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Duality of planar and spacial curves: new insight. (English) Zbl 1330.14043

This paper addresses the equiclassical strata in the space of plane (complex) curves of given degree \(d\). More precisely, let \(V_{d,g,c}\) denote the set of plane curves of degree \(d\), geometric genus \(g\), and class \(c\). (Recall that the class is equal to the degree of the dual curve.) Sending a curve to its dual curve induces an isomorphism \(V_{d,g,c}\cong V_{c,g,d}\). Exploiting this fact enables the authors to obtain new numerical criteria, in the form of linear and quadratic inequalities involving \(d\), \(c\), and \(g\), that suffice to ensure “regularity” (non-emptiness, irreducibility, …) of a given stratum \(V_{d,g,c}\). In the last section the authors consider curves in \(\mathbb P^n\) and the duality theory for such curves. (Contrary to what the authors say, this theory was not invented by the reviewer, but only studied by her [R. Piene, in: Real and compl. Singul., Proc. Nordic Summer Sch., Symp. Math., Oslo 1976, 475–495 (1977; Zbl 0375.14017)]). In particular, they revisit H. Weyl’s proof of the duality theorem, as was also done by the reviewer in [loc. cit.]. The \(k\)th associated curve is the curve in the Grassmann variety of \(k\)-spaces in \(\mathbb P^n\); the (strict) dual curve is the \((n-1)\)th associated curve. The authors give a condition for when a curve in a Grassmann variety is an associated curve. They also give a tentative definition of equiclassical curves in \(\mathbb P^n\) and end with some interesting open problems related to this notion.
Reviewer: Ragni Piene (Oslo)

MSC:

14H10 Families, moduli of curves (algebraic)
14H20 Singularities of curves, local rings
14H50 Plane and space curves

Citations:

Zbl 0375.14017
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References:

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