In the book "Toric Varieties" by Cox, Little and Schenck, page 39 one can read that the affine surface $Y \subset \mathbb C^4$ parametrized by $(s^4,s^3t, st^3, t^4)$ is not normal.

I heard that normal means for a variety, being no singular in codimension 1. So I was waiting to find a curve $C \subset Sing(Y)$ but I couldn't, and found that the singular locus of $Y$ is only the origin : the singular points of $Y$ are the $(s,t)$ such that $rank \pmatrix{ 4s^3 & 3s^2t & t^3 & 0 \\ 0 & s^3 & 3t^2s & 4t^3 } \leq 1$ .

Where am I mistaken ?

  • 1
    $\begingroup$ You are not mistaken about the singular locus in your case is not of codimension one, but two. You are mistaken in that normality is not just non-singular in codimension one. Look up Serre's criterion for normality. $\endgroup$ – Mohan Dec 26 '16 at 15:09
  • $\begingroup$ Ok thanks a lot again, your comment is really useful ! $\endgroup$ – user378546 Dec 26 '16 at 15:11
  • $\begingroup$ @Mohan : so is there is simple geometric interpretation of normality ? $\endgroup$ – user378546 Dec 26 '16 at 15:12
  • 1
    $\begingroup$ What does geometric mean here? Serre's criterion is algebraic (depth). $\endgroup$ – Mohan Dec 26 '16 at 15:15
  • $\begingroup$ Ok I see. Thanks again for your answer. Since I don't know much about commutative algebra, is it true for example that no singularities in codimension $1$ implies that $X$ is normal ? $\endgroup$ – user378546 Dec 26 '16 at 15:19

Your Answer

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