Algebraic Geometry Santa Cruz 1995, Part 2 by Kollar J., Lazarsfeld R., Morrison D. (eds.)

By Kollar J., Lazarsfeld R., Morrison D. (eds.)

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Extra resources for Algebraic Geometry Santa Cruz 1995, Part 2

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We have ⎧ √ 2 ⎪ ⎨1 + 45q + q − 10(q + 1) q if q ≥ 64; √ N1 (F(X)) ≥ 1 + 13q + q2 − 6(q + 1) q if 16 ≤ q ≤ 61; ⎪ √ ⎩ if q ≤ 13. 1 − 3q + q2 − 2(q + 1) q In particular, X contains at least 10 Fq -lines if q ≥ 11. Moreover, for all q, √ N1 (F(X)) ≤ 1 + 45q + q2 + 10(q + 1) q. Proof As we saw in Sect. 1, we can write the roots of Q1 (F(X), T ) as ω1 , . . , ω5 , √ √ ω 1 , . . , ω 5 . 1, we have rj + 5q + N1 F(X) = 1 − 1≤j≤5 qrj + q2 (ωj ωk + ω j ωk + ωj ω k + ω j ω k ) − 1≤j≤5 1≤j

In this paper we take this correspondence further. We have described this correspondence in Table 1. In this paper we describe a technology for finding such “good” flat families of perverse sheaves of categories. This is done by deforming LG models as sheaves of categories. The main geometric outcomes of our work are: Classical Categorical W = P equality for tropical varieties “W = P” for perverse sheaves of categories Voisin theory of deformations Good flat deformations of PSC Canonical deformations and compactification HN and additional filtrations of perverse of moduli spaces sheaves of categories We will briefly discuss our procedure.

13 In the coordinates x2 , x4 , x5 , an equation of the discriminant quintic 3 3 3 2 L ⊂ P is x2 x4 (x2 + x4 + 4x5 ) = 0. Therefore, • in characteristics other than 2 and 3, it is a nodal quintic which is the union of two lines and an elliptic curve, all defined over the prime field; • in characteristic 2, it is the union of 5 lines meeting at the point (0, 0, 1); 3 of them are defined over F2 , the other 2 over F4 . 5 (although the choice of coordinates is different). If x = (0, x2 , 0, x4 , x5 ) ∈ P2 , the residual conic Cx is defined by the equation 1 3 y + (x2 y1 − y2 )3 + y33 + (x4 y1 − y3 )3 + y13 x53 = y12 (x22 + x42 + x52 ) − 3x22 y1 y2 − 3x42 y1 y3 + 3x2 y22 + 3x4 y32 y1 2 in the coordinates (y1 , y2 , y3 ).

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