Varieties n-Covered by Curves of a Fixed Degree and the XJC Correspondence

Abstract

We introduce and study projective varieties \(X^{r+1} \subset \mathbb{P}^{N}\) of dimension r + 1 which are n-covered by irreducible curves of degree δ ≥ n − 1 ≥ 1, that is, varieties such that through n-general points there passes an irreducible curve of degree δ contained in X, denoted by Xr+1(n, δ). We present the sharp Pirio–Trepreau bound for the embedding dimension N in terms of r, n, δ in Theorem 6.2.3, which is obtained geometrically via the iteration of projections from general osculating spaces to \(X^{r+1}(n,\delta ) \subset \mathbb{P}^{N}\), determined by the irreducible curves of degree δ which n-cover the variety. The varieties extremal for the previous bound are subject to even stronger restrictions—e.g. they are rational and through n general points there passes a unique rational normal curve of degree δ, see Theorem 6.3.2 and Theorem 6.3.3. The main result of Pirio and Trepreau (Bull Soc Math Fr 141:131–196, 2013) ensures that the examples of Castelnuovo type are the only extremal varieties except possibly when n > 2, r > 1 and δ = 2n − 3. The first open case, that is, the classification of extremal varieties \(X = X^{r+1}(3,3) \subset \mathbb{P}^{2r+3}\) not of Castelnuovo type, is considered in Sect. 6.4, where it is proved that these varieties are in one-to-one correspondence, modulo projective transformations, with quadro-quadric Cremona transformations on \(\mathbb{P}^{r}\), Pirio and Russo (Commentarii Math Helv 88:715–756, 2013, Theorem 5.2) and Theorem 6.4.5 here. We deduce from this that a quadro-quadric Cremona transformation is, modulo projective transformations acting on the domain and on the codomain, an involution, see Corollary 6.4.6. We obtain the classification of smooth extremal varieties \(X^{r+1}(3,3) \subset \mathbb{P}^{2r+3}\) showing that there are two infinite series: smooth rational normal scrolls and \(\mathbb{P}^{1} \times Q^{r}\) Segre embedded; and four isolated examples appearing for r = 5, 8, 14 and 26 whose \(\mathcal{L}_{x} \subset \mathbb{P}^{r}\) is one the four Severi varieties. In Sect. 6.5 we include a self-contained presentation of the basics of the theory of power associative algebras and of their subclass of Jordan algebras, generalizing to this setting the usual Laplace formulas for inversion of a square matrix. In Theorem 6.5.22 we recall that every quadro-quadric Cremona transformation is linearly equivalent to the cofactor or adjoint map of a suitable rank three Jordan algebra, see loc. cit. for details. We end the chapter by surveying the recent results in Pirio and Russo (J. Reine Angew. Math, 2014, to appear) showing that extremal Xr+1(3, 3) and quadro-quadric Cremona transformations are also in bijection with the isotopy classes of rank three complex Jordan algebras, see Sect. 6.6 for precise formulations of these equivalences leading to the so-called XJC-correspondence, defined in Pirio and Russo (J. Reine Angew. Math, 2014, to appear).