## ANGEL DAVID RIOS ORTIZ

PhD Graduate**PhD program:**: XXXIV

**supervisor**: D. Fiorenza

**advisor**: K. O'Grady

**Thesis title:**Linear Systems on Hyperkähler Manifolds

This thesis studies linear systems of divisors on Hyperkähler manifolds. A relationship between Hilbert squares on a K3 surface and Gaussian maps is established, then is used for the study of Gaussian maps on canonical curves. An infinite family of non-divisorial base loci for ample divisors is constructed for Hilbert schemes of points on K3 surfaces. The formulae for the Euler characteristic of divisors is completed for the last two known examples of Hyperkähler manifolds. In addition, a general theorem is proved on the asymptotic base loci of big divisors on Hyperkähler manifolds.