# Minimal resolution of singularities of surfaces

Let $$X$$ be a normal projective irreducible surface over an algebraically closed field $$k$$. Let $$\pi\colon Y\to X$$ be a birational morphism, such that $$Y$$ is a smooth projective surface, and assume that no $$(-1)$$-curve of $$Y$$ is contracted by $$\pi$$. How do we prove that there is an isomorphism $$\pi^{-1}(X_s)\to X_s$$, where $$X_s$$ denotes the smooth locus of $$X$$, and that $$\pi^{-1}(x)$$ is a connected curve for each $$x\in X\setminus X_s$$?

I would say that it follows from the fact that there exists a unique minimal resolution and that on the smooth locus we only have blow-ups. Does someone has a reference for the statement?

The statement that $$\pi^{-1}(x)$$ is (geometrically) connected for all $$x \in X$$ is Stein factorisation or Zariski's main theorem; see e.g. [Hartshorne, Cor. III.11.5].

For the other statement, it suffices to show that if $$\pi \colon Y \to X$$ is a proper birational morphism of smooth surfaces, then $$\pi$$ is obtained as a sequence of blow-ups in points. Indeed, applying this to the smooth locus shows that $$\pi$$ is an isomorphism there, as otherwise the last blowup introduces a $$(-1)$$-curve contracted by $$\pi$$.

This statement is for instance proved in [Beauville, Thm. II.11] (stated under the hypotheses that $$k = \mathbf C$$ and $$X$$ is projective, but this is not used in the proof).

References.

[Beauville] A. Beauville, Complex algebraic surfaces, second edition. London Mathematical Society Student Texts 34. Cambridge Univ. Press, 1996. ZBL0849.14014.

[Hartshorne] R. Hartshorne, Algebraic geometry. Graduate Texts in Mathematics 52. Springer-Verlag, 1977. ZBL0367.14001.

• Thanks for your answer. Why is $\pi^{-1}(x)$ never a point for $x\in X$ singular? [Hartshorne, Cor. III.11.5] shows that the preimage is connected, but why not only one point? Can we show that we would have a local isomorphism in this case? May 3 at 12:39
• If $\pi^{-1}(x) = \{y\}$, then $Y \times_X \operatorname{Spec} \mathcal O_{X,x} \cong \operatorname{Spec} \mathcal O_{Y,y}$, so $\mathcal O_{X,x} \to \mathcal O_{Y,y}$ is finite (since it is proper and quasi-finite). Since it is also birational extension, normality of $\mathcal O_{X,x}$ shows that $\mathcal O_{X,x} \stackrel\sim\to \mathcal O_{Y,y}$. May 3 at 20:01