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Given a family $\pi:\mc{X} \rightarrow B$ of smooth projective varieties, aclosed fiber $\mc{X}_o$ and an invertible sheaf $\mc{L}$ on $\mc{X}_o$, wecompare the Hodge locus in $B$ corresponding to the Hodge class $c_1(\mc{L})$with the locus of points $b\,\in\, B$ such that $\mc{L}$ deforms to aninvertible sheaf $\mc{L}_b$ on $\mc{X}_b$ with at least$h^0(\mc{L})$--dimensional space of global sections (it is a Brill-Noether typelocus associated to $\mc{L}$). We finally give an application by comparing theBrill-Noether locus to a family of curves on a surface passing through a fixedset of points.

Hodge locus and Brill-Noether type locus