• Brill-noether theory
• Green's conjecture
• Hilbert scheme
• K3 surfaces
• Koszul cohomology
• Mukai-Lazarsfeld bundle
• Voisin's theorem

Since 1984, year in which it was rst formulated, Green's conjecture has been object of study in algebraic geometry research: my master thesis is devoted to Clair Voisin's solution to such problem.
In particular, Green's conjecture solution stems, as a corollary, from a more general theorem by Voisin.
The most relevant concepts this thesis touches upon are:
Hilbert schemes of point, the Cayley-Bacharach property, the Mukai-Lazarsfeld bundle, the Koszul cohomology and the K3 surfaces.
Being this work a descriptive thesis, it does not lead to any innovative results. However, having built and proved myself most of the theorems and demonstrations covered in the first three chapters, I managed to get in-depth knowledge and get considerably familiarised with many tools and topics typical of modern algebraic geometry.
This thesis contributes to the literature in that it is, leaving Aprodu & Nagel ([2010] chapters 4-6) and Voisin ([2002]) aside, the rst paper fully dedicated to the theorem.
This means that it extensively articulates crucial aspects of the theory that Voisin ([2002]) only quickly touches upon.
In particular, to work on this thesis allowed me to study concepts and focus on some research areas that I had not have the chance to study during my bachelor and master classes.