4
$\begingroup$

Consider an affine variety $Y$ equipped with a morphism $\pi: Y \rightarrow \mathbb{C}$. The conditions we have are that $\pi^{-1}(0)=\mathbb{C}$, and for any $x$ not equal to zero, $\pi^{-1}(x)=\mathbb{C}^*$. Is it possible to find a complete list of such $Y$?

For example, the affine blow up of $({\mathbb{C}^*})^2$ at $(1,1)$, which could be defined as $Y=\{(x,y,u)\in\mathbb{C}^3\mid x^2-uy^2=y\}$ with projection to $\mathbb{C}_u$.

Improve this question
$\endgroup$
6
  • 1
    $\begingroup$ Not sure about a classification, but an easier example is $\mathbf A^2 \setminus V(xy-1)$ projecting (via first or second projection) to $\mathbf A^1$. If you want to embed into an affine space, take $\{(x,y,u) \in \mathbf A^3\ |\ u(xy-1) = 1\}$, which looks a little similar (but not identical) to your example. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Jan 25 at 11:24
  • $\begingroup$ @R.vanDobbendeBruyn Thank you for providing an example. Could you please explain why they are not identical? $\endgroup$
    – Yunsong WEI
    Commented Jan 25 at 12:45
  • $\begingroup$ By 'not identical' I really mean that: not identical (i.e. the equations differ). I don't know whether they are isomorphic. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Jan 25 at 13:05
  • $\begingroup$ There exists a unique projective model of the generic fiber. The generic fiber is an open subset of this projective model. The closed complement of this open subset is a finite, etale extension of the function field $\mathbb{C}(x)$ of degree $2$. By the hypotheses, this extension is either $\mathbb{C}(x)\times \mathbb{C}(x)$ or $\mathbb{C}(x)[y]/\langle y^2 -x\rangle$. Both of these occur. $\endgroup$
    – Jason Starr
    Commented Jan 25 at 23:56
  • $\begingroup$ Actually the comment is okay (I forgot that the OP wants $Y$ to be affine). $\endgroup$
    – Jason Starr
    Commented Jan 26 at 1:48

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .