Search

The purpose of this paper is fourfold. The first is to develop the theory of tropical differential algebraic geometry from scratch; the second is to present the tropical fundamental theorem for differential algebraic geometry, and show how it may be used to extract combinatorial information about the set of power series solutions to a given system of differential equations, both in the archimedean (complex analytic) and in the non-Archimedean (e.g., p-adic) setting. A third and subsidiary aim is to show how tropical differential algebraic geometry is a natural application of semiring theory, and in so doing, contribute to the valuative study of differential algebraic geometry. We use this formalism to extend the fundamental theorem of partial differential algebraic geometry to the differential fraction field of the ring of formal power series in arbitrarily (finitely many variables; in doing so we produce new examples of non-Krull valuations that merit further study in their own right. [ABSTRACT FROM AUTHOR]

We classify projective manifolds X of dimension n ≥ 3 with nef anticanonical class and index n – 1 such that . [ABSTRACT FROM AUTHOR]

In this paper we improve a classical result of Van de Ven which characterizes linear subspaces of as the only smooth closed subvarieties of for which the normal sequence splits (see [A. Van de Ven, A property of algebraic varieties in complex projective spaces. In: Colloque Géom. Diff. Globale ( Bruxelles, 1958), 151–152, Centre Belge Rech. Math., Louvain 1959. MR0116361 (22 #7149) Zbl 0092.14004]). Precisely we prove the following: Let X be a submanifold of of dimension ≥ 3, and Y ⊆ X a submanifold of X of dimension ≥ 2. Assume that Span( Y) = Span( X), where Span( X) is the smallest linear subspace of containing X. Then the exact sequence: splits if and only if X is a linear subspace of . [ABSTRACT FROM AUTHOR]