The problem of resolution of singularities in positive characteristic can be reformulated as follows: fix a hypersurface X, embedded in a smooth scheme, with points of multiplicity at most n. Let an n-sequence of transformations of X be a finite composition of monoidal transformations with centers included in the n-fold points of X, and of its successive strict transforms. The open problem (in positive characteristic) is to prove that there is an n-sequence such that the final strict transform of X has no points of multiplicity n (no n-fold points). In characteristic zero, such an n-sequence is defined in two steps. The first consists of the transformation of X to a hypersurface with n-fold points in the so-called monomial case. The second step consists of the elimination of these n-fold points (in the monomial case), which is achieved by a simple combinatorial procedure for choices of centers. The invariants treated in this work allow us to present a notion of strong monomial case which parallels that of monomial case in characteristic zero: if a hypersurface is within the strong monomial case we prove that a resolution can be achieved in a combinatorial manner.
We give an overview of invariants of algebraic singularities over perfect fields. We then show how they lead to a synthetic proof of embedded resolution of singularities of 2-dimensional schemes.
Abstract. We show that locally acyclic cluster algebras have (at worst) canonical singularities. In fact, we prove that locally acyclic cluster algebras of positive characteristic are strongly F -regular. In addition, we show that upper cluster algebras are always Frobenius split by a canonically defined splitting, and that they have a free canonical module of rank one. We also give examples to show that not all upper cluster algebras are F-regular if the local acyclicity is dropped.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.