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Collisions of fat points and applications to interpolation theory. (English) Zbl 1446.14034

Let \(p\) be a smooth point of the integral projective variety \(X\). For all \(m>0\) let \(mp\) denote the closed subscheme of \(X\) with \((\mathcal {I}_p)^m\) as its ideal sheaf. A fat point scheme of \(X\) is a finite union of different fat points \(m_ip_i\). Key geometric problems were solved computing \(h^0(X,\mathcal{I}_{m_1p_1\cup \cdots \cup m_rp_r}\otimes L)\) for certain line bundles \(L\) on \(X\) and for general \((p_1,\dots ,p_r)\in X^r\). Since [A. Hirschowitz, Manuscrip. Math. 50, 337–388 (1985; Zbl 0571.14002)] it was used taking a flat limit of fat point schemes and using the semicontinuity theorem for cohomology. Soon, J. Roé [Trans. Amer. Math. Soc. 366, No. 2, 857–874 (2014; Zbl 1291.14019)] found stronger tools, essentially virtual limits, instead of a scheme which is a flat limit. But the problem to compute the flat limits, at least in some cases, is very interesting and this is the content of the paper under review.This is done for small multiplicities and small dimension. This is important for interpolation in \(\mathbb {P}^n\) and it gives some cases, \(r\le 15\) and \(d\ge 3m\) of a conjecture in [A. Laface and L. Ugaglia, Trans. Am. Math. Soc. 358, No. 12, 5485–5500 (2006; Zbl 1160.14003)] giving the exact value of \(h^0(\mathbb {P}^3,\mathcal {I}_Z(d))\), where \(Z\) is a general union of \(r\) fat points of multiplicity \(m\).

MSC:

14N07 Secant varieties, tensor rank, varieties of sums of powers
14N05 Projective techniques in algebraic geometry
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References:

[1] Alexander, J.; Hirschowitz, A., The blown-up Horace method: application to fourth-order interpolation, Invent. Math., 107, 3, 585-602 (1992) · Zbl 0784.14002
[2] Ballico, E.; Brambilla, M. C., Postulation of general quartuple fat point schemes in \(P^3\), J. Pure Appl. Algebra, 213, 6, 1002-1012 (2009) · Zbl 1170.14039
[3] Ballico, E.; Brambilla, M. C.; Caruso, F.; Sala, M., Postulation of general quintuple fat point schemes in \(P^3\), J. Algebra, 363, 113-139 (2012) · Zbl 1262.14067
[4] Bocci, C., Special effect varieties in higher dimension, Collect. Math., 56, 3, 299-326 (2005) · Zbl 1093.14009
[5] Brambilla, M. C.; Dumitrescu, O.; Postinghel, E., On a notion of speciality on linear system of \(P^n\), Trans. Amer. Math. Soc., 367, 8, 5447-5473 (2015) · Zbl 1331.14007
[6] Brambilla, M. C.; Dumitrescu, O.; Postinghel, E., On linear systems of \(P^3\) with nine base points, Ann. Mat. Pura Appl., 195, 5, 1551-1574 (2016) · Zbl 1359.14007
[7] Catalisano, M. V.; Geramita, A. V.; Gimigliano, A., Higher secant varieties of Segre-Veronese varieties, (Projective Varieties with Unexpected Properties (2005), Walter de Gruyter GmbH and Co. KG: Walter de Gruyter GmbH and Co. KG Berlin), 81-107 · Zbl 1102.14037
[8] Ciliberto, C., Geometric aspects of polynomial interpolation in more variables and of Waring’s problem, (Proceedings of the European Congress of Mathematics 1. Proceedings of the European Congress of Mathematics 1, Barcelona, 2000. Proceedings of the European Congress of Mathematics 1. Proceedings of the European Congress of Mathematics 1, Barcelona, 2000, Progress in Math., vol. 201 (2001), Birkhäuser: Birkhäuser Basel), 289-316 · Zbl 1078.14534
[9] Ciliberto, C.; Miranda, R., Degenerations of planar linear systems, J. Reine Angew. Math., 501, 191-220 (1998) · Zbl 0943.14002
[10] Ciliberto, C.; Miranda, R., Matching conditions for degenerating plane curves and applications, (Projective Varieties with Unexpected Properties (2005), Walter de Gruyter GmbH and Co. KG: Walter de Gruyter GmbH and Co. KG Berlin), 177-197 · Zbl 1109.14009
[11] De Volder, C.; Laface, A., On linear systems of \(P^3\) through multiple points, J. Algebra, 310, 1, 207-217 (2007) · Zbl 1113.14036
[12] Dumnicki, M., Cutting diagram method for systems of plane curves with base points, Ann. Polon. Math., 90, 131-143 (2007) · Zbl 1107.14007
[13] Dumnicki, M.; Jarnicki, W., New effective bounds on the dimension of a linear system in \(P^2\), J. Symbolic Comput., 42, 621-635 (2007) · Zbl 1126.14008
[14] Eisenbud, D.; Harris, J., 3264 and All That (2016), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1341.14001
[15] Evain, L., Calculs de dimensions de systèmes linéaires de courbes planes par collisions de gros points, C. R. Acad. Sci. Paris Sér. I Math., 325, 12, 1305-1308 (1997) · Zbl 0905.14005
[16] Evain, L., La fonction de Hilbert de la reunion de \(4^h\) points generiques de \(P^2\) de même multiplicité, J. Algebraic Geom., 8, 4, 787-796 (1999) · Zbl 0953.14027
[17] Galuppi, F.; Mella, M., Identifiability of homogeneous polynomials and Cremona transformations, J. Reine Angew. Math. (2017) · Zbl 1437.14023
[18] Laface, A.; Ugaglia, L., On a class of special linear systems on \(P^3\), Trans. Amer. Math. Soc., 358, 12, 5485-5500 (2006) · Zbl 1160.14003
[19] Nesci, M., Collisions of Fat Points (2009), Università Roma III, PhD thesis
[20] Roé, J., Maximal rank for schemes of small multiplicity by Evain’s differential Horace method, Trans. Amer. Math. Soc., 366, 857-874 (2014) · Zbl 1291.14019
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