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I have some questions about section 3 of Atiyah's "On analytic surfaces with double points," a short 9 page paper. Section 3 is all dedicated to proving lemma 4.

  1. Near the end of section 3, Atiyah claims that $\widetilde{W}$, the preimage of $W$ in $\widetilde{Q}_3$, is the standard non-singular model of $W$, and that it is isomorphic to $W'$. Both assertions, that the preimage of $W$ in $\widetilde{Q}_3$ is the standard non-singular model, and that it is isomorphic to $W'$, are not at all clear to me. Can someone provide clarification?

  2. At the very end of section 3, Atiyah writes a note: "$V'$ is not unique. At each node of $V$, we have to make a choice of one system of generators, so that there are $2^p$ possible varieties $V'$." What does he mean by this? What generators is he talking about?

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  1. The strict transform of $W$ in the blowup $\tilde{Q}_3$ of $Q_3$ at $O$ is equal to the blowup of $W$ at $O$, and since $W$ has a node at $O$, this blowup is the standard nonsingular model.

  2. To construct of $E$ at the beginning of this section, Atiyah chose one of the two factors $\mathbb{P}^1$ of $V_2$. He refers to this choice as choice of a system of generators.

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    $\begingroup$ Ah. The strict transform clarifies everything. Thank you! $\endgroup$
    – maxo
    Commented Nov 29 at 18:09
  • $\begingroup$ One more question, if I may: throughout the paper, Atiyah talks about some variety not "touching" another variety, even though they may "intersect." I have a guess as to what "touching" means, but can't seem to write down what he precisely means. Any idea? $\endgroup$
    – maxo
    Commented Nov 29 at 18:11
  • $\begingroup$ Atiyah calls it "counter-image". $\endgroup$
    – Sasha
    Commented Nov 29 at 18:11
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    $\begingroup$ I would guess that "to touch" means "to have a common tangent line". $\endgroup$
    – Sasha
    Commented Nov 29 at 18:12

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