×

3-dimensional sundials. (English) Zbl 1237.14066

A \(3\)-dimensional sundial in \(\mathbb P^n\) is the scheme defined by a pair of intersecting lines, with a non reduced structure at the common point \(p\). The structure at \(p\) is determined by taking the square of the ideal in some linear three-dimensional space which contains the lines. Sundials are useful, since they are the flat limit, in the Hilbert scheme, of a pair of skew lines degenerating to a reducible conic. From this point of view, sundials were used by Hartshorne and Hirschowitz, to prove that a general union of lines in \(\mathbb P^3\) has the expected Hilbert function. The authors generalize the previous result: they prove that the Hilbert function of a general union of lines and \(3\)-dimensional sundials in \(\mathbb P^n\), has the expected behaviour.

MSC:

14N20 Configurations and arrangements of linear subspaces
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
14C05 Parametrization (Chow and Hilbert schemes)

Software:

CoCoA
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Abo H., Ottaviani G., Peterson C., Induction for secant varieties of Segre varieties, Trans. Amer. Math. Soc., 2009, 361(2), 767-792 http://dx.doi.org/10.1090/S0002-9947-08-04725-9; · Zbl 1170.14036
[2] Alexander J., Hirschowitz A., Polynomial interpolation in several variables, J. Algebraic Geom., 1995, 4(2), 201-222; · Zbl 0829.14002
[3] Bürgisser P., Clausen M., Shokrollahi M.A., Algebraic Complexity Theory, Grundlehren Math. Wiss., 315, Springer, Berlin, 1997; · Zbl 1087.68568
[4] Carlini E., Catalisano M.V., Geramita A.V., Bipolynomial Hilbert functions, J. Algebra, 2010, 324(4), 758-781 http://dx.doi.org/10.1016/j.jalgebra.2010.04.008; · Zbl 1197.13016
[5] Carlini E., Catalisano M.V., Geramita A.V., Reduced and non-reduced linear spaces: lines and points (in preparation); · Zbl 1356.14046
[6] Carlini E., Chiantini L., Geramita A.V., Complete intersections on general hypersurfaces. Michigan Math. J., 2008, 57, 121-136 http://dx.doi.org/10.1307/mmj/1220879400; · Zbl 1181.14057
[7] Catalisano M.V., Geramita A.V., Gimigliano A., Ranks of tensors, secant varieties of Segre varieties and fat points, Linear Algebra Appl., 2002, 355, 263-285 http://dx.doi.org/10.1016/S0024-3795(02)00352-X; · Zbl 1059.14061
[8] Catalisano M.V., Geramita A.V., Gimigliano A., Erratum to: “Ranks of tensors, secant varieties of Segre varieties and fat points” [Linear Algebra Appl. 355 (2002) 263-285], Linear Algebra Appl., 2003, 367, 347-348 http://dx.doi.org/10.1016/S0024-3795(03)00455-5; · Zbl 1059.14061
[9] Catalisano M.V., Geramita A.V., Gimigliano A., Higher secant varieties of Segre-Veronese varieties, In: Projective Varieties with Unexpected Properties, Walter de Gruyter, Berlin, 2005, 81-107; · Zbl 1102.14037
[10] Catalisano M.V., Geramita A.V., Gimigliano A., Secant varieties of Grassmann varieties, Proc. Amer. Math. Soc., 2005, 133(3), 633-642 http://dx.doi.org/10.1090/S0002-9939-04-07632-4; · Zbl 1077.14065
[11] Catalisano M.V., Geramita A.V., Gimigliano A., Segre-Veronese embeddings of ℙ1×ℙ1×ℙ1 and their secant varieties, Collect. Math., 2007, 58(1), 1-24; · Zbl 1122.14037
[12] Catalisano M.V., Geramita A.V., Gimigliano A., Secant varieties of ℙ1 ×…×ℙ1 (n-times) are not defective for n ≥ 5, J. Algebraic Geom., 2011, 20(2), 295-327; · Zbl 1217.14039
[13] Comon P., Mourrain B., Decomposition of quantics in sums of powers of linear forms, Signal Process., 1996, 53(2), 93-107 http://dx.doi.org/10.1016/0165-1684(96)00079-5; · Zbl 0875.94079
[14] CoCoA: a system for making Computations in Commutative Algebra, available at http://cocoa.dima.unige.it;
[15] Hartshorne R., Hirschowitz A., Droites en position générale dans l’espace projectif. In: Algebraic Geometry, La Rábida, 1981, Lecture Notes in Math., 961, Springer, Berlin, 1982, 169-188 http://dx.doi.org/10.1007/BFb0071282; · Zbl 0555.14011
[16] Pistone G., Riccomagno E., Wynn H.P., Algebraic Statistics, Monogr. Statist. Appl. Probab., 89, Chapman & Hall/CRC, Boca Raton, 2001; · Zbl 0960.62003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.