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Products of homogeneous forms. (English) Zbl 1054.14514

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
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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
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