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Subspace arrangements, configurations of linear spaces and the quadrics containing them. (English) Zbl 1258.14064

Fix integers \(s\geq 2\) and \(n > n_1\geq \cdots \geq n_s >0\). Let \(X\subset \mathbb {P}^n\) be a general union of \(s\) linear subspaces of dimension \(n_1,\dots ,n_s\). R. Hartshorne and A. Hirschowitz [Lect. Notes Math. 961, 169–188 (1982; Zbl 0555.14011)] proved that if \(n_1=1\) and \(n\geq 3\) (i.e. if \(X\) is the union of \(s\) disjoint lines), then the Hilbert function of \(X\) is the expected one: for each integer \(d>0\) \(X\) is contained in exactly \(\min \{\binom{m+d}{m} -s(d+1),0\}\) linearly independent degree \(d\) hypersurfaces. If \(n=3\), then the homogeneous ideal of \(X\) is known by M. Idà [J. Reine Angew. Math. 403, 67–153 (1990; Zbl 0681.14032)]. If \(n_1\geq 2\) only a few cases for the Hilbert function or the minimal free resolution or the Castelnuovo-Mumford’s regularity of \(X\) are known and listed in the paper under review: A. Björner, I. Peeva and J. Sidman [J. Lond. Math. Soc., II. Ser. 71, No. 2, 273–288 (2005; Zbl 1111.13018)]; E. Carlini, M. V. Catalisano and A. V. Geramita [J. Algebra 324, No. 4, 758–781 (2010; Zbl 1197.13016)]; H. Derksen [J. Pure Appl. Algebra 209, No. 1, 91–98 (2007; Zbl 1106.13016)]; H. Derksen and J. Sidman [Adv. Math. 172, No. 2, 151–157 (2002; Zbl 1040.13009)]; [J. Sidman, Int. Math. Res. Not. No. 15, 713–727 (2004; Zbl 1082.52504), and Lect. Notes Pure Appl. Math. 254, 249–265 (2007; Zbl 1132.14046)]. In the paper under review the authors compute for every \(n_i\) the Hilbert function in degree \(2\) of \(X\), i.e. the number of linearly independent quadric hypersurfaces containing \(X\) (it is not an easy result, the case \(n_1+n_2\geq n\) is quite complicated even to be stated). There are algebraic applications, applications in statistics (the quoted paper by Derksen), to data modeling and segmentation [Y. Ma et al., SIAM Rev. 50, No. 3, 413–458 (2008; Zbl 1147.52010)], for incidence properties of rational normal curves and linear spaces [E. Carlini and M.V. Catalisano, J. Lond. Math. Soc., II Ser. 76, No. 1, 380–388 (2006; Zbl 1181.14035); J. Lond. Math. Soc., II Ser. 80, No. 1, 1–17 (2009; Zbl 1134.14022)], for the higher secant varieties of Segre-Veronese varieties. Here the authors find another application to a special kind of polynomial decomposition.

MSC:

14N05 Projective techniques in algebraic geometry
52C35 Arrangements of points, flats, hyperplanes (aspects of discrete geometry)
13D02 Syzygies, resolutions, complexes and commutative rings
13P25 Applications of commutative algebra (e.g., to statistics, control theory, optimization, etc.)
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References:

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