UC San Diego

An additive action is an effective action of Gn a on an n-dimensional variety with an open orbit, and an additive variety is a normal variety admitting an additive action. In the first half of this dissertation, we study additive actions on projective space, which are classified by certain local Artinian algebras, and we provide a geometric explanation for the unexpected appearance of some moduli spaces within this classification. We also describe the G2 a -equivariant Sarkisov program for smooth additive surfaces. In the second half, motivated by the fact that not all additive varieties are Mori dream spaces, we present some conjectures about extremal rays in the Kleiman–Mori cone of additive varieties, and we prove them in specific cases.