Projective minus-one contraction and crepant canonical-surface resolution

Choose an ample Cartier divisor D on T and put $d=DE>0$ and $B=D+dE$. Then B $E=0$, $B^2=D^2+d^2>0$, and $BC>0$ for every irreducible curve C≠E, because $DC>0$ and E C≥0. Adjunction gives $K_{T}E=-1$.

For each $j=0,1,2$ and all sufficiently large integers k, the divisor $kB-jE-K_T$ is ample. Indeed choose k so that $kD-K_T$ is ample and kd−j≥0. On C≠E its intersection is $(kD-K_T)C+(kd-j)EC>0$; on E it is $j+1>0$. Its square has leading term $k^2{B}^2>0$ for sufficiently large k. The actual surface Nakai--Moishezon criterion proves ampleness. Choose one k working for all three j and with kD very ample.

Smooth adjoint vanishing gives H¹(T,kB−E)=H¹(T,kB−2E)=0. Hence sections of kB surject onto $H^0(E,O_E)=C$, and sections of kB−E surject onto $H^0(E,O_E(1))$. The complete |kB| is globally generated along E. Off E, multiply all sections of kD by the canonical section of kdE: these give a subsystem of |kB| generating and separating points and tangent vectors on T minus E. These sections all vanish on E and have no common zero off E, so no point outside E has the same image as E.

Thus the complete |kB| gives a projective morphism with fibre E over one point and an immersion on the complement. Take the normalization $T_1$ of its projective image. Normalization is finite; the morphism factors through it because T is normal. This is a projective birational morphism $c:T\to T_1$, an isomorphism outside E, with exactly E over the contracted point q. It remains to prove q smooth; normality alone does not supply that assertion.

Choose $s_0\in H^0(T,kB)$ restricting to a nonzero constant on E. Choose $s_1,s_2\in H^0(T,kB-E)$ whose restrictions form a basis of $H^0(E,O_E(1))$. Regard these as sections of kB vanishing along E. On the chart $s_0\mathrm{\ne }0$ their ratios $u=s_1\mathrm{/}{s}_0$ and $v=s_2\mathrm{/}{s}_0$ are regular functions descending to $T_1$: they are ratios of the actual coordinates of the complete projective image.

Near E the ideal generated by u,v is exactly $O_T(-E)$. Their images generate its conormal quotient $O_E(1)$, and Nakayama gives generation along E and hence on an open neighbourhood. Write locally $u=t$ a, $v=t$ b for a local equation t of E; a and b have no simultaneous zero there. The two blowup charts (u,v/u) and (u/v,v) therefore define a morphism to $B{l}_0(A^2)$ near E. Its restriction to E is the degree-one map [a:b]:P¹→P¹ and hence an isomorphism. On a chart where a is a unit, du is nonzero in the normal direction and the derivative of b/a is nonzero in the tangent direction along E. The analogous calculation holds where b is a unit. Thus this morphism is etale at every point of E.

The two functions define $h:T_1\to A^2$ near q. Their common zero set pulls back to E near E, so q is an isolated point of $h^{-1}(0)$. Shrink around q using properness of c: the image of the closed complement of the etale neighbourhood of E does not contain q. On the resulting punctured neighbourhood of q, h is etale, since c and $B{l}_0(A^2)\to A^2$ are isomorphisms off their exceptional curves. This is an actual coordinate calculation, not a formal-neighbourhood assertion.

Apply the exact finite-after-elementary-etale-localization theorem to h at the isolated fibre point q. It gives an elementary etale neighbourhood (U',0')→(A²,0), with the same residue field C, and an affine open V in the base change of the chosen neighbourhood of $T_1$ that is finite over U' and has just one point q' over 0', with residue field C. Localize at 0'. The base ring A' is regular of dimension two; V is normal, and its restriction to Spec A' minus its closed point is finite etale by Section 2.

The supplied purity theorem for finite covers of regular local rings extends that punctured finite etale cover to a finite etale cover of Spec A'. The existing finite normal V is that extension: both are the integral closure of A' in the same finite generic algebra, so they agree. Therefore V is etale over A' also at q', and its local ring is regular.

The map from V to $T_1$ is etale at q' with the same complex residue field. Its completed local ring is therefore the completed local ring of $T_1$ at q. Completion preserves dimension and embedding dimension, so regularity of the former implies regularity of $O_{T_1,q}$. This proves the contraction conclusion with the exact purity dependency exposed. It does not call a normal contraction target smooth without a proof.

Suppose also π:T→S is a projective morphism and π(E) is one point. It factors through $c:T\to T_1$. To see this without Stein factorization, take the closure of the graph of the induced rational map $T_1⇢S$ in $T_1\times S$. The morphism from T to this graph exists by (c,π), is onto, and the fibre of the graph over q is the image of π(E), a single point. The projection of the graph to normal $T_1$ is thus proper birational and quasi-finite at that point. The supplied proper-birational-normal-target theorem makes it an isomorphism near q; elsewhere it already is the graph of a morphism. Hence the rational map extends and π factors through the smooth $T_1$.

Such a contraction strictly lowers the numerical Picard rank: pullbacks of numerical divisor classes from $T_1$ are orthogonal to E, while E²=−1. Pullback is injective for a birational morphism, so its image is a proper subspace of the finite-dimensional numerical divisor space of T. Thus only finitely many consecutive such contractions can occur. Finite dimensionality follows also by mapping first Chern classes into the finite-dimensional H² of the compact projective surface; homologically trivial classes are numerically trivial by the intersection pairing.

Start with a smooth projective resolution of S and contract any exceptional smooth rational minus-one curves, using Section 4 to remain over S. Termination follows from the rank drop. On the terminal smooth resolution write $K_T={\pi }^{\ast }{K}_S+C$ with C effective integral exceptional by the actual canonical hypothesis. The exceptional intersection form is negative definite: the pullback of an ample divisor on normal S has positive square, is orthogonal to all exceptional curves, and the actual Hodge-index equality criterion gives negative definiteness by the nonnegative-off-diagonal argument used in the nonvanishing packet.

If C≠0 then $C^2<0$. Since C²=Σ $a_i(C{E}_i)$, some component with $a_i>0$ has $K_T{E}_i=C{E}_i<0$, since ${\pi }^{\ast }{K}_S{E}_i=0$. Adjunction gives $K_T{E}_i=2p_a(E_i)-2-E_i^2$.

The negative integer $E_i^2$ and $p_a(E_i)\ge 0$ force $p_a(E_i)=0$ and $E_i^2=-1$. An integral projective curve of arithmetic genus zero over C is smooth rational: the normalization genus and the finite nonnormality length are nonnegative and sum to its arithmetic genus. Thus $E_i$ is an exceptional smooth rational minus-one curve, contradicting termination. Consequently $C=0$ and $K_T={\pi }^{\ast }{K}_S$. This argument uses only Cartier $K_S$ and canonical discrepancies, not triviality of $K_S$.

Let H be an ample Cartier divisor on S and $L={\pi }^{\ast }H$. For all sufficiently large integers n, $nH-K_S$ is ample. Its pullback $N={\pi }^{\ast }(nH-K_S)$ is nef and big. The actual smooth nef-big adjoint vanishing theorem gives $H^1(T,K_T+N)=H^1(T,nL)=0$, using the proved crepancy $K_T={\pi }^{\ast }{K}_S$. For n sufficiently large, Serre vanishing and the Leray sequence identify $H^0(S,R^1{\pi }_{\ast }{O}_T\otimes O_S(nH))$ with H¹(T,nL), because ${\pi }_{\ast }{O}_T=O_S$ by normality and the outside coherent cohomology groups vanish. A nonzero coherent sheaf has nonzero sections after sufficiently high ample twist, so $R^1{\pi }_{\ast }{O}_T=0$.

The low-degree Leray sequence now identifies $H^1(T,O_T)\cong H^1(S,O_S)$. If $K_S\sim 0$ and $H^1(S,O_S)=0$, the resolution has $K_T\sim 0$ and $H^1(O_T)=0$, exactly as required by the nef-big K3 argument. If $K_S\sim -qH$ with integer $q>0$, then $K_T+qL\sim 0$ and qL is nef and big; the same adjoint vanishing gives $H^i(T,O_T)=0$ for every $i>0$. This covers $q=1,2,3$ weak del-Pezzo/quadric/plane resolution inputs without assuming a rational-singularity theorem. No stronger classification, proper formal-functions theorem or simple connectivity is asserted.