The blowup of a variety at a point (Tag D-VARBLOW)
The blowup of \(\AA^n\) at the origin is \[ \Bl_0 \AA^n \da \ts{ (x, \ell) \in \AA^n \times \PP^{n-1} \st x \in \ell } , \] with \(\pi\) the first projection. For \(X\subseteq\AA^n\) a closed subvariety through \(p=0\), \(\Bl_p X\) is the closure of \(\inverseof{\pi}(X \sm \ts{p})\) inside \(\Bl_0 \AA^n\); for a variety \(X\) and \(p\in X\), it is defined through an affine neighbourhood of \(p\) embedded in some \(\AA^n\). The exceptional divisor is \(E \da \inverseof{\pi}(p)\cap\Bl_pX\), and the proper transform of a subvariety \(C \subseteq X\) is the closure of \(\inverseof{\pi}(C \sm \ts{p})\).
\(\pi\) is an isomorphism away from \(E\), hence birational, and \(\Bl_p X\) is again a variety, projective over \(X\). The exceptional divisor \(E\subseteq\PP^{n-1}\) is the projectivization of the tangent cone of \(X\) at \(p\). If \(p\) is a smooth point of \(X\) and \(\dim X=d\), then \(E\cong\PP^{d-1}\) is the space of tangent directions at \(p\); for \(X=\AA^n\), \(E\cong\PP^{n-1}\).
Since \(E\) parametrizes tangent directions at \(p\), the proper transforms of two smooth branches through \(p\) with distinct tangent lines are disjoint over \(p\). The nodal cubic \(y^2 = x^2(x+1)\) has two branches at the origin with tangent lines \(y=\pm x\), and its proper transform after one blowup is smooth. The cuspidal cubic \(y^2 = x^3\) has proper transform \(y_1^2 = x\) in the chart \(y = xy_1\), which is smooth and tangent to \(E\); two further blowups make the total transform a divisor with normal crossings.
The scheme-theoretic definition \(\Bl_Z X = \Proj \bigoplus_{d \geq 0} \mci_Z^d\) also applies to non-reduced centers and makes the exceptional divisor Cartier by construction.
The exceptional curve has self-intersection $-1$ (Tag PR-VAREM1)
Let \(X\) be a smooth projective surface, \(\pi: \Bl_p X \to X\) the blowup at a point, \(E\) the exceptional curve. Then \[ E \cong \PP^1, \qquad E^2 = -1, \qquad K_{\Bl_p X} = \pi^* K_X + E . \] In dimension \(n\) the last formula reads \(K_{\Bl_p X} = \pi^* K_X + (n-1)E\).
(of the self-intersection)
Pullback preserves intersection numbers, and \(\pi^* A . E = 0\) for every divisor \(A\) on \(X\), since \(\pi_* E = 0\). For \(C \subseteq X\) a curve smooth at \(p\), the proper transform satisfies \(\pi^* C = \tilde{C} + E\) and \(\tilde{C}\) meets \(E\) transversally in the one point recording the tangent direction of \(C\). Intersecting with \(E\): \[ 0 = \pi^* C . E = (\tilde{C} + E).E = \tilde{C}.E + E^2 = 1 + E^2 . \]
An alternative geometric computation uses two lines in \(\PP^2\): when they meet transversally away from \(p\), \(\pi^* L_1 . \pi^* L_2 = L_1 . L_2 = 1\); moving them to meet at \(p\) gives \(\pi^* L_i' = \tilde{L}_i' + E\), while the proper transforms miss each other, so expanding gives \(1 = 0 + 1 + 1 + E^2\). Both arguments run on the same two facts, \(\pi^*A.E = 0\) and \(\pi^*C = \tilde{C} + E\).
The consequence is Castelnuovo’s criterion — a smooth rational curve with \(E^2 = -1\) on a surface is the exceptional curve of a blowup and can be contracted — which is what makes minimal models of surfaces possible. The canonical formula is the reason blowing up strictly increases \(K^2\)-deficiency: \(K^2\) drops by one each time, and \(p_a\) and \(p_g\) do not change.
Resolution and minimal models
For curves, normalization resolves singularities because a normal one-dimensional variety is regular.
Normal domains and schemes, and their relation to regularity (Tag D-QJ5M9)
An integral domain is normal if it is integrally closed in its fraction field. A scheme \(X\) is normal if every local ring \(\OO_{X,p}\), for \(p \in X\), is a normal domain.
A regular local ring is normal, and the converse fails in dimension \(\geq 2\). In dimension \(1\) the two agree: a Noetherian local domain of dimension \(1\) is normal exactly when it is a discrete valuation ring, that is, regular.
(Normalization)
Let \(X\) be an integral scheme. A normalization of \(X\) is a normal integral scheme \(\widetilde{X}\) with a dominant morphism \(\nu \colon \widetilde{X} \to X\) such that every dominant morphism \(f \colon Y \to X\) from a normal integral scheme \(Y\) factors uniquely as \(f = \nu \circ \tilde{f}\) for a morphism \(\tilde{f} \colon Y \to \widetilde{X}\).
(Existence of the normalization)
Every integral scheme \(X\) has a normalization, unique up to unique isomorphism. If \(X = \Spec A\) is affine, \(\widetilde{X} = \Spec \widetilde{A}\) for the integral closure \(\widetilde{A}\) of \(A\) in its fraction field, and in general \(\widetilde{X}\) is obtained by gluing these over an affine cover. If \(X\) is a variety over a field \(k\), then \(\nu\) is finite and birational, and \(\widetilde{X}\) is projective when \(X\) is projective [@Har10a, Exercise II.3.8].
The normalization of an integral curve is regular, hence smooth over a perfect field. Thus normalization resolves singularities of curves over perfect fields.
The quadric cone \(V(xy - z^2) \subseteq \AA^3\) is normal and singular at the origin: it is regular in codimension one and Cohen–Macaulay, so it is normal by Serre’s criterion \(R_1 + S_2\). The cuspidal cubic, with coordinate ring \(k[t^2,t^3] \subseteq k[t]\), is not normal, and its normalization is \(\Spec k[t]\).
By Castelnuovo’s theorem, an integral curve on a smooth projective surface is the exceptional curve of the blowup of a smooth surface at a point exactly when it is a \((-1)\)-curve; contracting \((-1)\)-curves until none remain gives a minimal surface. Over an algebraically closed field, the del Pezzo surfaces are \(\PP^1\times\PP^1\) and the blowups of \(\PP^2\) at \(r\le8\) points in general position.
Fano, Calabi--Yau, and general type (Tag D-VARFANO)
Let \(X\) be a smooth complete variety with canonical divisor \(K_X\).
\(X\) is Fano if \(-K_X\) is ample.
\(X\) is Calabi–Yau if \(K_X \sim_\QQ 0\); the stricter convention demands \(\omega_X \cong \OO_X\) together with \(h^j(\OO_X) = 0\) for \(1 \leq j \leq \dim X - 1\).
\(X\) is of general type if \(K_X\) is big.
A del Pezzo surface is a Fano variety of dimension two.
A smooth projective curve of genus \(g\) is Fano if and only if \(g = 0\), Calabi–Yau if and only if \(g = 1\), and of general type if and only if \(g \geq 2\), since \(\deg K=2g-2\). A complete variety with an ample divisor is projective, so every Fano variety is projective.
The two conventions for Calabi–Yau are not equivalent: abelian varieties have \(K_X=0\) but do not satisfy the intermediate-cohomology vanishing, so they satisfy the first convention and not the second. The del Pezzo surfaces are \(\PP^1\times\PP^1\) and \(\PP^2\) blown up at \(r\le8\) general points, with \(K^2=9-r\); the cubic surface is the case \(r=6\) and contains \(27\) lines.
For a surface, \(K_{\Bl_p X} = \pi^* K_X + E\) gives \(K_{\Bl_pX}^2=K_X^2-1\), while \(p_g\) and \(q\) are unchanged; a surface obtained from a surface \(X_0\) by \(r\) point blowups has \(K^2=K_{X_0}^2-r\).