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A family of algebraic curves covering a projective variety $X$ is called a web of curves on $X$ if it has only finitely many members through a general point of $X$. A web of curves on $X$ induces a web-structure, in the sense of local differential geometry, in a neighborhood of a general point of $X$. We study how the local differential geometry of the web-structure affects the global algebraic geometry of $X$. Under two geometric assumptions on the web-structure, the pairwise non-integrability condition and the bracket-generating condition, we prove that the local differential geometry determines the global algebraic geometry of $X$, up to generically finite algebraic correspondences. The two geometric assumptions are satisfied, for example, when $X subset {bf P}^N$ is a Fano submanifold of Picard number 1, and the family of lines covering $X$ becomes a web. In this special case, we have a stronger result that the local differential geometry of the web-structure determines $X$ up to biregular equivalences. As an application, we show that if $X, X subset {bf P}^N, dim X geq 3,$ are two such Fano manifolds of Picard number 1, then any surjective morphism $f: X to X$ is an isomorphism.
A nilmanifold is a (left) quotient of a nilpotent Lie group by a cocompact lattice. A hypercomplex structure on a manifold is a triple of complex structure operators satisfying the quaternionic relations. A hypercomplex nilmanifold is a compact quoti
In this paper, we adapt the differential signature construction to the equivalence problem for complex plane algebraic curves under the actions of the projective group and its subgroups. Given an action of a group $G$, a signature map assigns to a pl
We explain that in the study of the asymptotic expansion at the origin of a period integral like $gamma$z $omega$/df or of a hermitian period like f =s $rho$.$omega$/df $land$ $omega$ /df the computation of the Bernstein polynomial of the fresco (fil
Using Galois theory, we construct explicitly (in all complex dimensions >1) an infinite family of simple complex tori of algebraic dimension 0 with Picard number 0.
This is an expanded version of the two papers Interpolation of Varieties of Minimal Degree and Interpolation Problems: Del Pezzo Surfaces. It is well known that one can find a rational normal curve in $mathbb P^n$ through $n+3$ general points. More r