Mathematics – Algebraic Geometry
Scientific paper
1999-12-11
Michigan Math. J. 48 (2000), 1-37
Mathematics
Algebraic Geometry
33 pages, AmS-TeX 2.1
Scientific paper
10.1307/mmj/1030132706
The `linear orbit' of a plane curve of degree $d$ is its orbit in $\P^{d(d+3)/2}$ under the natural action of $\PGL(3)$. In this paper we obtain an algorithm computing the degree of the closure of the linear orbit of an arbitrary plane curve, and give explicit formulas for plane curves with irreducible singularities. The main tool is an intersection@-theoretic study of the projective normal cone of a scheme determined by the curve in the projective space $\P^8$ of $3\times 3$ matrices; this expresses the degree of the orbit closure in terms of the degrees of suitable loci related to the limits of the curve. These limits, and the degrees of the corresponding loci, have been established in previous work.
Aluffi Paolo
Faber Carel
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