Varieties with small discriminant variety

Let $X$ be a smooth complex projective variety, let $L$ be an ample and spanned line bundle on $X$, $V\subseteq H^{0}(X,L)$ defining a morphism $\phi _{V}:X \to \mathbb{P}^{N}$ and let $\mathcal{D}(X,V)$ be its discriminant locus, the variety parameterizing the singular elements of $\vert V\vert$. We present two bounds on the dimension of $\mathcal{D}(X,V)$ and its main component relying on the geometry of $\phi _{V}(X) \subset \mathbb{P}^{N}$. Classification results for triplets $(X,L,V)$ reaching the bounds as well as significant examples are provided.