Geometric Invariant Theory for Polarized Curves by Gilberto Bini Fabio Felici Margarida Melo & Filippo Viviani

Author:Gilberto Bini, Fabio Felici, Margarida Melo & Filippo Viviani
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

Fig. 9.1The basin of attraction of a curve X 0 with a special cuspidal elliptic tail F 0

Our second result concerns tacnodes with a line (in the sense of 1.4).

Theorem 9.3

If and with X connected, then X does not have tacnodes with a line.

Proof

This follows from [Gie82, Prop. 1.0.6, Case 2]; however, for the reader’s convenience and also because we will need it later, we give a sketch of the proof.

Using the hypothesis on , we get that X is quasi-wp-stable by Corollary 5.​6(ii) and that is very ample, non-special and balanced of degree d by the Potential pseudo-stability Theorem 5.​1. Suppose that X has a tacnode with a line, i.e. that we can write with , a tacnode of X and degL  | E  = 1. We want to show, by contradiction, that if .

Since E and Y are tangent in , we can choose coordinates {x 1, …, x r+1} of H 0(X, L) so that

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