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In this article we are going to address the following issues: (1) the first is a rigidity property for pairs (S, C) consisting of a general projective K 3 surface S, and a curve C obtained as the normalization of a nodal, hyperplane sectionĈ → S. We prove that a non-trivial deformation of a pair (S, C) induces a non-trivial deformation of C; (2) the second question concerns the Wahl map of curves C obtained as above. We prove that the Wahl map of the normalization of a nodal curve contained in a general projective K 3 surface is non-surjective. In both cases, we impose upper bounds on the number of nodes ofĈ.

The goal of this article is to define partially ample subvarieties of projective varieties, generalizing Ottem's work on ample subvarieties, and also to show their ubiquity. As an application, we obtain a connectedness result for pre-images of subvarieties by morphisms, reminiscent to a problem posed by Fulton-Hansen in the late 1970s. Similar criteria are not available in the literature.

Let π : Y →X be a surjective morphism between two irreducible, smooth complex projective varieties with dim Y > dim X>0. We consider polarizations of the form Lc = L+c·π * A on Y , with c > 0, where L, A are ample line bundles on Y, X respectively.For c sufficiently large, we show that the restriction of a torsion free sheaf F on Y to the generic fibre Φ of π is semi-stable as soon as F is Lc-semi-stable; conversely, if F ⊗ O Φ is L-stable on Φ, then F is Lc-stable. We obtain explicit lower bounds for c satisfying these properties. Using this result, we discuss the construction of semi-stable vector bundles on Hirzebruch surfaces and on P 2 -bundles over P 1 , and establish the irreducibility and the rationality of the corresponding moduli spaces.

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