Let $V$ be linear vector space of dimension $n$ over an algebraically closed field $k$. Fix $X$ an affine algebraic variety $X$ and a finite morphism of algbebraic varieties: $\mu: X \rightarrow V$.

A line $l \subset V$ is $\mu$ regular if the composition morphism $pr\circ \mu: X \rightarrow V/l$ is finite.

Let $S$ denote the set of $\mu$ regular lines. Then I want to show that $S$ contains a dense open subset in $\mathbb{P}(V)$, the projective space.

Any references or answers or hints are welcomed.

Thanks in advance.

edit: I forgot to mention that I suppose that $X$ has $\dim X <n$.


1 Answer 1


I will start with a hint.

Write $k[x_1,...,x_n]$ for the coordinate ring of $V$ and $X=Spec(R)$. The map $\mu^*: k[x_1,...,x_n] \rightarrow R$ has kernel $I$. It will be enough to show this in the case $X=Spec(k[x_1,...,x_n]/I)$ and $\mu^*$ is the quotient map.

Suppose the line $l$ is of the form $(a_1 t, a_2 t, ..., a_n t)$ as $t$ varies in $k$. The elements $y_i = a_nx_i - a_ix_n$ for $1 \leq i < n$ are in the image of the map $$k[V/l] \rightarrow k[V].$$

Why do the images of $y_i$ in $k[x_1,...,x_n]/I$ generate the algebra $k[x_1,...,x_n]/I$ for generic choices of $a_1,...,a_n$?

  • $\begingroup$ I understand that they generate this algebra, because $y_i$ are combinations of $x_n,x_i$ for $1\leq i <n$, and $y_i$ never vanish (I mean I can pick coeffecients $a_i$ such that they never get vanish). Can you please elaborate how construct this open dense set? Thanks in advance. $\endgroup$ Feb 6, 2014 at 10:24
  • $\begingroup$ Try this example. If $n=2$ and $I = (x_1-x_2)$ what are the conditions I'd need to require on $a_1$ and $a_2$? $\endgroup$
    – SomeEE
    Feb 6, 2014 at 18:00
  • $\begingroup$ $a_1=a_2=1$, I think. $\endgroup$ Feb 6, 2014 at 19:51
  • $\begingroup$ $a_1=-1, a_2 =1$ works. Remember you want it to be an open condition, not a closed one. What other choices of $a_1$ and $a_2$ will result in $a_2x_1-a_1x_2$ generating the $k$-algebra $k[x_1,x_2]/(x_1-x_2)$? (Caveat: $char(k)$ not 2 here) $\endgroup$
    – SomeEE
    Feb 6, 2014 at 19:56
  • $\begingroup$ $a_1 = -a_2$. Can you please elaborate how does this relate to the question I asked, i.e that $S$ has an open dense set in $\mathbb{P}(V)$; Thanks. $\endgroup$ Feb 6, 2014 at 20:19

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.