# Number of conditions imposed by general points

I encountered with a problem when I read the part of Enriques-Babbage Theorem of the book Geometry of Algebraic Curves Vol. I by ACGH. It is stated on page 112-113 that all subsets of $$m$$ points of a hyperplane section $$\Gamma=H\cap C$$ of a curve $$C\subset\mathbb{P}^r$$ impose the same number of conditions on a linear system $$\mathcal{D}$$ on $$C$$.

My first question is what the definition of the number of conditions on the linear system $$\mathcal{D}$$ is.

In the proof of Proposition 3.1 of Chapter III, a hyperplane section $$\Gamma=H\cap C$$ of a canonical curve $$C$$, which consists of $$2g-2$$ points, together with a point on $$H$$ outside $$C$$ form $$2g-1$$ points in general position that impose $$2g-3$$ conditions on quadrics by the Uniform Position Theorem. How is the number $$2g-3$$ obtained? How is the Uniform Position Theorem applied?

• "The number of conditions" is the codimension of the linear subsystem that satisfies these conditions. Feb 13 at 7:35
• Can you explain more accurately about what "satisfies these conditions" means? I guess if the linear system is that of all the quadrics on $C$, then we need to count the codimension of the quadrics passing through certain given collection of points. Is that right?@Sasha Feb 14 at 16:14
• Yes, if you are talking about the number of conditions imposed by a given set of points on the linear system of quadrics. Feb 14 at 18:44
• So, how can I see that the dimension of quadrics on $C$ passing through $2g-1$ points in general position is $g-1$(after projectivization)?@Sasha Feb 16 at 4:44