Let be a projective variety which is covered by rational curves, for instance a Fano manifold over the complex numbers. In this paper, we give sufficient conditions which guarantee that every tangent vector at a general point of is contained in at most one rational curve of minimal degree. As an immediate application, we obtain irreducibility criteria for the space of minimal rational curves.
We survey recent work on arithmetic analogues of ordinary and partial differential equations.
We prove an arithmetic analogue of Fujita’s approximation theorem in Arakelov geometry, conjectured by Moriwaki, by using measures associated to -filtrations.
We give an estimation for the arithmetic genus of an integral space curve which is not contained in a surface of degree . Our main technique is the Bogomolov-Gieseker type inequality for proved by Macrì.
Let be a rationally connected algebraic variety, defined over a number field We find a relation between the arithmetic of rational points on and the arithmetic of zero-cycles. More precisely, we consider the following statements: (1) the Brauer-Manin obstruction is the only obstruction to weak approximation for -rational points on for all finite extensions (2) the Brauer-Manin obstruction is the only obstruction to weak approximation in some sense that we define for zero-cycles of degree...