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I'm studying the book of Rick Miranda; Algebraic Curves and Riemann Surfaces. I'm studying about degree of projective curves and I find a term used very often and that is very important by the amount of times it appears:

"general hyperplane $H$ in $\mathbb{P}^n$"

I need to understand what this means, but in the context of the subject. I searched all over Rick Miranda's book, but I could not find it.

Thamk you!

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Usually, that means that there is a nonempty open subset $U$ of the dual projective space $(\mathbb P^n)^*$ (whose points are the hyperplanes in $\mathbb P^n$) such that $H$ can be an arbitrary element of $U$.

For example, suppose that $p$ is a point. Then a "general hyperplane does not go through $p$". Indeed, the set of hyperplanes which go through $p$ is a proper closed subset of $(\mathbb P^n)^*$.

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  • $\begingroup$ $H$ to be a hyperplane in general position of $\mathbb{P}^n$ = general hyperplane $H$ in $\mathbb{P}^n$ ??? $\endgroup$ – Manoel Dec 29 '17 at 16:46
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    $\begingroup$ I do not understand what you are asking. $\endgroup$ – Mariano Suárez-Álvarez Dec 29 '17 at 16:48
  • $\begingroup$ There is a definition for hyperplane in general position of $\mathbb{P}^n$, I'm wondering if it matches what you replied to general hyperplane $H$ in $\mathbb{P}^n$ $\endgroup$ – Manoel Dec 29 '17 at 16:53
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    $\begingroup$ «Hyperplane in general position in $P^n$» does not mean anything. It has to be in general position with respect to something. $\endgroup$ – Mariano Suárez-Álvarez Dec 29 '17 at 16:55
  • $\begingroup$ so when I say, for example: "...the general tangent hyperplane $H$ to $X$ at $p$ is such that $div(H) = 2p + q_3 + • • • + q_d$ with all $q_i$ distinct and unequal to $p$ ", means that the set of hyperplane $H$ tangent to $X$ at $p$ such that $div(H) = 2p + q_3 + • • • + q_d$ with all $q_i$ distinct and unequal to $p$, is dense in $(\mathbb{P}^{n})^*$??? is that interpretation correct? $\endgroup$ – Manoel Dec 30 '17 at 0:37

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