Varieties with small discriminant variety

Lanteri, Antonio; Muñoz, Roberto

doi:10.1090/S0002-9947-06-03915-8

Varieties with small discriminant variety
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by Antonio Lanteri and Roberto Muñoz PDF

Trans. Amer. Math. Soc. 358 (2006), 5565-5585 Request permission

Abstract:

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 $|V|$. 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.

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