By Ivan Arzhantsev, Hubert Flenner (auth.), Fedor Bogomolov, Brendan Hassett, Yuri Tschinkel (eds.)

This e-book positive aspects fresh advancements in a speedily becoming region on the interface of higher-dimensional birational geometry and mathematics geometry. It specializes in the geometry of areas of rational curves, with an emphasis on functions to mathematics questions. Classically, mathematics is the examine of rational or crucial suggestions of diophantine equations and geometry is the examine of strains and conics. From the fashionable perspective, mathematics is the examine of rational and critical issues on algebraic types over nonclosed fields. an important perception of the 20 th century used to be that mathematics homes of an algebraic sort are tightly associated with the geometry of rational curves at the kind and the way they range in families.

This choice of solicited survey and examine papers is meant to function an creation for graduate scholars and researchers attracted to getting into the sphere, and as a resource of reference for specialists engaged on comparable difficulties. issues that may be addressed comprise: birational houses reminiscent of rationality, unirationality, and rational connectedness, life of rational curves in prescribed homology periods, cones of rational curves on rationally attached and Calabi-Yau types, in addition to similar questions in the framework of the minimum version Program.

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In the cone generated by X3,1 , X2,2 , and X3,2 , the base locus is Z(1, 0; 3) ∪ Z(0, 1; 4). In the cone generated by X2,1 , X3,1 , and X2,2 , the base locus is Z(1, 0; 3) ∪ Z(0, 1; 4) ∪ Z(1, 1; 5). Finally, in the cone spanned by X2,1 , X1,2 , and X2,2 , the base locus is Z(1, 0; 3) ∪ Z(0, 1; 3) ∪ Z(1, 1; 5). 1 4H1 + 4H2 − B 2 X1,2 3 B 2 Fig. 4 The stable base locus decomposition of (P1 × P1 )[5] Proof. We proved (1) before stating the decomposition. Parts (2), (4), (7), (8), and (9) follow from Theorems 1, 16 and Proposition 6.

The effective cone contains the cone spanned by B, E[2] and X0,1 . In view of the discussion preceding Theorem 24, to prove (1), it suffices to exhibit a moving curve dual to the face spanned by E[2], X0,1 . Let R be a curve in the class E + F. Then the curve R(2) defined in Construction 4 is the required moving curve. Since the base loci described in parts (5)–(9) all contain a fixed divisor and the base locus in (4) is not divisorial; parts (4)–(9) imply (3). The curves E(1, 2), E(2, 2) and F(2, 2) defined in Construction 5 are dual to the faces spanned by [B, E[2] + F[2]], [E[2] + F[2], X0,1], and [F[2], X1,1 ], respectively.

The map f restricted to the zero locus C of a general section of −KD2 gives a two-to-one map from the elliptic curve C onto a line in P2 . Hence, a line on Y parameterizes the fibers of a hyperelliptic map on a section of −KD2 . Since (−KD2 )2 = 2 and the hyperelliptic map on an elliptic curve has four branch points, we conclude that the intersection of a line in Y with −K − B2 is 0. Therefore, the restriction of −K − B2 to Y is trivial. Consider the line bundle ε H[2] − K − B2 . Since H[2] is base-point-free, the base locus of this line bundle is contained in Y .