# Do there exist double points on an algebraic surface in $\mathbb{P}_{\mathbb{C}}^3$ that are not rational?

The title explains it all.

I'm familiar with the du val singularities on surfaces, also known as rational double points. In http://homepages.warwick.ac.uk/~masda/surf/more/DuVal.pdf, 2.1, they are characterized as those isolated double points that admit a resolution that is given by stepwise blowup of isolated double points.

My question now is whether other type of multiplicity two points exist, i.e. nonrational double points? If so, can they occur on a surface embedded in $\mathbb{P}^3$?

Due to Reids characterization i would say these are isolated double points such that somewhere in the blowup process either a singular point of multiplicity $>2$ or a singular curve appears. Both would seem weird to me, but perhaps my intuition is not correct in this case. Any comments on this aspect are highly appreciated!

• By the way, i asked this also on MSE a few days ago, but got no replies whatsoever. See math.stackexchange.com/questions/561247/… Nov 13, 2013 at 21:39
• What about the singularity $z^6-y^2-x^3-1$ . These occur on a sextic surface that is a degree 6 cover of $P^2$ branched along a sextic with cusps. The resolution of such a singularity is an elliptic curve. Unless I'm misunderstanding, that is a solution.
– meh
Nov 13, 2013 at 22:54
• @aginensky, is the formula correct? Looking for a singular point and taking derivatives i get $x=y=z=0$ but that point is not on the surface.. Or am i being a complete idiot? Nov 14, 2013 at 18:36
• @ Joachim- oops . Remove the "-1" .
– meh
Nov 15, 2013 at 13:25
• @aginensky, thanks. You could have posted it as an answer actually! Dec 25, 2013 at 15:57

In this case the singularity $x^2+y^3+z^6=0$ is a double point which is an ellitpic hypersuface singularity in $\mathbb C^3$, it is not rational. For more examples you can check Chapter 4 in Miles Reid's notes http://arxiv.org/pdf/alg-geom/9602006.pdf