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Session S35 - Moduli Spaces in Algebraic Geometry and Applications

Thursday, July 15, 13:20 ~ 14:00 UTC-3

Global Prym-Torelli theorem for ramified double coverings

Angela Ortega

Humboldt Universität zu Berlin, Germany   -   This email address is being protected from spambots. You need JavaScript enabled to view it. document.getElementById('cloak70f055826c93d6d432f977e95dbbc2f9').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addy70f055826c93d6d432f977e95dbbc2f9 = '&#111;rt&#101;g&#97;' + '&#64;'; addy70f055826c93d6d432f977e95dbbc2f9 = addy70f055826c93d6d432f977e95dbbc2f9 + 'm&#97;th' + '&#46;' + 'h&#117;-b&#101;rl&#105;n' + '&#46;' + 'd&#101;'; var addy_text70f055826c93d6d432f977e95dbbc2f9 = '&#111;rt&#101;g&#97;' + '&#64;' + 'm&#97;th' + '&#46;' + 'h&#117;-b&#101;rl&#105;n' + '&#46;' + 'd&#101;';document.getElementById('cloak70f055826c93d6d432f977e95dbbc2f9').innerHTML += '<a ' + path + '\'' + prefix + ':' + addy70f055826c93d6d432f977e95dbbc2f9 + '\'>'+addy_text70f055826c93d6d432f977e95dbbc2f9+'<\/a>';

Given a finite morphism between smooth curves one can canonically associate it a polarised abelian variety, the Prym variety. This induces a map from the moduli space of coverings to the moduli space of polarised abelian varieties, known as the Prym map. It is a classical result that the Prym map is generically injective for étale double coverings.

In this talk we will give an introduction to the Prym varieties and the Prym maps. We will then consider the Prym map between the moduli space $\mathcal{R}_{g,r}$ of double coverings over a genus g curve ramified at r points and $\mathcal{A}^{\delta}_{g-1+r/2}$ the moduli space of polarised abelian varieties of dimension $g-1+r/2$ with polarisation of type $\delta$. Unexpectedly, in the ramified case the Prym map is everywhere an embedding when $r \geq 6$ and $g>0$. We will present a constructive proof of this result.

Joint work with Juan Carlos Naranjo (Universidad de Barcelona).