Limits of PGL(3)-translates of plane curves, I

Details Limits of PGL(3)-translates of plane curves, I Limits of PGL(3)-translates of plane curves, I

2007-06-09

arxiv.org/abs/0706.1301v4

J. Pure Appl. Algebra 214 (2010), no. 5, 526-547

Mathematics

Algebraic Geometry

28 pages. Minor revision. Final version

Scientific paper

10.1016/j.jpaa.2009.06.010

We classify all possible limits of families of translates of a fixed, arbitrary complex plane curve. We do this by giving a set-theoretic description of the projective normal cone (PNC) of the base scheme of a natural rational map, determined by the curve, from the $P^8$ of 3x3 matrices to the $P^N$ of plane curves of degree $d$. In a sequel to this paper we determine the multiplicities of the components of the PNC. The knowledge of the PNC as a cycle is essential in our computation of the degree of the PGL(3)-orbit closure of an arbitrary plane curve, performed in our earlier paper "Linear orbits of arbitrary plane curves".

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