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Asymptotic Growth of Symbolic Powers, Mixed Multiplicities, and Convex Bodies

08-21-2020 - The symbolic powers of an ideal encode important algebraic and geometric information of the ideal and the variety it defines. The PI proposes to investigate the asymptotic behavior of symbolic powers by studying the growth of three important sequences, the number of generators, the (Castelnuovo-Mumford) regularity, and the projective dimension. For the number of generators, the PI plans to show that the sequence has polynomial complexity. This result would have important consequences on the arithmetic rank, Frobenius complexity, and Kodaira dimension of divisors. He plans to achieve this by using the theory of cohomological degrees, a generalization of the Hilbert-Samuel multiplcity. For the other two sequences, regularity and projective dimension, the PI plans to focus on regular rings of positive characteristic. In such rings, a new class of ideals is defined for which the PI has previously shown the limit of these sequences exist; this class includes several types of determinantal ideals, as well as the square-free monomial ideals. The PI proposes possible ways to show the existence of these limits for more general classes of ideals. The relation between multiplicities and convex bodies is an important research topic lying in the interaction of commutative algebra, algebraic geometry, and combinatorics. In recent years, this line of research has gained much activity due to the introduction of Newton-Okounkov bodies and mixed multiplicities of filtrations of ideals. As a first project in this framework, the PI proposes a conjecture that establishes conditions for the non-vanishing of mixed multiplicities of multigraded algebras. A positive answer for this conjecture would unify results in the literature coming from several different contexts. As a second project, the PI proposes a notion of mixed multiplicities for filtrations of not necessarily zero dimensional ideals. Moreover, he proposes a relation of these new multiplicities with the mixed volumes of certain Newton-Okounkov bodies.