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By Fedor Bogomolov, Brendan Hassett, Yuri Tschinkel

Based at the Simons Symposia held in 2015, the court cases during this quantity concentrate on rational curves on higher-dimensional algebraic forms and functions of the idea of curves to mathematics difficulties. there was major development during this box with significant new effects, that have given new impetus to the learn of rational curves and areas of rational curves on K3 surfaces and their higher-dimensional generalizations. One major contemporary perception the publication covers is the concept the geometry of rational curves is tightly coupled to homes of derived different types of sheaves on K3 surfaces. The implementation of this concept ended in proofs of long-standing conjectures bearing on birational houses of holomorphic symplectic types, which in flip may still yield new theorems in mathematics. This lawsuits quantity covers those new insights intimately.

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14) √ Finally, since |rj | ≤ 2 q for each j, we also have |b| = |r1 r2 r3 r4 | ≤ 16q2 and |a| = | 4j=1 b/rj | ≤ 32q3/2 . A computer search done with these bounds shows that polynomials of the form (14) with four real roots and q ∈ {2, 3, 4, 5, 7} only exist for q ≤ 3, which proves the corollary. ) T −4T −9T +47T −32 ( ) 4 3 2 ⎪ ⎩ T 4 −4T 3 −9T 2 +46T −29 ( ) . ) The nodal cubics of Sect. 5, defined over F2 and F3 , correspond to the polynomials T 4 − 3T 3 − 6T 2 + 24T − 15 and T 4 − 4T 3 − 9T 2 + 47T − 32, respectively.

T 5 , hence by the numbers N1 (X), . . , N5 (X). The direct computation of these numbers is possible (with a computer) when q is small (see Sect. 5 for examples), but the amount of calculations quickly becomes very large. We will explain a method for computing directly the numbers M1 (X), . . , M5 (X). It was first introduced in [5] and uses a classical geometric construction which expresses the blow up of X along a line as a conic bundle. It is valid only in characteristics = 2 and requires X to contain an Fq -line L.

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