Automorphisms of smooth hypersurfaces and Gizatullin's problem

Let $X_d \subset \mathbb{P}^{n+1}$ be a smooth hypersurface of degree d. Classical theorems of Matsumura-Monsky and Chang state that every automorphism of $X_d$ is the restriction of a linear automorphism of the ambient space, except in the two exceptional cases (n,d) = (1,3) and (2,4). Nonetheless, when (n,d) = (1,3), every automorphism of $X_3$ can be realized as the restriction of a birational map of $\mathbb{P}^{2}$.  Gizatullin’s problem concerns the last open case, i.e. which automorphisms of a smooth quartic surface $X_4 \subset \mathbb{P}^{3}$ are restrictions of birational selfmaps of the ambient space.
In this talk we will employ the machinery of log Calabi-Yau pairs, and more specifically that of the volume preserving Sarkisov program, to completely address Gizatullin’s problem for the general non-trivial case: smooth quartics of Picard rank 2. We will see that the answer can be either positive or negative, the dependence being on the existence of certain curves on the quartic. This is joint work with Carolina Araujo and Daniela Paiva.