Sums of three squares and Noether–Lefschetz loci

Abstract

We show that the set of real polynomials in two variables that are sums of three squares of rational functions is dense in the set of those that are positive semidefinite. We also prove that the set of real surfaces in P 3 whose function field has level 2 is dense in the set of those that have no real points. Strategy of the proofOur starting point is Colliot-Thélène's Hodge-theoretic proof of the Cassels-Ellison-Pfister theorem [15]: he associates to a polynomial f its homogenization F ∈ R[X 0 , X 1 , X 2 ], and the real algebraic surface defined by S := {Z 2 + F (X 0 , X 1 , X 2 ) = 0}. He then interpretes the polynomials f that are sums of three squares in R(x 1 , x 2 ) as those for which the complex surface S C carries an extra line bundle of a particular kind, and concludes by applying the Noether-Lefschetz theorem.As a consequence, Q d may be viewed as a union of Noether-Lefschetz loci in P d . Over C, density results for Noether-Lefschetz loci have been first obtained by Ciliberto-Harris-Miranda and Green [13], and we adapt these arguments over R.We rely on a real analogue of Green's infinitesimal criterion ([13, §5], [37, §17.3.4]), which was developed for other purposes in a joint work with Olivier Wittenberg [4]. Section 1 is devoted to establishing this criterion in a form suitable for our needs: Proposition 1.3. One way to verify the hypothesis of Green's criterion is to construct Noether-Lefschetz loci of the expected dimension. Following Ciliberto and Lopez [14], we do so in Section 2 by considering Noether-Lefschetz loci associated to determinantal curves, a strategy independently adopted by Bruzzo, Grassi and Lopez in [11]. Finally, Section 3 contains the proof of Theorem 0.1. Level of function fieldsThe argument described above may be adapted to other families of real surfaces: here is another application of it. Recall that if K is a field, Pfister [32, Satz 4] has shown that the smallest integer s such that −1 is a sum of s squares in K is a power of 2 (or +∞): it is the level s(K) of K. Moreover, if X is an integral variety over R of dimension n without real points, s(R(X)) 2 n [33, Theorem 2].Let us restrict to varieties that are smooth degree d surfaces S ⊂ P 3 R , defined by a degree d homogeneous equation F ∈ R[X 0 , X 1 , X 2 , X 3 ] d . Let Θ d ⊂ P(R[X 0 , X 1 , X 2 , X 3 ] d ) be the set of those surfaces that have no real points. As before, we assume that d is even, since otherwise Θ d = ∅. If d = 2, any such surface S is isomorphic to the anisotropic quadric {X 2 0 +X 2 1 +X 2 2 +X 2 3 = 0}, and s(R(S)) = 2. On the other hand, it follows from the Noether-Lefschetz theorem applied as in [15] that if d 4, a very general S ∈ Θ d satisfies s(R(S)) = 4. In section 4, we will show: Conventions about algebraic varieties over RAn algebraic variety X over R is a separated scheme of finite type over R. We denote its complexification by X C . Its set of complex points X(C) is endowed with an action of G := Gal(C/ R) ≃ Z /2 Z such that the complex conjugation σ ∈ G acts antiholomorphically. The r...

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