Erratum to: “On surfaces with zero vanishing cycles”

1 Erratum to: manuscripta math. 145, 235–242 (2014) https://doi.org/10.1007/s00229-014-0672-z

In the proof of Proposition 2.3 of the paper [1] I referred to a result form the paper [2] which is valid only in characteristic zero, while later in the paper I use Proposition 2.3 for varieties over an arbitrary algebraically closed field. What follows is a proof of this proposition that is valid in any characteristic.

Proof

Suppose that X is a smooth projective surface and $C\subset X$ is a curve such that $C\cong \mathbb {P} ^1$, the self-intersection index (C, C) equals 1, and $C\subset X$ is an ample divisor. We are to prove that $X\cong \mathbb {P} ^2$.

Observe that any flat deformation $C'\subset X$ of the curve $C\subset X$ is isomorphic to $\mathbb {P} ^1$. Indeed, $\chi (\mathscr {O} _{C'})=1$ and $C'$ is irreducible since $(C',C)=1$ and C is ample. Hence, $h^0(N_{X|C'})=2$, $h^i(N_{X|C'})=0$ for $i>0$, so if B is the connected component of the Hilbert scheme of curves on X which (the component) contains the point corresponding to C, then B is a (smooth) projective surface. If

is the standard diagram representing the family of curves on X parameterized by B, then, for a general (closed) point $x\in X$, one has $\dim q^{-1}(x)=1$.

Let $\sigma :{\tilde{X}}\rightarrow X$ be the blowup of X at x. Proper transforms (with respect to $\sigma$) of the curves from the family B passing through x, are isomorphic to $\mathbb {P} ^1$ and have zero self-intersection. Arguing as in the proof of Proposition 2.2 in [1], we conclude that ${\tilde{X}}$ admits a morphism $\pi :{\tilde{X}}\rightarrow C$ onto a smooth curve C such that the fibers of $\pi$ are the above-mentioned proper transforms, all isomorphic to $\mathbb {P} ^1$. Restricting $\pi$ to the exceptional curve $E=\sigma ^{-1}(x)\subset {\tilde{X}}$, one concludes that there exists a surjective morphism $E\rightarrow C$, whence $C\cong \mathbb {P} ^1$ by Lüroth’s theorem. Besides, E is a section of the morphism $\pi$, so $\tilde{X}$ is a $\mathbb {P} ^1$-bundle over $\mathbb {P} ^1\cong C$, so $X\cong \mathbb {P} (\mathscr {O} _{\mathbb {P} ^1}\oplus \mathscr {O} _{\mathbb {P} ^1}(d))$ for some $d\ge 0$. Since this $\mathbb {P} ^1$-bundle has a section E with self-intersection equal to $-1$, one concludes that $d=1$; the blowdown of such a section of $\mathbb {P} (\mathscr {O} _{\mathbb {P} ^1}\oplus \mathscr {O} _{\mathbb {P} ^1}(1))$ is isomorphic to $\mathbb {P} ^2$, and we are done. $\square$