Let $X=\mathbb P^3_k$ ($k=\bar k$) and $l_1,l_2,l_3$ three distinct lines such that $l_1\cap l_2\neq \emptyset$,$l_2\cap l_3\neq \emptyset$, $l_1\cap l_3=\emptyset$ and $l_2\cap l_3\neq l_1\cap l_2$ (a chain). Is there a nice presentation (for example in using the normal bundle of the lines) of the normal bundle $\mathcal N_{\Gamma/X}$ of the locally complete intersection curve $\Gamma=l_1\cup l_2\cup l_3$.

  • $\begingroup$ Up to projective linear transformations, there is only one such chain. So if you work out the answer in one case, you will have worked it out in every case. One approach: use the fact that such a chain is a Cartier divisor of type $(2,1)$ in a smooth quadric surface in $\mathbb{P}^3$. $\endgroup$ – Jason Starr Sep 23 '15 at 15:00
  • $\begingroup$ Using my previous comment, the normal bundle $N$ appears to be $E\otimes_{\mathcal{O}_{\Gamma}}\mathcal{O}(2,2)|_{\Gamma}$, where $E$ is the unique nontrivial extension of $\mathcal{O}_{\Gamma}$ by $\omega_{\Gamma}$, cf. my preprint with de Jong "Divisor classes and the virtual canonical bundle for genus 0 maps". In particular, there is a short exact sequence, $$0 \to \mathcal{O}(2,1)|_{\Gamma} \to N \to \mathcal{O}(2,2)|_{\Gamma} \to 0.$$ $\endgroup$ – Jason Starr Sep 23 '15 at 15:14
  • $\begingroup$ Thank you for your answer. It should be an other question but are there similar methods to compute the normal bundle of a Néron polygon in $\mathbb P^n_k$ (whose ideal is generated by quadrics) ? $\endgroup$ – user3001 Jan 27 '16 at 10:54
  • $\begingroup$ "... are there similar methods to compute the normal bundle ..." Perhaps, but I will need to check. Are you interested in every integer $n$, or primarily in the case $n=5$ (so that the curve is a Pfaffian quintic curve of arithmetic genus $1$)? $\endgroup$ – Jason Starr Jan 27 '16 at 11:17
  • $\begingroup$ Unfortunately, I am interested in the case $n=4$. $\endgroup$ – user3001 Jan 27 '16 at 11:18

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