# A geometric definition of the addition law on abelian surfaces

Most people will have see a geometric "explanation" of the addition law on elliptic curves given by embedding it as a cubic in the projective plane and cutting it with lines.

Is there a similar explicit, geometric definition of the addition law on (a family of?) abelian surfaces?

So the question is really: Give a nice embedding of abelian surfaces into projective space and then define the addition law using this embedding - if not for all abelian surfaces, at least for some non trivial family. In fact, it would be really nice if we could do this for the embedding that realizes the surface as a degree 10 variety using the Horrocks-Mumford bundle.

• There is one on $E\times E'$ in $P^2\times P^2\subset P^8$ at least. – wnx Sep 15 '20 at 14:25
• There is one on the Jacobian of a hyperelliptic curve of genus 2 by intersecting with $y=cubic$. – Henri Cohen Sep 15 '20 at 20:45
• @HenriCohen That sounds great, do you have a reference? – Asvin Sep 15 '20 at 21:56
• Some Jacobians of genus 2 curves are in fact HM surfaces; see mathoverflow.net/questions/321368/… – SWS Sep 17 '20 at 17:52
• Here are some more references: Leitenberger - About the group law for the Jacobi variety of a hyperelliptic curve ; The addition on Jacobian varieties from a geometric viewpoint - Yaacov Kopeliovich, Tony Shaska ; Costello, Lauter - Group Law Computations on Jacobians of Hyperelliptic Curves. – Watson May 19 at 17:53

Jacobians of genus-2 curves - and abelian surfaces in general, I suppose - can be realized as the variety of lines on the intersection of two quadrics in $$\mathbb{P}^5$$ (once you've chosen a line to act as the neutral element). This is analogous to seeing an elliptic curve as the variety of 0-dimensional spaces (i.e. points) on the intersection of two quadrics in $$\mathbb{P}^3$$ (which is sometimes called the "Jacobi" model of an elliptic curve). The group law has a really nice geometric expression.

This is covered at length in Chapter 17 ("A neoclassical approach") of Cassels and Flynn's Prolegomena to a middlebrow arithmetic of curves of genus 2, and in even more length in Chapter 6 of Principles of algebraic geometry by Griffiths and Harris (specifically Section 6.3, "Lines on the quadric line complex").

Edit (bonus): If you're interested in higher dimensions, then let $$X$$ be the intersection of two quadrics in $$\mathbb{P}^{2g+1}$$, and let $$S$$ be the variety of $$(g-1)$$-planes in $$X$$. Then $$S$$ is a homogeneous space under the Jacobian of a hyperelliptic curve $$C$$ of genus $$g$$. The relationship between $$X$$, $$S$$, and $$C$$ (and the action of $$\mathrm{Jac}(C)$$ on $$S$$) is very explicit. Chapter 4 of Miles Reid's PhD thesis (The complete intersection of two or more quadrics) has the details.

• See also R. Donagi, Group law on the intersection of two quadrics Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, tome 7, no 2 (1980), p. 217-239 – SWS Sep 17 '20 at 17:53
• The Cassels and Flynn book is great! – Asvin Sep 20 '20 at 3:38

This must be standard, I don't have a reference but the construction is easy: let $$y^2=f(x)$$ be a genus 2 hyperelliptic curve with $$f$$ squarefree of degree $$5$$ or $$6$$. As a set the Jacobian is the symmetric square of the curve, so let $$(A,B)$$ and $$(C,D)$$ be 4 points on the curve. Generically (apart from special configurations) there is a unique $$y=g(x)$$ with $$g$$ of degree 3 which passes through the 4 points (4 linear equations in 4 unknowns). Replacing in the equation of the curve gives (again generically) a sixth degree equation, 4 of the roots being the abcissas of $$A$$, $$B$$, $$C$$, $$D$$. The other two roots define your addition law, as usual after changing the sign of $$y$$.

• Aha, I see! The cubic is in an embedding of the curve, not the Jacobian. Thanks a lot! – Asvin Sep 16 '20 at 21:28
• One decent reference for this is Mumford's Tata Lectures on Theta II (Chapter IIIa). – Ben Smith Sep 17 '20 at 7:46