Registered users receive a variety of benefits including the ability to customize email alerts, create favorite journals list, and save searches.
Please note that a Project Euclid web account does not automatically grant access to full-text content. An institutional or society member subscription is required to view non-Open Access content.
Contact firstname.lastname@example.org with any questions.
Building on our earlier results on tropical independence and shapes of divisors in tropical linear series, we give a tropical proof of the maximal rank conjecture for quadrics. We also prove a tropical analogue of Max Noether’s theorem on quadrics containing a canonically embedded curve, and state a combinatorial conjecture about tropical independence on chains of loops that implies the maximal rank conjecture for algebraic curves.
We present an effective criterion to determine if a normal analytic compactification of with one irreducible curve at infinity is algebraic or not. As a byproduct we establish a correspondence between normal algebraic compactifications of with one irreducible curve at infinity and algebraic curves contained in with one place at infinity. Using our criterion we construct pairs of homeomorphic normal analytic surfaces with minimally elliptic singularities such that one of the surfaces is algebraic and the other is not. Our main technical tool is the sequence of key forms — a “global” variant of the sequence of key polynomials introduced by MacLane  to study valuations in the “local” setting — which also extends the notion of approximate roots of polynomials considered by Abhyankar and Moh [19 73].
We solve the local lifting problem for the alternating group , thus showing that it is a local Oort group. Specifically, if is an algebraically closed field of characteristic , we prove that every -extension of lifts to characteristic zero.
We show that the logarithmic version of the syntomic cohomology of Fontaine and Messing for semistable varieties over -adic rings extends uniquely to a cohomology theory for varieties over -adic fields that satisfies -descent. This new cohomology — syntomic cohomology — is a Bloch–Ogus cohomology theory, admits a period map to étale cohomology, and has a syntomic descent spectral sequence (from an algebraic closure of the given field to the field itself) that is compatible with the Hochschild–Serre spectral sequence on the étale side and is related to the Bloch–Kato exponential map. In relative dimension zero we recover the potentially semistable Selmer groups and, as an application, we prove that Soulé’s étale regulators land in the potentially semistable Selmer groups.
Our construction of syntomic cohomology is based on new ideas and techniques developed by Beilinson and Bhatt in their recent work on -adic comparison theorems.
For any root system and any commutative ring, we give a relatively simple presentation of a group related to its Steinberg group . This includes the case of infinite root systems used in Kac–Moody theory, for which the Steinberg group was defined by Tits and Morita–Rehmann. In most cases, our group equals , giving a presentation with many advantages over the usual presentation of . This equality holds for all spherical root systems, all irreducible affine root systems of rank , and all -spherical root systems. When the coefficient ring satisfies a minor condition, the last condition can be relaxed to -sphericity.
Our presentation is defined in terms of the Dynkin diagram rather than the full root system. It is concrete, with no implicit coefficients or signs. It makes manifest the exceptional diagram automorphisms in characteristics and , and their generalizations to Kac–Moody groups. And it is a Curtis–Tits style presentation: it is the direct limit of the groups coming from - and -node subdiagrams of the Dynkin diagram. Over nonfields this description as a direct limit is new and surprising. Our main application is that many Steinberg and Kac–Moody groups over finitely generated rings are finitely presented.