Rational Curves on Algebraic Varieties

This book provides an introduction to algebraic cycles on complex algebraic varieties, to the major conjectures relating them to cohomology, and even more precisely to Hodge structures on cohomology. The book is intended for both students and researchers, and not only presents a survey of the geometric methods developed in the last thirty years to understand the famous Bloch-Beilinson conjectures, but also examines recent work by the author. It focuses on two central objects: the diagonal of a variety—and the partial Bloch-Srinivas type decompositions it may have depending on the size of Chow groups—as well as its small diagonal, which is the right object to consider in order to understand the ring structure on Chow groups and cohomology. An exploration of a sampling of recent works by the author looks at the relation, conjectured in general by Bloch and Beilinson, between the coniveau of general complete intersections and their Chow groups and a very particular property satisfied by the Chow ring of K3 surfaces and conjecturally by hyper-Kähler manifolds. In particular, the book delves into arguments originating in Nori's work that have been further developed by others.

Download Full-text

Geometric Discriminate Methods of a Class of Quartic Offset-rational Curves

Journal of Computer-Aided Design & Computer Graphics ◽

10.3724/sp.j.1089.2018.16368 ◽

2018 ◽

Vol 30 (3) ◽

pp. 499

Author(s):

Xiaojuan Duan ◽

Guozhao Wang

Keyword(s):

Rational Curves

Download Full-text

Non Kählerian surfaces with a cycle of rational curves

Complex Manifolds ◽

10.1515/coma-2020-0114 ◽

2021 ◽

Vol 8 (1) ◽

pp. 208-222

Author(s):

Georges Dloussky

Keyword(s):

Deformation Theory ◽

Complex Surface ◽

Rational Curves ◽

Intersection Form ◽

Connected Component ◽

Hirzebruch Surface ◽

Compact Complex Surface ◽

A Chain

Abstract Let S be a compact complex surface in class VII0 + containing a cycle of rational curves C = ∑Dj . Let D = C + A be the maximal connected divisor containing C. If there is another connected component of curves C ′ then C ′ is a cycle of rational curves, A = 0 and S is a Inoue-Hirzebruch surface. If there is only one connected component D then each connected component Ai of A is a chain of rational curves which intersects a curve Dj of the cycle and for each curve Dj of the cycle there at most one chain which meets Dj . In other words, we do not prove the existence of curves other those of the cycle C, but if some other curves exist the maximal divisor looks like the maximal divisor of a Kato surface with perhaps missing curves. The proof of this topological result is an application of Donaldson theorem on trivialization of the intersection form and of deformation theory. We apply this result to show that a twisted logarithmic 1-form has a trivial vanishing divisor.

Download Full-text