Timeline for Obstruction to rationality of del Pezzo surfaces of degree 4
Current License: CC BY-SA 3.0
4 events
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| May 11, 2016 at 11:09 | comment | added | Daniel Loughran | I have realised that you can prove non-rationality in this case using a result of Iskovskih on conic bundles. Your DP4 has Picard number $1$. The blow-up in a point off a line is a cubic surface of Picard number $2$ with a line. Such cubic surfaces are non-rational by Corollary 2.6 of "Iskovskih - Rational surfaces with a sheaf of rational curves and with a positive square of canonical class". | |
| May 11, 2016 at 9:43 | comment | added | Martin Bright | OK, maybe I need a little more. I just computed the Galois group of the field of definition of the 16 lines in one random example, and it was the whole Weyl group $W(D_5)$. I hope that's enough to guarantee non-rationality. And if that happens in one example, then I think it has to be the generic behaviour: specialising can only make the Galois action smaller. | |
| May 11, 2016 at 9:29 | comment | added | Daniel Loughran | Hi Martin. I like your answer, but can you provide more details how you show non-rationality of such a surface? I mean a DP5 can have irreducible Fano scheme of lines, yet is still rational. | |
| May 11, 2016 at 9:19 | history | answered | Martin Bright | CC BY-SA 3.0 |