MSCA.24.Mallory

Objective:

SIPOCAG aims to understand and classify singularities occurring in positive-characteristic from two points of view: First, to understand the indecomposable summands arising via the Frobenius morphism of a ring or a variety, and second, to understand the relation between the theory of arc/jet schemes and the "standard" singularities of positive-characteristic commutative algebra. Towards the first objective, given a ring R in characteristic p, one obtains a new module by restricting scalars along the Frobenius self-map on R. The properties of resulting module reflect the singularities of R; for example, Kunz showed that this module is flat if and only if R is regular. I seek to understand what the indecomposable summands of this module tell us about R, especially as one restricts scalars along higher iterates of the Frobenius. In particular, SIPOCAG focuses on the question of how many different isomorphism classes of indecomposables show up in this process: I propose to understand when this number is finite, and what the types and nature of the summands tell us about the ring R. Towards the second objective, the arc scheme of a variety X and its truncation, the jet schemes, are moduli spaces of infinitesimal curves on X, and have proved useful in studying singularities in birational geometry. However, the relation between the arc/jet schemes and the usual notions of singularities in commutative algebra, such as F-purity, remains unknown. SIPOCAG will characterize such classes of singularities via the study of their arc/jet schemes, and leverage these connections to better understand the behavior of these singularities. Finally, I propose to develop useful new computational tools for working with concrete examples of the preceding two phenomena, which will be useful not only for SIPOCAG but also to many other researchers. These goals will be approached by combining my existing geometric approach to these topics with the algebraic expertise of the host.