Counterexamples to Bertini Theorems for Test Ideals: In this paper I examine a conditions on which the test ideal does not descend to a general hyperplane section of a variety, something that holds true by analogy in characteristic 0. This answers questions of Hochster-Huneke and Schwede-Zhang in the negative.

SectionRing.m2: A Macaulay2 package meant to compute the regularity of a sheaf, the power for which a divisor is globally generated, and the section ring associated to an ample (not necessarily globally generated) sheaf using Castelnuovo-Mumford regularity.

TestForHypersurfaceBehaviorSigma.m2 Some Macaulay2 code to check whether or not one can find a particular degree (or specific equation) for which there are lines through the origin which violate a restriction theorem for sigma ideal or test ideals in other specific scenarios.