Example 89.9.3. There exists an excellent $2$-dimensional Noetherian local ring and a modification $X \to S = \mathop{\mathrm{Spec}}(A)$ which is not a scheme. We sketch a construction. Let $X$ be a normal surface over $\mathbf{C}$ with a unique singular point $x \in X$. Assume that there exists a resolution $\pi : X' \to X$ such that the exceptional fibre $C = \pi ^{-1}(x)_{red}$ is a smooth projective curve. Furthermore, assume there exists a point $c \in C$ such that if $\mathcal{O}_ C(nc)$ is in the image of $\mathop{\mathrm{Pic}}\nolimits (X') \to \mathop{\mathrm{Pic}}\nolimits (C)$, then $n = 0$. Then we let $X'' \to X'$ be the blowing up in the nonsingular point $c$. Let $C' \subset X''$ be the strict transform of $C$ and let $E \subset X''$ be the exceptional fibre. By Artin's results ([ArtinII]; use for example [Mumford-topology] to see that the normal bundle of $C'$ is negative) we can blow down the curve $C'$ in $X''$ to obtain an algebraic space $X'''$. Picture
We claim that $X'''$ is not a scheme. This provides us with our example because $X'''$ is a scheme if and only if the base change of $X'''$ to $A = \mathcal{O}_{X, x}$ is a scheme (details omitted). If $X'''$ where a scheme, then the image of $C'$ in $X'''$ would have an affine neighbourhood. The complement of this neighbourhood would be an effective Cartier divisor on $X'''$ (because $X'''$ is nonsingular apart from $1$ point). This effective Cartier divisor would correspond to an effective Cartier divisor on $X''$ meeting $E$ and avoiding $C'$. Taking the image in $X'$ we obtain an effective Cartier divisor meeting $C$ (set theoretically) in $c$. This is impossible as no multiple of $c$ is the restriction of a Cartier divisor by assumption.
To finish we have to find such a singular surface $X$. We can just take $X$ to be the affine surface given by
in $\mathbf{A}^3_\mathbf {C} = \mathop{\mathrm{Spec}}(\mathbf{C}[x, y, z])$ and singular point $(0, 0, 0)$. Then $(0, 0, 0)$ is the only singular point. Blowing up $X$ in the maximal ideal corresponding to $(0, 0, 0)$ we find three charts each isomorphic to the smooth affine surface
which is nonsingular with exceptional divisor $C$ given by $x = 0$. The reader will recognize $C$ as an elliptic curve. Finally, the surface $X$ is rational as projection from $(0, 0, 0)$ shows, or because in the equation for the blowup we can solve for $x$. Finally, the Picard group of a nonsingular rational surface is countable, whereas the Picard group of an elliptic curve over the complex numbers is uncountable. Hence we can find a closed point $c$ as indicated.
Comments (0)