Luis Giraldo

On the existence of Enriques-Fano threefolds of index greater than one

Let X C P-N be an irreducible threefold having a hyperplane section Y that is a smooth Enriques surface and such that X is not a cone over Y. In 1938 Fano claimed a classification of such threefolds; however, due to gaps in his proof, the problem still remains open. In this article we solve the case when Y is the r-th Veronese embedding, for r greater than or equal to 2, of another Enriques surface, by proving that there are no such X. The latter is achieved, among other things, by a careful study of trisecant lines to Enriques surfaces. As another consequence we get precise information on the ideal of an Enriques surface. In a previous paper we had proved that any smooth linearly normal Enriques surface has homogeneous ideal generated by quadrics and cubics. Here we are able to specify when the quadrics are enough, at least scheme-theoretically.

New proof and generalizations of the Demidowitsch–Schneider criterion

A new proof of the Demidowitsch–Schneider criterion on the absence of closed trajectories of R3 is given. The new proof is generalized and applied in several directions: Rn vector fields (n>3),… .

Completeness is determined by any non-algebraic trajectory

Abstract It is proved that any polynomial vector field in two complex variables which is complete on a non-algebraic trajectory is complete.

Jacobian mates for non-singular polynomial maps in C^n with one-dimensional fibers

Let (F2, . . . , Fn) : Cn → Cn−1 be a non-singular polynomial map. We introduce a non-singular polynomial vector field X tangent to the foliation F having as leaves the fibers of the map (F2, . . . , Fn). Suppose that the fibers of the map are irreducible in codimension ≥ 2, that the one forms of time associated to the vector field X are exact along the leaves, and that there is a finite set at the hyperplane at infinity containing all the points necessary to compactify the affine curves appearing as fibers of the map. Then, there is a polynomial F1 (a Jacobian mate) such that the completed map (F1, F2, . . . , Fn) is a local biholomorphism. Our proof extends the integration method beyond the known case of planar curves (introduced by Ilyashenko [Ily69]).

Chow forms of congruences

For X PN an n-dimensional variety the set of linear spaces of dimension N − n − 1 meeting X defines a hypersurface, H, in the Grassmann variety G(N − n,N + 1). The homogeneous form in the Pl¨ucker coordinates defining H or H itself is called the Chow form of X. This notion was defined by Cayley [A. Cayley, “On a new analytical representation of curves in space”, Q. J. Pure Appl. Math. 3, 225-236 (1860), and 5, 81-86 (1862); for a modern treatment see M. Green and I. Morrison, Duke Math. J. 53, 733-747 (1986; Zbl 0621.14028)]. In the present paper the authors study Chow forms of integral surfaces in G(2, 4) following the approach of M. Green and I. Morrison. Let V be a fixed 4-dimensional space and F P3 סP3, the flag variety parametrizing all chains V1 V3, where Vi is a subspace of V with dim Vi = i. F parametrizes the lines of G and to each integral surface Y in G there corresponds, in a natural way, an integral hypersurface X in F. The main result in this paper is a characterization of integral hypersurfaces X in F that are Chow forms of integral surfaces in G, in terms of some differential equations.