When is the natural projection of the HIlbert flag scheme a flat morphism - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T08:29:41Z http://mathoverflow.net/feeds/question/104666 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/104666/when-is-the-natural-projection-of-the-hilbert-flag-scheme-a-flat-morphism When is the natural projection of the HIlbert flag scheme a flat morphism Naga Venkata 2012-08-14T06:25:45Z 2012-08-14T13:29:59Z <p>Let ${Hilb_{P,Q}}_{red}$ be the reduced scheme associated to the Hilbert flag scheme parametrizing all pairs $(C,X)$ with $C \subset X \subset \mathbb{P}^3$, where $C$ is a curve and $X$ a degree $d$ surface in $\mathbb{P}^3$. When is the natural projection map $$pr_1:{Hilb_{P,Q}}_{red} \to pr_1({Hilb_{P,Q}}_{red}) \subset {Hilb_P}_{red}$$ a flat morphism?</p> http://mathoverflow.net/questions/104666/when-is-the-natural-projection-of-the-hilbert-flag-scheme-a-flat-morphism/104692#104692 Answer by Sasha for When is the natural projection of the HIlbert flag scheme a flat morphism Sasha 2012-08-14T12:07:31Z 2012-08-14T12:07:31Z <p>The fiber of the map over a curve $C$ is just $P(H^0(P^3,I_C(d)))$. So, the sufficient and necessary condition is that $\dim H^0(P^3,I_C(d))$ is constant on $Hilb_P$.</p>