Let $X$ and $Y$ be smooth del Pezzo surfaces of the same degree $K_X^2=K_Y^2$.
Are the sets $X(\mathbb{C})$ and $Y(\mathbb{C})$ homeomorphic, or at least homotopy equivalent?
MathOverflow is a question and answer site for professional mathematicians. It only takes a minute to sign up.
Sign up to join this communityLet $X$ and $Y$ be smooth del Pezzo surfaces of the same degree $K_X^2=K_Y^2$.
Are the sets $X(\mathbb{C})$ and $Y(\mathbb{C})$ homeomorphic, or at least homotopy equivalent?
Yes, with precisely one exception.
If $K^2 \neq 8$, then the del Pezzo surface is the blow-up of the plane at $9-K^2$ points, so it is homeomorphic to the connected sum of $\mathbb{CP}^2$ with $9-K^2$ copies of $\overline{\mathbb{CP}^2}$.
If $K^2=8$, then we have either the quadric $\mathbb{P}^1 \times \mathbb{P}^1$, which is clearly homeomorphic to $S^2 \times S^2$, or the blow-up of $\mathbb{P}^2$ at one point, which is homeomorphic to $\mathbb{CP}^2 \# \, \overline{\mathbb{CP}^2}$. These surfaces are not homeomorphic, since in the former case the class of $K$ is $2$-divisible in (co)homology whether in the latter case is not, and the divisibility of the canonical class is known to be a topological invariant.