By a line I mean a (-1)-curve. Given a weak del Pezzo surface $X$ of degree $d$, how many lines would $X$ contain?
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$\begingroup$ What kind of answer do you expect? This depends of course on the configuration of $(-2)$-curves on $X$. A case by case analysis is not difficult but tedious. $\endgroup$– abxCommented Mar 23, 2022 at 13:04
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$\begingroup$ @abx I was hoping that the number of lines would be the same as for del pezzos of degree $d$, but this might of been too optimistic of me. How would you determine the number of lines depending on the configuration of $(-2)$-curves? $\endgroup$– H UCommented Mar 23, 2022 at 13:08
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$\begingroup$ Let me give an example: the blowing up of 6 points on $\mathbb{P}^2$. If the points are general, you get the (in)famous 27 lines: the 6 exceptional divisors and the strict transforms of the 15 lines through two of the points and of the 5 conics through 5 of the points. But if your points lie on a smooth conic, the latter disappear and you have only 21 lines. $\endgroup$– abxCommented Mar 23, 2022 at 13:53
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$\begingroup$ @abx I understand! So this would be the case of a weak del pezzo of degree 3 with a single minus $(-2)$-curves (ie it is the minimal desingularisation of a cubic surface with an $A_1$ singularity)? Thanks for the great answer! $\endgroup$– H UCommented Mar 23, 2022 at 13:58
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$\begingroup$ H U: Yes, exactly. $\endgroup$– abxCommented Mar 23, 2022 at 14:37
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