Emre Sertöz

We compute the Picard groups of smooth surfaces in P3 using high precision computation of periods and lattice reduction techniques. Our approach applies more generally to the computation of the lattice generated by Hodge cycles of middle dimension on smooth projective hypersurfaces. We will explain the lattice reduction technique and tie the possibility of numerical error to an intrinsic measure of complexity of the hypersurfaces. As an application of the Picard group computation, we will count smooth rational curves on smooth quartic surfaces (K3) and compute the endomorphism ring of their transcendental lattices. This is joint work with Pierre Lairez.