Sofie Angere: Inverse Systems with Applications to Ideals of Projective Points

Abstract.

We study Macaulay’s concept of an inverse system of a polynomial ideal, largely in the form it was given by Emsalem and Iarrobino in their paper “Inverse system of a symbolic power, I”. One of our main goals is to present a version of their theorem giving the inverse system of the intersection of ideals, each of which describes a projective point and is raised to a power. At the same time, we wish to make a deeper exploration of the concepts involved. In particular, we investigate some of the linear algebraic properties of the operators that are used to define inverse systems, and also highlight the simple form these systems take for monomial ideals, as well as for ideals that can be reduced to monomial ones through suitable linear transformations. Finally, we use the results we have gathered to briefly explore two new problems: the inverse system of an ideal of several projective points that is together raised to a power, and the relationship between inverse systems and coordinate rings.