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In the group of polynomial automorphisms of the plane, the conjugacy class of an element is closed if and only if the element is diagonalisable. In this article, we show that this does not hold for the group of special automorphisms, giving a first step in the direction of showing that this group is not simple, as an infinite-dimensional algebraic group.
We prove several results about the behavior of Du Bois singularities and Du Bois pairs in families. Some of these generalize existing statements about Du Bois singularities to the pair setting while others are new even in the nonpair setting. We also prove a new inversion of adjunction result for Du Bois and rational pairs. In the nonpair setting this asserts that if a family over a smooth base has a special fiber with Du Bois singularities and the general fiber has rational singularities, then the total space has rational singularities near .
For a flat commutative -algebra such that the enveloping algebra is noetherian, given a finitely generated bimodule , we show that the adic completion of the Hochschild cohomology module is naturally isomorphic to . To show this, we make a detailed study of derived completion as a functor over a nonnoetherian ring , prove a flat base change result for weakly proregular ideals, and prove that Hochschild cohomology and analytic Hochschild cohomology of complete noetherian local rings are isomorphic, answering a question of Buchweitz and Flenner. Our results make it possible for the first time to compute the Hochschild cohomology of over any noetherian ring , and open the door for a theory of Hochschild cohomology over formal schemes.
We prove the equivalence of dynamical stability, preperiodicity, and canonical height 0, for algebraic families of rational maps , parameterized by in a quasiprojective complex variety. We use this to prove one implication in the if-and-only-if statement of a certain conjecture on unlikely intersections in the moduli space of rational maps (see “Special curves and postcritically finite polynomials”, Forum Math. Pi 1 (2013), e3). We present the conjecture here in a more general form.
The behavior of the Frobenius map is investigated for valuation rings of prime characteristic. We show that valuation rings are always F-pure. We introduce a generalization of the notion of strong F-regularity, which we call F-pure regularity, and show that a valuation ring is F-pure regular if and only if it is Noetherian. For valuations on function fields, we show that the Frobenius map is finite if and only if the valuation is Abhyankar; in this case the valuation ring is Frobenius split. For Noetherian valuation rings in function fields, we show that the valuation ring is Frobenius split if and only if Frobenius is finite, or equivalently, if and only if the valuation ring is excellent.
One of the most significant discrete invariants of a quadratic form over a field is its (full) splitting pattern, a finite sequence of integers which describes the possible isotropy behavior of under scalar extension to arbitrary overfields of . A similarly important but more accessible variant of this notion is that of the Knebusch splitting pattern of , which captures the isotropy behavior of as one passes over a certain prescribed tower of -overfields. We determine all possible values of this latter invariant in the case where is totallysingular. This includes an extension of Karpenko’s theorem (formerly Hoffmann’s conjecture) on the possible values of the first Witt index to the totally singular case. Contrary to the existing approaches to this problem (in the nonsingular case), our results are achieved by means of a new structural result on the higher anisotropic kernels of totally singular quadratic forms. Moreover, the methods used here readily generalize to give analogous results for arbitrary Fermat-type forms of degree over fields of characteristic .
The zeta function of a K3 surface over a finite field satisfies a number of obvious (archimedean and -adic) and a number of less obvious (-adic) constraints. We consider the converse question, in the style of Honda–Tate: given a function satisfying all these constraints, does there exist a K3 surface whose zeta-function equals ? Assuming semistable reduction, we show that the answer is yes if we allow a finite extension of the finite field. An important ingredient in the proof is the construction of complex projective K3 surfaces with complex multiplication by a given CM field.