On the dimension of voisin sets in the moduli space of abelian varieties

We study the subsets $$V_k(A)$$ V k ( A ) of a complex abelian variety A consisting in the collection of points $$x\in A$$ x ∈ A such that the zero-cycle $$\{x\}-\{0_A\}$$ { x } - { 0 A } is k-nilpotent with respect to the Pontryagin product in the Chow group. These sets were introduced recently by Voisin and she showed that $$\dim V_k(A) \le k-1$$ dim V k ( A ) ≤ k - 1 and $$\dim V_k(A)$$ dim V k ( A ) is countable for a very general abelian variety of dimension at least $$2k-1$$ 2 k - 1 . We study in particular the locus $${\mathcal {V}}_{g,2}$$ V g , 2 in the moduli space of abelian varieties of dimension g with a fixed polarization, where $$V_2(A)$$ V 2 ( A ) is positive dimensional. We prove that an irreducible subvariety $${\mathcal {Y}} \subset {\mathcal {V}}_{g,2}$$ Y ⊂ V g , 2 , $$g\ge 3$$ g ≥ 3 , such that for a very general $$y \in {\mathcal {Y}}$$ y ∈ Y there is a curve in $$V_2(A_y)$$ V 2 ( A y ) generating A satisfies $$\dim {\mathcal {Y}}\le 2g - 1.$$ dim Y ≤ 2 g - 1 . The hyperelliptic locus shows that this bound is sharp.

Concomitants of ternary quartics and vector-valued Siegel and Teichmüller modular forms of genus three

We show how one can use the representation theory of ternary quartics to construct all vector-valued Siegel modular forms and Teichmüller modular forms of degree 3. The relation between the order of vanishing of a concomitant on the locus of double conics and the order of vanishing of the corresponding modular form on the hyperelliptic locus plays an important role. We also determine the connection between Teichmüller cusp forms on $$\overline{\mathcal {M}}_{g}$$ M ¯ g and the middle cohomology of symplectic local systems on $${\mathcal {M}}_{g}\,$$ M g . In genus 3, we make this explicit in a large number of cases.

Trigonal Deformations of Rank One and Jacobians

We study the infinitesimal deformations of a trigonal curve that preserve the trigonal series and such that the associate infinitesimal variation of Hodge structure is of rank $1.$ We show that if $g\geq 8$ or $g=6,7$ and the curve is Maroni general, this locus is zero dimensional. Moreover, we complete the result [10, Theorem 1.6]. We show in fact that if $g\geq 6$, the hyperelliptic locus ${{\mathcal{M}}}^1_{g,2}$ is the only $2g-1$-dimensional sub-locus ${{\mathcal{Y}}}$ of the moduli space ${{\mathcal{M}}}_g$ of curves of genus $g$, such that for the general element $[C]\in{{\mathcal{Y}}}$, its Jacobian $J(C)$ is dominated by a hyperelliptic Jacobian of genus $g^{\prime}\geq g$.