Are you an EPFL student looking for a semester project?
Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.
Concept
Cone of curves
Summary
In mathematics, the cone of curves (sometimes the Kleiman-Mori cone) of an algebraic variety is a combinatorial invariant of importance to the birational geometry of .
Let be a proper variety. By definition, a (real) 1-cycle on is a formal linear combination of irreducible, reduced and proper curves , with coefficients . Numerical equivalence of 1-cycles is defined by intersections: two 1-cycles and are numerically equivalent if for every Cartier divisor on . Denote the real vector space of 1-cycles modulo numerical equivalence by .
We define the cone of curves of to be
where the are irreducible, reduced, proper curves on , and their classes in . It is not difficult to see that is indeed a convex cone in the sense of convex geometry.
One useful application of the notion of the cone of curves is the Kleiman condition, which says that a (Cartier) divisor on a complete variety is ample if and only if for any nonzero element in , the closure of the cone of curves in the usual real topology. (In general, need not be closed, so taking the closure here is important.)
A more involved example is the role played by the cone of curves in the theory of minimal models of algebraic varieties. Briefly, the goal of that theory is as follows: given a (mildly singular) projective variety , find a (mildly singular) variety which is birational to , and whose canonical divisor is nef. The great breakthrough of the early 1980s (due to Mori and others) was to construct (at least morally) the necessary birational map from to as a sequence of steps, each of which can be thought of as contraction of a -negative extremal ray of . This process encounters difficulties, however, whose resolution necessitates the introduction of the flip.
The above process of contractions could not proceed without the fundamental result on the structure of the cone of curves known as the Cone Theorem. The first version of this theorem, for smooth varieties, is due to Mori; it was later generalised to a larger class of varieties by Kawamata, Kollár, Reid, Shokurov, and others.
Official source
About this result
This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.