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Hilbert scheme of a pair of codimension two linear subspaces. (English) Zbl 1238.14012
The Hilbert scheme \(\text{Hilb}^{p(m)}(\mathbb P ^n)\) parametrizes closed subschemes of \(\mathbb P^n\) with given Hilbert polynomial \(p(m)\). By a result of Groethendieck, it is known that \(\text{Hilb}^{p(m)}(\mathbb P ^n)\) is a projective scheme and by a result of Hartshorne, it is connected. In the paper under review, the authors study the component \(H_n\) of the Hilbert scheme whose general point parametrizes a pair of codimension two linear subspaces of \(\mathbb P ^n\) for \(n\geq 3\). In particular they show that \(H_n\) is smooth and isomorphic to the blow up of the symmetric square of \(\mathbb G (n-2,n)\) along the diagonal, and that \(H_n\) intersects only one other component of \(\text{Hilb}^{p(m)} (\mathbb P ^n)\). Moreover, they show that \(H_n\) is a Mori dream space.

14E05 Rational and birational maps
14E30 Minimal model program (Mori theory, extremal rays)
14M15 Grassmannians, Schubert varieties, flag manifolds
14D22 Fine and coarse moduli spaces
14C05 Parametrization (Chow and Hilbert schemes)
Full Text: DOI arXiv
[1] Birkar C., J. Amer. Math. Soc. 23 pp 405– (2010) · Zbl 1210.14019 · doi:10.1090/S0894-0347-09-00649-3
[2] Grothendieck , A. ( 1995 ). Techniques de construction et théorèmes d’existence en géométrie algébrique IV. Les schémas de Hilbert, Séminaire Bourbaki, Vol. 6, Exp. No. 221, 249–276,Soc. Math. France, Paris .
[3] Harris J., Curves in Projective Space (1982) · Zbl 0511.14014
[4] Hartshorne R., Inst. Hautes Études Sci. Publ. Math. 29 pp 5– (1966)
[5] Hu Y., Michigan Math. J. 48 pp 331– (2000) · Zbl 1077.14554 · doi:10.1307/mmj/1030132722
[6] Lazarsfeld R., Positivity in Algebraic Geometry I, Classical Setting: Line Bundles and Linear Series (2004) · Zbl 1093.14501
[7] Lee , Y.H. A. ( 2000 ). The Hilbert schemes of curves in \(\mathbb{P}\)3. Senior Thesis, Harvard University .
[8] Migliore J., Trans. Amer. Math. Soc. 294 pp 177– (1986) · doi:10.1090/S0002-9947-1986-0819941-3
[9] Nollet S., Ann. Sci. École Norm. Sup 4 pp 367– (1997)
[10] Piene R., Amer. J. Math. 107 pp 761– (1985) · Zbl 0589.14009 · doi:10.2307/2374355
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