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Abstract. In this paper we study the Brill Noether locus of rank 2, (semi)stable vector bundles with at least two sections and of suitable degrees on a general ν-gonal curve. We classify its reduced components whose dimensions are at least the corresponding Brill-Noether number. We moreover describe the general member F of such components just in terms of extensions of line bundles with suitable minimality properties, providing information on the birational geometry of such components as well as on the very-ampleness of F .

In this paper we consider curves on a cone that pass through the vertex and are also triple covers of the base of the cone, which is a general smooth curve of genus γ and degree e in P e−γ . Using the free resolution of the ideal of such a curve found by Catalisano and Gimigliano, and a technique concerning deformations of curves introduced by Ciliberto, we show that the deformations of such curves remain on cones over a deformation of the base curve. This allows us to prove that for γ ≥ 3 and e ≥ 4γ + 5 there exists a non-reduced component H of the Hilbert scheme of smooth curves of genus 3e + 3γ and degree 3e + 1 in P e−γ+1 . We show that dim T [X] H = dim H + 1 = (e − γ + 1) 2 + 7e + 5 for a general point [X] ∈ H.

Let I d,g,r be the union of irreducible components of the Hilbert scheme whose general points correspond to smooth irreducible non-degenerate curves of degree d and genus g in P r . We use families of curves on cones to show that under certain numerical assumptions for d, g and r, the scheme I d,g,r acquires generically smooth components whose general points correspond to curves that are double covers of irrational curves. In particular, in the case ρ(d, g, r) := g − (r + 1)(g − d + r) ≥ 0 we construct explicitly a regular component that is different from the distinguished component of I d,g,r dominating the moduli space M g .

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