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Abstract The Brill–Noether theory of curves plays a fundamental role in the theory of curves and their moduli and has been intensively studied since the 19th century. In contrast, Brill–Noether theory for higher dimensional varieties is less understood. It is hard to determine when Brill–Noether loci are nonempty and these loci can be reducible and of larger than the expected dimension. Let $$E$$ be a semistable sheaf on $${\mathbb{P}}^{2}$$. In this paper, we give an upper bound $$\beta _{r, \mu }$$ for $$h^{0}(E)$$ in terms of the rank $$r$$ and the slope $$\mu $$ of $$E$$. We show that the bound is achieved precisely when $$E$$ is a twist of a Steiner bundle. We classify the sheaves $$E$$ such that $$h^{0}(E)$$ is within $$\lfloor \mu (E) \rfloor + 1$$ of $$\beta _{r, \mu }$$. We determine the nonemptiness, irreducibility and dimension of the Brill–Noether loci in the moduli spaces of sheaves on $${\mathbb{P}}^{2}$$ with $$h^{0}(E)$$ in this range. When they are proper subvarieties, these Brill–Noether loci are irreducible though almost always of larger than the expected dimension. 

Abstract Let$$\alpha \colon X \to Y$$be a finite cover of smooth curves. Beauville conjectured that the pushforward of a general vector bundle under$$\alpha $$is semistable if the genus ofYis at least$$1$$and stable if the genus ofYis at least$$2$$. We prove this conjecture if the map$$\alpha $$is general in any component of the Hurwitz space of covers of an arbitrary smooth curveY. 

Abstract Let $$\alpha \colon X \to Y$$ be a general degree $$r$$ primitive map of nonsingular, irreducible, projective curves over an algebraically closed field of characteristic zero or larger than $$r$$. We prove that the Tschirnhausen bundle of $$\alpha $$ is semistable if $$g(Y) \geq 1$$ and stable if $$g(Y) \geq 2$$.