Finite set of points impose conditions on quadrics I'm reading Castelnuovo Lemma in Griffiths & Harris p. 531 and I am having problems with the concept "impose conditions on quadrics".
As I understand it is in te following way: A finite set $X$ imposes independent conditions on quadrics if for all $p\in X$ there is a quadric that vanishes in all the points of $X$ except $p$.
I undestand it if we say, for example, that 5 points impose 5 conditions on quadrics, but if I say, for example, that 7 points impose 5 conditions on quadrics. What does that mean?
 A: The condition "goes through $p$" gives you a linear equation satisfied by the coefficients of your quadric. The condition that a finite set imposes independent conditions on the quadric means exactly that these equations on the coefficients are linearly independent: if a quadric passing through $X\setminus \{p\}$ must also pass through $p$, then the linear equation on the coefficients given by "goes through $p$" lies in the ideal determined by the other linear equations on coefficients from the rest of the points in $X$, which means it's a linear combination of those equations.
In general, "imposes a condition" can mean lots of different things - "condition" can be used in lots of different ways by lots of different authors. Sometimes it means an equation is satisfied, while other times it means an equation is not satisfied, or maybe a collection of such criteria (sometimes it means other things too - like "BLAH is an ADJECTIVE condition" means that the set of things satisfying BLAH is ADJECTIVE, with closed/open/constructible being popular choices for ADJECTIVE). My best guess for your example about "$7$ points imposing $5$ conditions" would be that the ideal generated by the equations on the coefficients coming from the $7$ points is generated by $5$ elements.
