Consider the moduli space $\mathcal{A}_{g,n}$ of abelian varieties with some level $n \geq 3$ structure. For simplicity, we just denote it by $\mathcal{A}_{g}$ and drop $n$. Denote the locus of hyperelliptic Jacobians by $H_{g}$ (i.e. the image of hyperelliptic curves under the Torelli map). Now, having a curve $C$ in $\mathcal{A}_{g}$, are there necessary and/or sufficient results to determine whether $C$ intersects $H_{g}$ ?(I am particularly interested in the case where $g=4$)
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2$\begingroup$ maybe if you told us how the curve arises, otherwise we have only to think about a curve and a codimension 3 locus in an abstract variety. what do you know of the theta divisors on the ppav's along your curve? of course Beauville characterized h.e. ones as having a curve of singular points, and the specific pattern of the theta nulls was also characterized classically, and generalized by Mumford. $\endgroup$ – roy smith Jan 2 '13 at 19:42

2$\begingroup$ well for g=4, g3 = 1 gives a curve. and Mumford's work is in his three volume work on theta functions, probably volume 2, where he generalizes a classical g=4 result by maybe Thomae? characterizing h.e. curves by the configuration of vanishing theta nulls on the Jacobian. $\endgroup$ – roy smith Jan 3 '13 at 19:07

2$\begingroup$ Mumford, Tata lectures on Theta II, chapter IIIa.9, page 3.137. – roy smith 0 secs ago $\endgroup$ – roy smith Jan 3 '13 at 19:13

1$\begingroup$ actually the unlikely event of a curve in A(4) meeting the codimension 3 h.e. locus is reminiscent of the interesting conjecture you mention that would imply the seemingly also unlikely event that whenever the locus singTheta has codimension 3 in J, then it must meet a large number of points of order 2. I am reaching back further now, but there is another similarity, since when g=4 perhaps the h.e. locus is a component of Sing.N(0), the singular points on the "discriminant" divisor N(0) parametrizing singular theta divisors in A(4)....Just playing around here.... $\endgroup$ – roy smith Jan 3 '13 at 19:30

1$\begingroup$ you might also consult Igusa's paper On the irreducibility of the Schottky locus, where he refers to Pringsheim, 1877, for the g=4 antecedent of Mumford's characterization of the h.e. locus. $\endgroup$ – roy smith Jan 3 '13 at 20:48