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?

  • $\begingroup$ "The number of conditions" is the codimension of the linear subsystem that satisfies these conditions. $\endgroup$
    – Sasha
    Feb 13 at 7:35
  • $\begingroup$ 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 $\endgroup$
    – Li Li
    Feb 14 at 16:14
  • $\begingroup$ Yes, if you are talking about the number of conditions imposed by a given set of points on the linear system of quadrics. $\endgroup$
    – Sasha
    Feb 14 at 18:44
  • $\begingroup$ 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 $\endgroup$
    – Li Li
    Feb 16 at 4:44


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