# General curves of genus 3 as plane sections of Kummer surfaces

Is it true that a general curve of genus 3 is a plane section of an appropriate Kummer surface in $\mathbb P^3$? By Kummer surface I mean image of a principally polarized Abelian surface w.r.t. the map defined by the complete linear system $|2\Theta|$, where $\Theta$ is a theta divisor of polarization.

This assertion agrees with the naive dimension count, but I have no idea whether it is true or false.

Serge

• So, just to check, your surfaces have 16 double points; you are not working with the desingularized K3 surface. – Jason Starr Aug 17 '13 at 19:52
• @Jason. Exactly so :) – Serge Lvovski Aug 17 '13 at 19:53

Let $C$ be a non hyperelliptic curve of genus 3, let $f\colon C'\to C$ be an e'tale double cover and let $A$ be the Prym variety of $f$. Then $A$ is a principally polarized surface and the Abel-Prym map embeds $C'$ in $A$ as an element of $|2\Theta|$. Multiplication by $-1$ on $A$ restricts on $C'$ to the involution induced by $f$, hence the image of $C'$ via the map given by $|2\Theta|$ is a plane section isomorphic to $C$.

A reference for these facts is chapter 12 of the book "Complex abelian varieties'' by Birkenhake-Lange.

This answer intends merely to build on the nice ones already given and summarize some of their content, in order to give an idea of what will be found in some of the references before consulting them. My apologies if memory has faded the details following into serious errors.

The original question as asked, is answered already by Wirtinger, e.g. in his classic work Untersuchungen uber Thetafunctionen, on page 113, section 54, Uber die Thetafunctionen von zwei Variablen, lines -13,-14. He says (possibly very roughly):

“Each system of “eigentliche” (non trivial?) everywhere tangent conics to a plane quartic curve $$C$$ defines a Kummer surface, on which the curve $$C$$ lies. There are thus $$63$$ such Kummer surfaces.”

The point is that a plane section of a Kummer surface determines not just a genus three curve but a connected double cover of it, for which the Kummer surface is associated to the corresponding Prym variety, and a genus 3 curve has $$2^{2g} -1 = 63$$ non trivial double covers corresponding to the 63 non canonical square roots of twice the canonical class.

In general, there are (rough) correspondences between any two of the following data:

1) a curve $$C$$ of genus 3 equipped with a double cover;

2) a curve $$C$$ of genus 3 equipped with a half period;

3) a plane section $$C$$ of a Kummer surface;

4) a principally polarized abelian surface containing a genus 5 curve $$C'$$ representing twice the “minimal” class, i.e. a divisor in the system $$|2\Theta|$$.

5) a (semi stable, even) $$\mathbb{P}^1$$ bundle over a genus 2 curve $$D$$.

E.g. a "non trivial everywhere tangent conic" (not necessarily effective but not twice a line) to a plane quartic $$C$$ determines a half period on the genus 3 curve C;

a half period (on $$\mathrm{Pic}(C)$$) determines a double cover of a section (isomorphic to $$C$$) of the trivial line bundle on the curve $$C$$;

a plane quartic section $$C$$ of a Kummer surface inherits a double cover from that of the Kummer;

a double cover of a genus 3 curve $$C$$ determines as in rita's answer a Prym variety $$P$$ and a plane section isomorphic to C of the Kummer surface of $$P$$;

a double cover of a genus 3 curve $$C$$ induces a $$\mathbb{P}^1$$ bundle over the genus 2 theta divisor $$D$$ of the associated Prym variety via the Abel - Prym map;

a $$\mathbb{P}^1$$ bundle on a genus 2 curve $$D$$ yields a curve $$C'$$ in the system $$|2\Theta|$$ on the Jacobian of $$D$$, which parametrizes effective twists of an associated rank $$2$$ vector bundle.

These statements from memory will surely require correction by current experts, but may be useful as a sketch.

edit: Indeed perusing the paper of Verra seems to show that some of these data (such as #4) must be considered modulo the action of the group of points of order two on the abelian variety.

http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002329107&IDDOC=161897 (Alessandro Verra, "The fibre of the Prym map in genus three", Math. Ann. 276, no. 3, 1987)

In addition to the references above to Wirtinger, Verra, and Birkenhake-Lange, one may consult Narasimhan - Ramanan: Moduli of vector bundles on a compact Riemann surface, Annals of Math (1969), and perhaps:

http://msp.org/pjm/1999/188-2/pjm-v188-n2-p09-s.pdf (R. Smith, R. Varley, "On the geometry of two dimensional Prym varieties", Pacific Journal of Mathematics, volume 188, no. 2, 1999)

• Dear @roy: There is no longer any reference elsewhere on this page to an article by d'Almeida, Gruson, and Perrin. – Ricardo Andrade Aug 19 '13 at 22:33

The paper by Alessandro Verra (Math. Ann. 276 (1987), no. 3, 433–448) gives a very precise description of all fibers of the Prym map $\mathcal{R}_3\to \mathcal{A}_2$ from which the answer to your question can be obtained.

Although your count of constants luckily works in this case, it does not work in general for nodal quartic surfaces. For example, it is not true that a general curve of genus 3 is realized as a plane section of a Weddle 6-nodal quartic surface. It is an open problem to find a condition when the map from the dual $\mathbb{P}^3$ to $\mathcal{}M_3$ given by taking a plane section of a fixed normal quartic surface is generically injective.

• Thank you, interesting indeed. Is this problem open even for the case of smooth quartics? – Serge Lvovski Aug 18 '13 at 19:30
• Sorry, do you really mean «generically injective»? If a quartic has automorphisms, then the mapping in question from $(\mathbb P^3)^*$ to $M_3$ cannot be generically injective. Did you actually mean «has 3-dimensional image»? – Serge Lvovski Aug 18 '13 at 19:38
• I am confused. Verra (p.434, lines 27-28) seems to say the map from the dual P^3 to the fiber of the Prym map is constant on orbits of the action of the group G of points of order 2 of the abelian surface on the system |2.(Theta)|, hence the map to M(3) would seem never to be of degree one. Did I misunderstand this? The images under the finite map R(3)-->M(3) however would always seem to be three dimensional. – roy smith Aug 20 '13 at 1:49
• Yes, I meant generically finite. – user37622 Aug 21 '13 at 12:42