Hristo Iliev
Institute of Mathematics and Informatics of the Bulgarian Academy of Sciences, Sofia, Bulgaria

In the present report we consider curves on a cone that pass through its vertex and are also triple covers of the base of the cone that is is a general smooth curve of genus $\gamma$ and degree $e$ in $\mathbb{P}^{e-\gamma}$.

Using the free resolution of the ideal of such a curve found by Catalisano and Gimigliano, as well as a technique involving very flat families introduced by Ciliberto, we show that the deformations of such curves remain on cones over a deformation of the base curve.

This allows us to prove that for every $\gamma \geq 3$ and $e \geq 4\gamma + 5$ there exists a non-reduced component $\mathcal{H}$ of the Hilbert scheme $\mathcal{I}_{3e+1, 3e + 3\gamma, e-\gamma+1}$ of smooth curves of genus $3e + 3\gamma$ and degree $3e+1$ in $\mathbb{P}^{e-\gamma+1}$. We show that $\dim T_{[X]} \mathcal{H} = \dim \mathcal{H} + 1 = (e - \gamma + 1)^2 + 7e + 5$ for a general point $[X] \in \mathcal{H}$.

The reported results are based on arXiv:2302.08707 [math.AG]. The last develops an approach of the same authors aimed at constructing non-reduced components of the Hilbert scheme of curves in projective spaces of high dimension introduced in arXiv:2208.12470 [math.AG].