Timeline for Smooth variety contained in another smooth variety

7 events

when toggle format	what		by	license	comment
Feb 23, 2012 at 7:23	vote	accept	Parsa
Feb 23, 2012 at 4:00	answer	added	Angelo		timeline score: 17
Feb 22, 2012 at 22:44	comment	added	Will Sawin		In case the complete intersection argument isn't obvious, if the variety is a complete intersection, then if one of the generators had vanishing derivatives somewhere on $Y$, the tangent space would have a dimension too high and form a singularity.
Feb 22, 2012 at 22:41	answer	added	Karl Schwede		timeline score: 4
Feb 22, 2012 at 22:36	comment	added	J.C. Ottem		For curves in $\mathbb{P}^3$, the following argument works: taking cohomology of the conormal sequence shows that global sections of $I_Y(m)$ give global generating sections of $N_Y^(m)$, which is a vector bundle on $Y$ since $Y$ is assumed smooth. But general sections of $N_Y^(m)$ do not vanish on on $Y$ (since such such loci have expected codimension 2).
Feb 22, 2012 at 22:35	comment	added	J.C. Ottem		Perhaps something like this would work? The ideal sheaf $I_Y(m)$ is generated by global sections for $m\gg 0$, by Serre's theorem. Fix such an $m>0$. Note that the general element in the linear system $H^0(I_Y(m))$ is smooth away from $Y$ by Bertini's theorem. So you are reduced to showing that the general element is smooth on $Y$.
Feb 22, 2012 at 21:44	history	asked	Parsa	CC BY-SA 3.0