Let a degree $6$ curve $C$ is given as $(Q_1 \cap Q_2) \cup l^2$ where $Q_i's$ are qudrics and $l$ is a line intersecting $Q_1 \cap Q_2$. Is it true that $C$ always lies on a quadric ?
$\begingroup$
$\endgroup$
4
-
$\begingroup$ No. The quadric must belong to the pencil spanned by $Q_1$ and $Q_2$; if a line is contained in such a quadric, it intersects $E=Q_1\cap Q_2$ in 2 points. Just take for $l$ a general line through a point of $E$. $\endgroup$– abxCommented May 24, 2020 at 14:49
-
$\begingroup$ @abx: "The quadric must belong to the pencil spanned by Q1 and Q2" - can you please explain why? Is it true that the space of quadrics containing a given curve is at most two dimensional? $\endgroup$– pinakiCommented May 24, 2020 at 15:23
-
$\begingroup$ Certainly not in general! But the space of quadrics containing $Q_1\cap Q_2$ is indeed the pencil spanned by $Q_1$ and $Q_2$ — this follows from an easy Koszul complex argument. $\endgroup$– abxCommented May 24, 2020 at 15:27
-
$\begingroup$ Yes. I realized that. Sorry for silly question. However if $Q_1 \cap Q_2$ is contained in a cubic which is singular along the line $l$, then there is a quadric containing $Q_1 \cap Q_2$ and $l$. $\endgroup$– user130022Commented May 24, 2020 at 16:00
Add a comment
|