The Tangent Cones at Double points of Prym-Canonical Divisors of Curves of genus 7

Let η be a line bundle on a smooth curve X with η^2=0 such that π_η, the double covering induced by η is an etale morphism. Assume also that X_η be the Prym-canonical model of X associated to K_X.η and Q is a rank 4 quadric containing X_η. After stablishing the projective normality of the prym-canonical models of curves X with Clifford index 2, we obtain in this paper a sufficient condition for Q to be a tangent cone. Under some circumstances, we obtain an inverse to this sufficient condition. We make our study complete by presenting an example of a curve of genus 7 and gonality 4 in chapter 4. The importance of our example stems from its elementary nature and the fact that it is generalizable.