We prove a conjecture of Gavril Farkas claiming that for all integers r ≥ 2 and g ≥ r+2 2there exists a component of the locus S r g of spin curves with a theta characteristic L such that h 0 (L) ≥ r + 1 and h 0 (L) ≡ r + 1(mod 2) which has codimension r+1 2 inside the moduli space S g of spin curves of genus g.

We prove uniruledness of some moduli spaces $\bar{M}_{g,n}$ of stable curves of genus $g$ with $n$ marked points using linear systems on nonsingular projective surfaces containing the general curve of genus $g$. Precisely we show that $\bar{M}_{g,n}$ is uniruled for $g=12$ and $n \leq 5$, $g=13$ and $n \leq 3$, $g=15$ and $n \leq 2$. We then prove that the pointed hyperelliptic locus $H_{g,n}$ is uniruled for $g \geq 2$ and $n \leq 4g+4$. In the last part we show that a nonsingular complete intersection surface does not carry a linear system containing the general curve of genus $g \geq 16$ and if it carries a linear system containing the general curve of genus $12 \leq g \leq 15$ then it is canonical.Comment: final version, minor improvements in the expositio

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Let V r d,g,δ be the Hilbert scheme of nodal curves in P r of degree d and arithmetic genus g with δ nodes. Under suitable numerical assumptions on d and g, for every 0 ≤ δ ≤ g we construct an irreducible component of V r d,g,δ having the expected number of moduli.

Let (S, L) be a smooth primitively polarized K3 surface of genus g and f : X → P 1 the fibration defined by a linear pencil in |L|. For f general and g ≥ 7, we work out the splitting type of the locally free sheaf Ψ * f T Mg , where Ψ f is the modular morphism associated to f . We show that this splitting type encodes the fundamental geometrical information attached to Mukai's projection map P g → M g , where P g is the stack parameterizing pairs (S, C) with (S, L) as above and C ∈ |L| a stable curve. Moreover, we work out conditions on a fibration f to induce a modular morphism Ψ f such that the normal sheaf N Ψ f is locally free.