Frobenius · Frobenius related invariants and singularities

Frobenius related invariants and singularities

This is a project in commutative algebra, which has also many connections with algebraic geometry. In fact, commutative algebra can be seen also as the algebraic approach to geometrical problems, that sometimes can be better solved in algebraic terms. In this project, a particular emphasis is given to problems related to positive characteristic, that is the study of rings, and the corresponding varieties, over fields of prime characteristic. Positive characteristic methods are difficult and require new approaches, quite different to the ones in characteristic zero. On the other hand, one often can use powerful techniques based on the Frobenius homomorphism, and therefore only available in positive characteristic, to get better results that afterwards can be lifted to characteristic zero. There are three main mathematical objects of investigation in this project: 1) the F-signature function; 2) the differential symmetric signature; 3) the Veronese compactification V_{d,n}. 1) The F-signature is a numerical function which can be defined only for rings of positive characteristic by looking at the asymptotic splitting properties of the Frobenius homomorphism. It has been introduced in 2004 by Huneke and Leuschke, who continued previous ideas of Smith and Van den Bergh. The focus so far has been mainly on the leading term of the function, simply called F-signature, which already encodes a significant amount of information about the ring and its singularities. Recently, Polstra and Tucker raised the question of whether a "second coefficient" for the function exists as well, i.e. whether the function can be written as f(x)=a*x^d+b*x^{d-1}+O(x^{d-2}), where a and b are real numbers. In this project, we investigate this question for specific classes of rings. 2) The differential symmetric signature is a new numerical invariant defined by the Experienced Researcher in his Ph.D. thesis in the attempt to find a characteristic-free version of the F-signature. The differential symmetric signature proved to be equal to the F-signature for two-dimensional Kleinian singularities and cones over elliptic curves. However, these examples are limited to the two-dimensional situation. In this project we investigate the differential symmetric signature for higher dimensional examples. 3) The Veronese compactification V_{d,n} is an algebraic projective variety which can be seen as the closure in the Zariski topology of the configuration space of n points in d-dimensional projective space lying on a common rational normal curve. The Veronese compactification is defined in every characteristic and has appeared in several settings in the last years, in particular in connection with certain moduli spaces. Recently, Speyer and Sturmfels raised the question of finding explicit equations that cut out V_{d,n} set-theoretically. In this project we investigate this question.

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