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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.more » « less
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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 $$g(Y) \geq 2$$.more » « less
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Let C be a general Brill–Noether curve. A classical problem is to determine when H^0(N_C(-2)) = 0, which controls the quadric section of C. So far this problem has only been solved in characteristic zero, in which case H^0(N_C(-2)) = 0 with finitely many exceptions. In this paper, we extend these results to positive characteristic, uncovering a wealth of new exceptions in characteristic 2.more » « less
