** Ananyo Dan**

TIFR

January 28, 2016

**Geometry of the Noether-Lefschetz locus**:
For an integer d ge 4, the Noether-Lefschetz theorem tells us that a
very general smooth degree d surface in mathbb{P}^3 has Picard number
equal to one. We define Noether-Lefschetz locus to be the space
parametrizes smooth degree d hypersurfaces in mathbb{P}^3 with Picard
number at least 2 i.e., violating the Noether-Lefschetz theorem. As part of my Ph.D., I studied various relations between the geometry of the Noether-Lefschetz locus and that of the Hilbert schemes of curves. I try to review existing and new results in this direction.