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We study intersection theory for differential algebraic varieties. Particularly, we study families of differential hypersurface sections of arbitrary affine differential algebraic varieties over a differential field. We prove the differential analogue of Bertini's theorem, namely that for an arbitrary geometrically irreducible differential algebraic variety which is not an algebraic curve, generic hypersurface sections are geometrically irreducible and codimension one. Surprisingly, we prove a stronger result in the case that the order of the differential hypersurface is at least one; namely that the generic differential hypersurface sections of an irreducible differential algebraic variety are irreducible and codimension one. We also calculate the Kolchin polynomials of the intersections and prove several other results regarding intersections of differential algebraic varieties. [ABSTRACT FROM AUTHOR]

We prove that if a symplectic 4-manifold X becomes a rational 4-manifold after applying rational blow-down surgery, then the symplectic 4-manifold X is originally rational. That is, a symplectic rational blow-up of a rational symplectic 4-manifold is again rational. As an application we show that a degeneration of a family of smooth rational complex surfaces is a rational surface if the degeneration has at most quotient surface singularities, which generalizes slightly a classical result of Bădescu [J. Reine Angew. Math. 367 (1986), pp. 76–89] in algebraic geometry under a mild additional condition. [ABSTRACT FROM AUTHOR]

Cross-ratio degrees count configurations of points z_1,\ldots,z_n\in \mathbb {P}^1 satisfying n-3 cross-ratio constraints, up to isomorphism. These numbers arise in multiple contexts in algebraic and tropical geometry, and may be viewed as combinatorial invariants of certain hypergraphs. We prove an upper bound on cross-ratio degrees in terms of the theory of perfect matchings on bipartite graphs. We also discuss several of the many perspectives on cross-ratio degrees—including a connection to Gromov-Witten theory—and give many example computations. [ABSTRACT FROM AUTHOR]