# How are Jacobians of genus $3$ curves different from one another?

There are two types of smooth projective (complex) curves of genus $3$: plane quartics, and hyperelliptic curves. The Torelli morphism $M_3\to A_3$, assigning a curve to its (principally polarized) Jacobian, is bijective on points. Thus $A_3$ consists of Jacobians of genus $3$ curves. Because of the "partition" of $M_3$ as {plane quartics} $\amalg$ {hyperelliptics}, I would expect this to be reflected somehow in $A_3$. Whence my questions:

Question 1. What are the main differences between Jacobians of plane quartics and of hyperelliptic curves?

(I am intentionally vague here, as I am curious about any sort of difference).

It would be also interesting to know if the deformation theory of the curve inside its Jacobian is sensibly different according to the curve itself; this leads to a second question:

Question 2. (Deformations inside the Jacobian) What do the normal bundles $\mathcal N_{C/Jac(C)}$ look like, when $C$ is a plane quartic or a hyperelliptic curve, respectively?

Thanks for any clue on this.

• Careful! The map $M_3\to A_3$ is actually not surjective. I believe it was Beauville who showed that the complement $A_3\backslash M_3$ consists of non-simple ppavs with product polarizations. – rfauffar Mar 18 '14 at 0:20
• As to your first question, all I can think of is a general property of hyperelliptic curves that is not shared by general curves, that says that the automorphism $-1$ of the Jacobian descends to the hyperelliptic involution on $C$ (if you embed $C$ the "right way" via a Weierstrass point). – rfauffar Mar 18 '14 at 0:21
• Dear @RobertAuffarth: thanks for your comments. In particular, I did not think about reducible polarizations, thanks for pointing that out. Regards – Brenin Mar 25 '14 at 8:59

## 1 Answer

Let $L$ be a principal polarization on the Jacobian of the genus $3$ curve. Then $L^2$ gives an embedding of the Kummer variety $X$ of this Jacobian (i.e., quotient by the action $x \mapsto -x$). Khaled showed that this embedding is projectively normal (i.e., the restriction map

$Sym^n(H^0(X, L^2)) \to H^0(X, L^{2n})$

is surjective for all $n$) if and only if $C$ is not hyperelliptic. This was actually useful in a paper I wrote (and I will also take the opportunity to advertise): see Proposition 6.9 of http://arxiv.org/abs/1203.2575 and also the reference for Khaled's paper.