# 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/… – Joachim Nov 13 '13 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 '13 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? – Joachim Nov 14 '13 at 18:36
• @ Joachim- oops . Remove the "-1" . – meh Nov 15 '13 at 13:25
• @aginensky, thanks. You could have posted it as an answer actually! – Joachim Dec 25 '13 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