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I am looking for references on theta characteristics. In particular I am interesting in understanding the isomorphism $\Omega_A^g\cong\mathcal{O}_A(\Theta)^2$ where $A$ is an abelian variety and $\Theta$ is the theta divisor. What is the geometric meaning/relation in the case in which $A$ is the Jacobian of a curve $C$ and $\Theta$ is induced by the "canonical" divisor $W^{g-1}$ image of $Pic^{g-1}(X)$? I mean $\Omega_A^g\cong\wedge^g H^0(X,\omega_X)\otimes\mathcal{O}$, so where I "read " "quadratic" theta functions on the differentials??


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I'm a little confused by your question. I don't know what you're looking for the geometric meaning of. Theta characteristics? The geometric meaning differs between an odd theta and an even theta. I also don't know what you mean when you put all these words in quotation marks. In general a reference I like is Dolgachev's Topics in Classical Algebraic Geometry, which is sadly no longer available on his webpage as he's publishing it. –  stankewicz Nov 14 '11 at 16:28

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In general, if $A$ is principally polarised, there is a canonical isomorphism $(\Omega_A^{g})^{\otimes 4}\simeq{\cal O}_A(\Theta)^{\otimes 8}$ (if you choose to divide by $4$, it is not canonical anymore). The justification for this isomorphism is the Grothendieck-Riemann-Roch theorem, applied to ${\cal O}(\Theta)$. For this, see the book by Moret-Bailly, "Pinceaux des variétés abéliennes" (Astérisque 129), whose principal aim is the investigation of (a generalization of) this canonical isomorphism. See also his article "Sur l'équation fonctionelle..." (Compositio Math. 75). Finally see his article "La formule de Noether sur les surfaces arithmétiques" (Invent. Math. 98), par. 2.3 for the case of jacobians that you are interested in. Other references are the book by Chai-Faltings, "Degeneration of abelian varieties", chap. I, Th. 5.1 and the article by V. Maillot and myself, "On the determinant bundles of abelian schemes" (Compositio 144).

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