Dimension-Theoretic Characterization of Maximal Irreducible Algebraic Systems of Plane Nodal Curves of a Given Order n and with a Given Number d of Nodes
Oscar Zariski · American Journal of Mathematics · 1982
Introduction. Let k be an algebraically closed ground field of characteristic zero. The following result is well-known (see Severi [1], Anhang F, p. 317). If En d is a complete (i.e., maximal) irreducible algebraic system, defined over k, of plane algebraic (not necessarily irreducible) curves of a given order n, such that the general curve C* of En d /k has d nodes (and no other singularities), then the dimension of En d is equal to 3n + p 1, where p = ((n 1)(n 2)/2) d is the genus of C*. We note that 3n + p 1 is equal to (n (n + 3)/2) d. Here N = (n(n + 3)/2) is the dimension of the (linear) system of all plane curves of order n. This is in agreement with the intuitive expectation that the requirement that a curve of order n possess d nodes (in non-assigned position) imposes d independent algebraic (non-linear) conditions on the curve. The above result is easily proved and will be included by us in a more general theorem, the proof of which is the main object of this paper (Theorem 2, Section 3) and which deals with irreducible algebraic systems E of curves of order n in which the general curve C* of E/k has arbitrary singularities (but is free from multiple components; equivalently: C* is a reduced curve). This theorem asserts that if p is the genus of C* (the definition of the effective genus of a reducible-but reduced-curve is given in Section 3, formula (2)), then dim E c 3n + p 1, with equality if and only if C* has only nodes as singularities. It is easily seen (see Severi, loc. cit.) that if 0 c d ' (n(n 1)/2) then there always exists curves of order n having d nodes (the maximum