Ballico, E. Products of homogeneous forms. (English) Zbl 1054.14514 J. Pure Appl. Algebra 173, No. 1, 1-5 (2002). Summary: In the paper a geometric proof of the following result is given: Let \(K\) be an algebraically closed field. Fix an integer \(s\geq 1\) and positive integers \(n_i\) and \(d_i\), \(1\leq i\leq s\). Set \(m_i=\min \{n_i,d_i+ 1\}\). For \(1\leq i\leq s\) and \(1\leq j\leq n_i\), take general homogeneous forms \(F_{ij}\in K[x,y]\) with \(\deg(F_{ij})=d_i\). Let \(I_i\subset K[x,y]\) be the homogeneous ideal generated by the forms \(F_{ij}\), for \(1\leq j\leq n_i\). Let \(d:=\sum^s_{i=1}d_i\) and denote by \((I_1\dots I_s)_d\) be the degree \(d\) part of the homogeneous ideal \(I_1\dots I_s\). Then \[ \dim (I_1\dots I_s)_d= \min\left\{\prod^s_{i=1} m_i,d+1 \right\}. \] MSC: 14H45 Special algebraic curves and curves of low genus 14N05 Projective techniques in algebraic geometry 13A02 Graded rings Keywords:Segre embedding; multiprojective space PDFBibTeX XMLCite \textit{E. Ballico}, J. Pure Appl. Algebra 173, No. 1, 1--5 (2002; Zbl 1054.14514) Full Text: DOI References: [1] Hartshorne, R., Algebraic Geometry (1977), Springer: Springer Berlin · Zbl 0367.14001 [2] R. Hartshorne, A. Hirschowitz, Smoothing algebraic space curves, in: E. Casas-Alvero, G.E. Welters, S. Xambo-Descamps (Eds.), Algebraic Geometry, Proceedings Sitges 1983, Lecture Notes in Mathematics, Vol. 1124, Springer, Berlin, 1985, pp. 98-131.; R. Hartshorne, A. Hirschowitz, Smoothing algebraic space curves, in: E. Casas-Alvero, G.E. Welters, S. Xambo-Descamps (Eds.), Algebraic Geometry, Proceedings Sitges 1983, Lecture Notes in Mathematics, Vol. 1124, Springer, Berlin, 1985, pp. 98-131. · Zbl 0574.14028 [3] Sernesi, E., On the existence of certain families of curves, Invent. Math., 75, 25-57 (1984) · Zbl 0541.14024 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.