org/euclid.ajm/1087840914