# Embedding abelian varieties into projective spaces of small dimension

Given a (complex) abelian variety $$A$$ of a fixed dimension $$g$$, let $$d(A)$$ be the dimension of the smallest complex projective space it embeds into.

Is $$d(A)$$ uniform over all abelian varieties of a fixed $$g$$? Or are there special ones that embed into even smaller projective spaces?

Can $$d(A)$$ be computed explicitly? I am particularly interested in the case $$g = 2$$.

Recall that any smooth projective variety of dimension $$g$$ embeds into $$\mathbf{P}^{2g+1}$$. Consider now an abelian variety $$A$$ of dimension $$g$$ which embeds into $$\mathbf{P}^{2g}$$. Van de Ven proves (essentially by applying the self-intersection formula to the normal bundle of $$A$$ in $$\mathbf{P}^{2g}$$) that the degree of $$A$$ in $$\mathbf{P}^{2g}$$ is given by $${2g+1\choose g}$$, and notes that the Riemann-Roch theorem implies that the degree has to be divisible by $$g!$$. This is possible only if $$g=1$$ or $$g=2$$. Of course elliptic curves embed into $$\mathbf{P}^2$$. One cannot embed abelian surfaces into $$\mathbf{P}^3$$, and if an abelian surface embeds into $$\mathbf{P}^4$$, then its degree is $$10$$. Abelian surfaces with this property exist (Mumford-Horrocks surfaces). This is all folklore. Maybe less known is the fact that Comessatti proved in 1919 (see this paper of Lange for a modern account) that for some curves of genus $$2$$ the Jacobian embeds into $$\mathbf{P}^4$$. More precisely: if $$C$$ is a curve of genus $$2$$ and $$J(C)$$ contains a curve $$D$$ with self-intersection $$2$$ and $$C\cdot D=3$$, then $$J(C)$$ embeds into $$\mathbf{P}^4$$ (with embedding given by $$|C+D|$$). But these should also be of the Mumford-Horrocks type (Theorem 5.2 in the paper of Mumford and Horrocks says that any abelian surface in $$\mathbf{P}^4$$ is projectively equivalent to one of theirs).
• Is it known what is the smallest degree that the image of $\mathbf{A}$ in $\mathbf{P}^{2g+1}$ can be?
• @Kim The degree $d$ has to be at least $d\geq g^2+g+2$ by the italicized statement on p. 44 here (numdam.org/article/ASNSP_1981_4_8_1_35_0.pdf).