A finiteness theorem for low-codimensional nonsingular subvarieties of quadrics. (English) Zbl 0874.14045
There are only finitely many families of codimension two nonsingular subvarieties not of general type of the projective space \(\mathbb{P}^n\), for \(n\geq 4\) [see G. Ellingsrud and C. Peskine, Invent. Math. 95, No. 1, 1-11 (1989; Zbl 0676.14009) and R. Braun, G. Ottaviani, M. Schneider and F. O. Schreyer in: Complex Analysis and Geometry, Univ. Ser. Math., 311-338 (1993; Zbl 0798.14023)]. More generally, a similar statement holds for the case of higher codimension [see M. Schneider, Invent. J. Math. 3, No. 3, 397-399 (1992; Zbl 0762.14022)]. In this paper we concentrate on the case of codimension two subvarieties of quadrics. Our main result is theorem 4.3:
There are only finitely many families of nonsingular codimension two subvarieties not of general type in the quadrics \({\mathcal Q}^n\), \(n=4,5\) or \(n\geq 7\).
The case \(n=4\) is proved by E. Arrondo and I. Sols, [“On congruences of lines in the projective space” Mém. Soc. Math. Fr., Nouv. Sér. 50 (1992; Zbl 0804.14016), §6]. The case of \(n=5\) is at the heart of the paper; the main tools are the semipositivity of the normal bundles of nonsingular subvarieties of quadrics, the double point formula, the generalized Hodge index theorem, bounds for the genus of curves on \({\mathcal Q}^3\). The case of codimension two with \(n\geq 7\) is covered by theorem 2.1; it also gives a finiteness result in codimension bigger than two.
There are only finitely many families of nonsingular codimension two subvarieties not of general type in the quadrics \({\mathcal Q}^n\), \(n=4,5\) or \(n\geq 7\).
The case \(n=4\) is proved by E. Arrondo and I. Sols, [“On congruences of lines in the projective space” Mém. Soc. Math. Fr., Nouv. Sér. 50 (1992; Zbl 0804.14016), §6]. The case of \(n=5\) is at the heart of the paper; the main tools are the semipositivity of the normal bundles of nonsingular subvarieties of quadrics, the double point formula, the generalized Hodge index theorem, bounds for the genus of curves on \({\mathcal Q}^3\). The case of codimension two with \(n\geq 7\) is covered by theorem 2.1; it also gives a finiteness result in codimension bigger than two.
MSC:
| 14M07 | Low codimension problems in algebraic geometry |
| 14N05 | Projective techniques in algebraic geometry |
| 14M15 | Grassmannians, Schubert varieties, flag manifolds |
Keywords:
Grassmannians; not of general type; quadrics; families of codimension two nonsingular subvarieties of projective spaceReferences:
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