Birational Geometry and Localisation of Categories With Appendices
نویسندگان
چکیده
We explore connections between places of function fields over a base field F and birational morphisms between smooth F varieties. This is done by considering various categories of fractions involving function fields or varieties as objects, and constructing functors between these categories. The main result is that in the localised category S b Sm(F ), where Sm(F ) denotes the usual category of smooth varieties over F and Sb is the set of birational morphisms, the set of morphisms between two objects X and Y , with Y proper, is the set of R-equivalence classes Y (F (X))/R. 2010 Mathematics Subject Classification: 14E05, 18F99
منابع مشابه
Birational Geometry and Localisation of Categories
The basic theme of this paper is to explore connections between places of function fields over a base field F of characteristic zero and birational morphisms between smooth F -varieties. This is done by considering various localised categories involving function fields or varieties as objects, and constructing functors between these categories. The main result is that in the localised category ...
متن کاملBirational motives, I: pure birational motives
In the preprint [19], we toyed with birational ideas in three areas of algebraic geometry: plain varieties, pure motives in the sense of Grothendieck, and triangulated motives in the sense of Voevodsky. These three themes are finally treated separately in revised versions. The first one was the object of [21]; the second one is the object of the present paper; we hope to complete the third one ...
متن کامل2 7 Fe b 20 09 BIRATIONAL MOTIVES , I : PURE BIRATIONAL MOTIVES
In the preprint [19], we toyed with birational ideas in three areas of algebraic geometry: plain varieties, pure motives in the sense of Grothendieck, and triangulated motives in the sense of Voevodsky. These three themes are finally treated separately in revised versions. The first one was the object of [21]; the second one is the object of the present paper; we hope to complete the third one ...
متن کاملA few localisation theorems
Given a functor T : C → D carrying a class of morphisms S ⊂ C into a class S ⊂ D, we give sufficient conditions in order that T induces an equivalence on the localised categories. These conditions are in the spirit of Quillen’s theorem A. We give some applications in algebaic and birational geometry. Introduction Let T : C → D be a functor and S ⊂ C, S ⊂ D two classes of morphisms containing id...
متن کاملBirational Calabi-Yau 3-folds and BPS state counting
This paper contains some applications of Bridgeland-Douglas stability conditions on triangulated categories, and Joyce’s work on counting invariants of semistable objects, to the study of birational geometry. We introduce the notion of motivic Gopakumar-Vafa invariants as counting invariants of D2-branes, and show that they are invariant under birational transformations between Calabi-Yau 3-fol...
متن کامل