Cioffi, Francesca; Orecchia, Ferruccio; Ramella, Luciana On rational varieties smooth except at a seminormal singular point. (English) Zbl 1242.14048 Commun. Algebra 40, No. 1, 26-41 (2012). The object of this paper is the study of a particular class of projective rational varieties \(X\); namely, \(X\) is the image of a birational morphism \(\phi : {\mathbb P}^m \rightarrow X \subset {\mathbb P}^n\), defined by a linear system \(\delta = <f_0>\oplus I(Z)_r \subset k[t_0,\dots,t_m]\), where \(Z=\{P_1,\dots,P_q\}\) is a scheme made of \(q\) simple points, \(f_0 \notin I(Z)\) and \(r\geq\mathrm{reg}(Z) - 1\) (where \(\mathrm{reg}(Z)\) is the Castelnuovo-Mumford regularity of \(Z\)). Such \(X\) is smooth outside a single point \(O=\phi(P_1)=\dots=\phi(P_q)\), \(({\mathbb P}^m,\phi)\) is the normalization of \(X\) and the tangent cone of \(X\) at \(O\) is made of \(q\) linear spaces \(\pi_i\), with \(\pi_i\cap \pi_j=O\), \(\forall i\neq j\).What is studied here is a condition of seminormality for \(X\), and it is showed that seminormality of \(X\) is equivalent to the condition that \(\mathrm{reg}(Z^{(2)}) \leq r+1\), where \(Z^{(2)}\) is the scheme of 2-fat points supported at \(P_1,\dots,P_q\). Moreover, if the \(q\) points are in generic position, then \(X\) is seminormal if and only if \(h^0({\mathcal I}_Z(r))\geq mq\), for all \((m,q,r)\notin \{(2,5,4),(2,9,4),(1,14,4),(4,7,3)\}\). Reviewer: Alessandro Gimigliano (Bologna) Cited in 1 Document MSC: 14M20 Rational and unirational varieties 14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) 14B05 Singularities in algebraic geometry Keywords:rational varieties; seminormality; branches; Hilbert function Software:Points; QPoints PDFBibTeX XMLCite \textit{F. Cioffi} et al., Commun. Algebra 40, No. 1, 26--41 (2012; Zbl 1242.14048) Full Text: DOI References: [1] Alexander J., J. Algebraic Geom. 4 pp 201– (1995) [2] Bigatti A. M., J. Pure and Applied Algebra 119 pp 237– (1997) · Zbl 0896.13013 [3] Chiantini L., J. Pure Appl. Algebra 186 pp 21– (2004) · Zbl 1071.14060 [4] Cioffi F., Collect. Math. 60 pp 89– (2009) · Zbl 1188.14020 [5] Conca A., Collect. Math. 54 pp 137– (2003) [6] d’Almeida J., Math. Z. 211 pp 479– (1992) · Zbl 0759.14004 [7] Davis E. D., Math. Ann. 279 pp 435– (1988) · Zbl 0657.14003 [8] De Paris A., Comm. Algebra 27 pp 6159– (1999) · Zbl 0958.14016 [9] De Paris A., Comm. Algebra 36 pp 2969– (2008) · Zbl 1156.13001 [10] Eisenbud D., Invent. Math. 136 pp 419– (1999) · Zbl 0943.13011 [11] Galligo , A. ( 1974 ). À propos du théorème de-préparation de Weierstrass,in’Fonctions de plusieurs variables complexes (Sém. François Norguet, octobre 1970–décembre 1973; à la mémoire d’André Martineau). Berlin, Springer, pp. 543–579. Lecture Notes in Math., Vol. 409. Thèse de 3ème cycle soutenue le 16 mai 1973 à l’Institut de Mathématique et Sciences Physiques de l’Université de Nice . [12] Geertsen J. A., Math. Z. 245 pp 155– (2003) · Zbl 1079.14057 [13] Geramita , A. V. ( 1990 ). Lectures on the non-singular cubic surface in \(\mathbb{P}\)3.Queen’s Papers in Pure and Appl. Math.85, Kingston, ON: Queen’s Univ . [14] Geramita A. V., J. Algebra 78 pp 36– (1982) · Zbl 0502.14001 [15] Gimigliano , A. ( 1991 ). Divisors on Bordiga & White surfaces, The curves seminar at Queen’s, Vol. VIII (Kingston, ON, 1990/1991), Exp. E, 10 pp. Queen’s Papers in Pure and Appl. Math., 88, Kingston, ON: Queen’s Univ . · Zbl 0751.14023 [16] Greco S., Compositio Math. 40 pp 325– (1980) [17] Green M. L., Generic Initial Ideals, in Six Lectures on Commutative Algebra (Bellaterra, 1996) 166 (1998) · Zbl 0933.13002 [18] Hartshorne R., Algebraic Geometry (1977) [19] Orecchia F., Boll. Un. Mat. Ital. B (5) 13 pp 588– (1976) [20] Orecchia F., J. Symbolic Computation 31 pp 343– (2001) · Zbl 0969.14041 [21] Pedrini C., Rend. Sem. Mat. Univ. Padova 48 pp 39– (1972) [22] Traverso C., Ann. Scuola Norm. Sup. Pisa (3) 24 pp 585– (1970) 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.