2.2.1. The Rees algebra and the affine blow-up🔗

For an ideal I of a ring R, the Rees algebra R[It] = \bigoplus_{n \geq 0} I^n t^n, a subring of R[t], is graded by the power of t, with R in degree zero; for I = 0 it is R, and for I = R it is R[t].

For a graded ring A = \bigoplus_{n \geq 0} A_n, the scheme \operatorname{Proj} A has as points the homogeneous prime ideals not containing the irrelevant ideal A_+ = \bigoplus_{n > 0} A_n. It is covered by the charts D_+(f) = \operatorname{Spec}(A_f)_0 for homogeneous f of positive degree, where (A_f)_0 is the degree-zero part of the localisation, and the inclusion A_0 \to (A_f)_0 makes it a scheme over \operatorname{Spec} A_0. The blow-up of \operatorname{Spec} R along V(I) is \operatorname{Proj} R[It], over \operatorname{Spec} R through the degree-zero piece. If I = (f_1, \dots, f_m), it is covered by the charts D_+(f_i t) = \operatorname{Spec} R[I / f_i], where R[I/f_i] is the subring of R_{f_i} generated by the fractions f_j / f_i (when f_i is a non-zero-divisor). In such a chart the inverse image of V(I) is generated by the single element f_i, because f_j = (f_j / f_i) f_i: the inverse image of the centre is a divisor, visibly. For R = k[x, y] and I = (x, y) the charts are k[x, y/x] and k[x/y, y], the two charts of the plane blown up at the origin: with v = y/x and u = x/y the blow-up map is (x, v) \mapsto (x, xv) on the first and (u, y) \mapsto (uy, y) on the second, and the exceptional divisor is x = 0 in the first. Geometrically, when V(I) is nowhere dense, \operatorname{Proj} R[It] is the closure in \operatorname{Spec} R \times \mathbb{P}^{m-1} of the graph of x \mapsto [f_1(x) : \dots : f_m(x)], defined off V(I); over a point of V(I) it records the limiting directions of approach. For I = 0 the irrelevant ideal is zero, every prime contains it, and \operatorname{Proj} is empty.

2.2.2. Gluing the affine blow-ups🔗

For a general scheme X and ideal sheaf \mathcal{I}, the blow-up is \operatorname{Proj}_X \bigoplus_n \mathcal{I}^n, obtained by gluing the affine blow-ups of \operatorname{Spec} \Gamma(X, U) along \mathcal{I}(U) over the affine opens U of X with the relative gluing lemma [TSPA, Tag 01LH], in Mathlib's form Scheme.Cover.RelativeGluingData. The lemma asks that for U \subseteq V the piece over U be the preimage of U in the piece over V (the natural transformation to the affine cover is equifibered, Mathlib's NatTrans.Equifibered); for blow-ups this holds because blowing up commutes with restriction to open subsets, a case of the flat pull-back of blow-ups (The universal property).

2.2.3. The universal property🔗

A morphism f \colon Y \to X is admissible for Z if the inverse image f^{-1}(Z) is an effective Cartier divisor on Y: its ideal sheaf f^{-1}\mathcal{I}_Z \cdot \mathcal{O}_Y is invertible, locally generated by one non-zero-divisor (Definition 2.1.4.1). The blow-up \pi \colon \mathrm{Bl}_Z X \to X is admissible for Z and terminal among admissible morphisms: every admissible f factors as f = \pi \circ \tilde f for a unique \tilde f \colon Y \to \mathrm{Bl}_Z X [TSPA, Tag 0806]; it is Kollár's B_Z X [Kol07, Notation 19]. Being so characterised, it is unique up to unique isomorphism over X, and the following facts, which the statements rely on, follow.

• If Z is already a Cartier divisor, \pi is an isomorphism. In particular the blow-up with empty centre (\mathcal{I}_Z = \mathcal{O}_X, the ideal sheaf ⊤) changes nothing, and \pi is an isomorphism over X \setminus Z.

• Blow-ups pull back. For h \colon Y \to X the blow-up of Y along h^{-1}(Z) maps to \mathrm{Bl}_Z X over h, and when h is flat (an open immersion, or a smooth morphism) the square is cartesian, \mathrm{Bl}_{h^{-1}Z} Y = \mathrm{Bl}_Z X \times_X Y [TSPA, Tag 0805]. This is the step behind the pull-back of a succession of blow-ups (Definition 2.4.3.1), and so behind the compatibility of the functorial theorems with smooth morphisms (Definition 4.2.4.3).

• Blowing up a whole component deletes it. If Z contains an irreducible component of a reduced X, then \mathcal{I}_Z vanishes there and \mathrm{Bl}_Z X has nothing over that component. Main Theorem II uses this in a degenerate case (Theorem 4.2.1.2).

• Smooth centre in smooth ambient. If X is smooth and Z is smooth of codimension c \geq 2, then \mathrm{Bl}_Z X is smooth and E = \pi^{-1}(Z) \to Z is the bundle of projective spaces \mathbb{P}^{c-1} of normal directions \mathbb{P}(N_{Z/X}). The principalization theorems blow up only such centres, and assert this of their successions (Definition 4.2.1.1, Definition 4.2.4.2); a resolution of a singular X is read off from such a succession by restriction to the strict transforms of X, whose centres need not be smooth [Kol07, Warning 23].

The centre is a subscheme, not a set. The blow-up depends on the ideal, not only on its zero set. Blowing up the plane along (x, y) gives a smooth surface. Blowing it up along (x^2, y), which has the same zero set, gives a surface with a chart \operatorname{Spec} k[x, y, u]/(yu - x^2), which is singular. This is why the library takes the centre as an ideal sheaf, D : X.IdealSheafData, and why "the centre is smooth" is a condition on the subscheme.

2.2.4. Transforms along a blow-up🔗

2.2.5. The exceptional divisor and the push-forward of a blow-up🔗

2.2.6. Example: resolving the cusp🔗

Take the cusp C = \{y^2 = x^3\} in the plane, singular at the origin, and blow up the plane at the origin. In the chart (x, v) \mapsto (x, xv) the equation becomes x^2 v^2 = x^3, that is, x^2 (v^2 - x) = 0. The factor x^2 is the exceptional divisor E_1 = \{x = 0\}, counted twice. What remains, x = v^2, is the strict transform C_1, the closure of the part of \pi^{-1}(C) lying off E_1: a parabola, which is smooth. In the other chart, (u, y) \mapsto (uy, y), the equation is y^2(1 - u^3 y) = 0 and C_1 does not meet E_1. Over \mathbb{C} the cusp is the image of t \mapsto (t^2, t^3), and C_1 is its smooth parametrising curve, with v = y/x = t.

The curve on its own. The strict transform of C is the blow-up of C itself at the point [TSPA, Tag 080E]. So C_1 \to C is a succession of one blow-up of C that is a strong resolution (Definition 4.2.6.2, Theorem 4.2.7.1): C_1 is smooth, the map is an isomorphism away from the cusp point, and the preimage of \operatorname{Sing} C = \{0\} is one point on a smooth curve, a simple normal crossings divisor. Whether Kollár's resolution functor assigns exactly this blow-up to C depends on his construction; the theorems assert only these properties.

Inside the plane. Viewed in the plane, C_1 is tangent to E_1 at the origin of the chart (the parabola x = v^2 touches the line x = 0), in the horizontal direction, the tangent line y = 0 of the cusp. Making the curve and all the exceptional curves cross normally takes two more blow-ups of points, each computed, like the first, in local coordinates centred at the point.

1. Blow up the tangency point. In the chart x = vw the equation x^2(v^2 - x) becomes v^3 w^2 (v - w): the new exceptional curve E_2 = \{v = 0\} with multiplicity 3, the strict transform of E_1 as \{w = 0\} with multiplicity 2, and C_2 = \{v = w\}. The three curves meet transversally in pairs, but all three pass through one point, which is not normal crossings (Normal crossings).

2. Blow up the triple point. In the chart w = vs the equation becomes v^6 s^2 (1 - s): the new curve E_3 = \{v = 0\} with multiplicity 6, the strict transform of E_1 at s = 0, C_3 at s = 1, and E_2 at s = \infty (in the other chart). The curves E_1, C_3, E_2 each cross E_3 transversally at distinct points, and none of them meet each other. The total transform 6E_3 + 3E_2 + 2E_1 + C_3 has simple normal crossings.

This embedded problem is the one the proofs solve. Kollár's proof embeds X in a smooth ambient space and principalizes the ideal of X (Theorem 4.2.4.6), making its pull-back the ideal of a simple normal crossings divisor, as v^6 s^2 (1 - s) is above; the resolution of X is read off from that succession, restricted to X and stopped at the right moment (the proof sketch after Theorem 4.2.6.3).