1
$\begingroup$

Let $X$ be a real algebraic surface with a unique $A_1$ point at $p \in X$ (that is, the germ of $X$ at $p$ is isomorphic to the germ of $\{x^2+y^2=z^2\}$ at the origin). Let $Y \to X$ be the blow up of $X$ at $p$. Is it true that $X$ is homotopy equivalent to the connected sum of $X$ with $\mathbb{RP}^2$?

We know that the blow up of an affine cone is a copy of $\mathbb{RP}^2$ (see answer in Blow-up of the affine cone), but how does the gluing work?

$\endgroup$
4
  • $\begingroup$ I'm a little confused about the real algebraic geometry stuff, but in the connected sum, you would remove a disk from the projective plane, so topologically, there's not necessarily a closed copy of the projective plane inside of what you have left, right? $\endgroup$
    – hunter
    Commented Feb 22 at 18:40
  • $\begingroup$ @hunter Yes, but in connected sum you are gluing them along the boundary afterwards. $\endgroup$
    – Serge the Toaster
    Commented Feb 22 at 19:48
  • $\begingroup$ If $A$ and $B$ are manifolds, it is not obvious to me that $A \sharp B$ has a closed submanifold isomorphic to $A$. On the other hand, $Y$ always has a submanifold isomorphic to $\mathbb{RP}^2$. $\endgroup$
    – hunter
    Commented Feb 22 at 22:43
  • $\begingroup$ That is true in general. Look at the orientable surface of genus $g>1$ produced by gluing $g$ tori. $\endgroup$
    – Serge the Toaster
    Commented Feb 22 at 22:55

0

Reset to default

You must log in to answer this question.