On the distribution of mixed Hodge loci
Let $\mathbb{V}$ be an admissible and graded-polarized integral variation of mixed Hodge structures over a smooth and irreducible complex algebraic variety $S$. We show that if the typical Hodge locus $\mathrm{HL}(S,\mathbb{V}^\otimes)_\mathrm{typ}$ of $\mathbb{V}$ is non-empty, the full Hodge locus $\mathrm{HL}(S,\mathbb{V}^\otimes)$ is dense in $S$ for the Zariski topology. In an other direction, we show that if the associated graded variation $\mathrm{Gr}(\mathbb{V})$ for the weight filtration has large monodromy and level at least 3 in the sense of Baldi- Klingler-Ullmo, the typical Hodge locus of $\mathbb{V}$ is empty, and the full Hodge locus of $\mathbb{V}$ is a strict Zariski-closed subset of $S$, at least if one restricts to its factorwise positive dimensional part, improving a classical result of Brosnan-Pearlstein-Schnell in this situation. These results follow from a detailed study of the transverse part $\mathrm{HL}(S,\mathbb{V}^\otimes)_\mathrm{trans}$ of the Hodge locus of $S$ for $\mathbb{V}$, a subset which contains $\mathrm{HL}(S,\mathbb{V}^\otimes)_\mathrm{typ}$ and whose Zariski-density in $S$ is equivalent, under the Zilber-Pink conjecture for $\mathbb{V}$, to the Zariski-density of $\mathrm{HL}(S,\mathbb{V}^\otimes)_\mathrm{typ}$. We show that non-emptiness of $\mathrm{HL}(S,\mathbb{V}^\otimes)_\mathrm{trans}$ is equivalent to its Zariski-density in $S$, we completely classify variations whose transverse Hodge locus $\mathrm{HL}(S,\mathbb{V}^\otimes)_\mathrm{trans}$ is Zariski-dense, and we prove an independent criterion ensuring that $\mathrm{HL}(S,\mathbb{V}^\otimes)_\mathrm{trans}$ is empty.