Helly\u27s Theorem and its Equivalences via Convex Analysis

Helly\u27s theorem is an important result from Convex Geometry. It gives sufficient conditions for a family of convex sets to have a nonempty intersection. A large variety of proofs as well as applications are known. Helly\u27s theorem also has close connections to two other well-known theorems from Convex Geometry: Radon\u27s theorem and Carathéodory\u27s theorem. In this project we study Helly\u27s theorem and its relations to Radon\u27s theorem and Carathéodory\u27s theorem by using tools of Convex Analysis and Optimization. More precisely, we will give a novel proof of Helly\u27s theorem, and in addition we show in a complete way that these three famous theorems are equivalent in the sense that using one of them allows us to derive the others

Deformation of morphisms, varieties of low codimension and asymptotic limits

In this article, we study the deformations of a finite morphism $\varphi$ to a projective space factoring through an abelian cover of a complete intersection subvariety $Y$ and the intimately connected problem of existence or non existence of multiple structures called ropes on $Y$. From our more general main theorems we obtain the following interesting applications: (1) We construct infinitely many ropes supported on complete intersection subvarieties, embedded in the projective space. We smooth these ropes to obtain infinitely many families of smooth subvarieties with codimension in the range of Hartshorne's conjecture. In addition, we systematically construct smooth non-complete intersection subvarieties, embedded by complete linear series, outside the range of Hartshorne's conjecture, thereby showing many ropes that are not in the boundary of a component of the Hilbert scheme parameterizing smooth complete intersections subvarieties. (2) We go beyond Enriques' original question on constructing simple canonical surfaces in projective spaces, and construct simple canonical varieties in all dimensions. An infinite subset of these simple canonical varieties have finite birational canonical map which is not an embedding. The deformations of the finite morphisms studied in this article display a wide variety of behavior. In particular we show that there are infinitely many families of examples for which a general deformation of the associated finite morphism of degree $n$ remains of degree $n$, infinitely many families of examples for which degree $n$ becomes degree $n/2$ (when $n$ is even), and infinitely many families for which degree $n$ becomes degree $1$. This has potential applications to finding components of moduli spaces of varieties of general type in higher dimensions, as the first two authors and Gonz\'alez showed for surfaces in [GGP2].Comment: 28 pages, comments are welcom