Resolving the quadric cone by one blowup

By [L1], $\tilde{C}$ is a smooth surface and $f:\tilde{C}\to C$ is proper and birational, isomorphic over $C\setminus \{0\}$, with $f^{-1}(0)=Q=E{\mathrm{\mid }}_{\tilde{C}}$ a smooth conic. Thus one ambient blowup resolves the cone. Its source charts are $k[x,v]$, $k[y,t]$, and $k[p,p^{-1},z]$, so no subsequent normalization is required. The conic is a smooth divisor and the strict transform meets the ambient exceptional divisor transversally.

The regular and smooth loci of $C$ are both $C\setminus \{0\}$: $D(x)$ and $D(y)$ cover this complement with rings $k[x,x^{-1},z]$ and $k[y,y^{-1},z]$, while the chain $(0)⊊(x,z)⊊(x,y,z)$ and $\dim C=2$ give vertex local dimension two, whereas its embedding dimension is three because $xy-z^2$ has no linear term. Since $C$ is normal, integral, affine (hence separated), finite type, and two-dimensional, [L3] applies. Characteristic zero makes $k$ perfect: an irreducible polynomial of positive degree has nonzero derivative of smaller degree, hence is relatively prime to its derivative and is separable. The surface theorem therefore supplies a smooth proper birational resolution isomorphic off the vertex. The explicit map in step 1.1 realizes these existence properties with a single blowup and the displayed conic; those extra descriptions come from the chart computation.

The affine integral finite-type scheme $C$ also satisfies [L4], which supplies a canonical smooth proper birational resolution isomorphic over $C\setminus \{0\}$. Thus both general theorems give the existence properties exhibited in step 1.1. The chart calculation identifies the explicit map $f$, and no identification with the canonical resolution is needed for this comparison.

To exhibit the freedom to use a nonminimal resolution, take the explicit $k$-rational point $q=(1:0:0)$ of the exceptional conic $Q$, and blow it up: $b:S={\mathrm{Bl}}_q\tilde{C}\to \tilde{C}$. By [L5], $S$ is smooth, $b$ is proper, has exceptional curve ${\mathbb{P}}_k^1$, and is an isomorphism away from $q$. The source is integral: $q$ is the origin in the chart $\mathrm{Spec}k[x,v]$ of $\tilde{C}$, so its two point-blowup charts are affine planes with dense complement of the exceptional curve, and off $q$ the source agrees with the integral surface $\tilde{C}$. Consequently $f\circ b$ is proper, isomorphic over $C\setminus \{0\}$, and birational, since this same dense open identifies its function field with that of the integral target. This resolution is nonminimal because the nontrivial morphism $b$ contracts its new exceptional curve back to the smooth surface $\tilde{C}$, which already resolves $C$. The smooth positive-dimensional exceptional fibre asserted in the example is the conic of the original single blowup in step 1.1.