Given a sufficiently positive embedding $X\subset\mathbb{ P}^N$ of a smooth projective variety $X$, we consider its secant variety $\Sigma$ that comes equipped with the embedding $\Sigma\subset\mathbb{ P}^N$ by its construction. In this article, we determine the local cohomological dimension $\textrm{ lcd}(\mathbb{ P}^N,\Sigma)$ of this embedding, as well as the generation level of the Hodge filtration on the topmost non-vanishing local cohomology module $\mathcal{ H}^{ q}_{ \Sigma}(\mathcal{ O}_{ \mathbb{ P}^N})$, i.e., when $q=\textrm{ lcd}(\mathbb{ P}^N,\Sigma)$. Additionally, we show that $\Sigma$ has quotient singularities (in which case the equality $\textrm{ lcd}(\mathbb{ P}^N,\Sigma)=\textrm{ codim}_{ \mathbb{ P}^N}(\Sigma)$ is known to hold) if and only if $X\cong\mathbb{ P}^1$. We also provide a complete classification of $(X,L)$ for which $\Sigma$ has ($\mathbb{ Q}$-)Gorentein singularities. As a consequence, we deduce that if $\Sigma$ is a local complete intersection, then either $X$ is isomorphic to $\mathbb{ P}^1$, or an elliptic curve.
Categories: math.AG