The space of cubic surfaces equipped with a line
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Mathematische Zeitschrift
The space of cubic surfaces equipped with a line Ronno Das1 Received: 19 July 2019 / Accepted: 1 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract The Cayley–Salmon theorem implies the existence of a 27-sheeted covering space of the parameter space of smooth cubic surfaces, marking each of the 27 lines on each surface. In this paper we compute the rational cohomology of the total space of this cover, using the spectral sequence in the method of simplicial resolution developed by Vassiliev. The covering map is an isomorphism in cohomology (in fact of mixed Hodge structures) and the cohomology ring is isomorphic to that of PGL(4, C). We derive as a consequence of our theorem that over the finite field Fq the average number of lines on a smooth cubic surface equals 1 (away from finitely many characteristics); this average is 1+ O(q −1/2 ) by a standard application of the Weil conjectures. Mathematics Subject Classification Primary 55R80; Secondary 14N15 · 14J70
1 Introduction One of the first theorems of modern algebraic geometry and specifically enumerative geometry is the Cayley–Salmon theorem [6]. This classical theorem states that every smooth cubic surface (over an algebraically closed field, in particular C) contains exactly 27 lines. A cubic (hyper)surface in P3 = CP 3 is the zero set S = V (F) of a homogeneous polynomial F of degree 3 in 4 variables. The surface S is singular (i.e. not smooth) if and only if the 20 coefficients of F are a zero of a discriminant polynomial : C20 → C. Thus the space of smooth cubic surfaces is an open locus M = M3,3 := P19 \ V (). The Cayley–Salmon → M, where M is the incidence theorem can be reinterpreted as a covering map π : M variety of lines and smooth cubic surfaces (see (2.1) and the preceding discussion for precise definitions). The fiber π −1 (S) over S ∈ M is the set of 27 lines on S. The automorphism group of P3 is PGL(4, C) and this group acts on lines and cubic → M is PGL(4, C)surfaces, preserving smoothness. In particular the covering map π : M equivariant. It was shown by Vassiliev (in [20]) that the space M has the same rational cohomology as PGL(4, C), and it follows from the results of Peters–Steenbrink [16] that the orbit map given by g → g(S0 ) induces an isomorphism for any choice of S0 ∈ M (see Theorem 1.4). See also [17].
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Ronno Das [email protected] Department of Mathematics, University of Chicago, Chicago, IL 60637, USA
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also has the same rational The main result of this paper is that the covering space M cohomology. the orbit map PGL(4, C) → M given by g → Theorem 1.1 For a choice (S0 , L 0 ) ∈ M, g(S0 , L 0 ) induces an isomorphism Q) −∼ H ∗ ( M; → H ∗ (PGL(4, C); Q) ∼ = Q[a3 , a5 , a7 ]/(a32 , a52 , a72 ) , π − → M also induces where ai ∈ H i (PGL(4, C); Q). Since the composition PGL(4, C) → M ∗ an isomorphism on H (_; Q), the map
Q) π ∗ : H ∗ (M; Q) → H ∗ ( M; is an isomorphism. Since the orbit map and π are algebraic, the isomorphisms re
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