Scheme-Theoretic Blow-Up: Construction & Applications

• Scheme-theoretic blow-up is a construction in algebraic geometry that replaces a subscheme with a Cartier divisor by using the relative Proj of the Rees algebra.
• The method uses explicit affine charts and local models to compute the blow-up, as seen in the classical example of blowing up A² at the origin.
• Applications include resolving singularities, analyzing morphisms and transforms, and generalizing to conductor blow-ups and families of sections.

A scheme-theoretic blow-up is a fundamental operation in algebraic geometry that resolves subschemes by replacing them with Cartier divisors, governed universally by an explicit construction via the relative Proj of the Rees algebra. It serves as a crucial tool for desingularization, birational geometry, and the analysis of morphism behavior. Its definition, properties, and applications admit generalizations to contexts such as conductor blow-ups and families of sections.

1. Universal Property and Construction

Let $X$ be a scheme and $Z \subset X$ a closed subscheme defined by a quasi-coherent ideal sheaf $I \subset \mathcal{O}_X$. The scheme-theoretic blow-up of $X$ along $Z$, denoted $\operatorname{Bl}_Z X$, is a scheme equipped with a morphism $\pi: \operatorname{Bl}_Z X \to X$ satisfying:

• $\pi^{-1}(Z)$ is a Cartier divisor in $\operatorname{Bl}_Z X$.
• Universality: For any scheme $Y$ with morphism $f: Y\to X$ such that $f^{-1}(Z)$ is a Cartier divisor in $Y$, there exists a unique $g: Y \to \operatorname{Bl}_Z X$ with $f = \pi \circ g$.

Explicitly, this blow-up is constructed as the relative Proj of the Rees algebra: $$R = \bigoplus_{n \geq 0} I^n t^n \subset \mathcal{O}_X[t], \qquad \operatorname{Bl}_Z X = \operatorname{Proj}_X R.$$ The morphism $\pi$ is inherently projective and birational onto the schematic image of $X$, and $$\pi^{-1}(I)\cdot \mathcal{O}_{\operatorname{Bl}_Z X}$$ becomes invertible, reflecting the Cartier property. The blow-up is unique up to unique isomorphism over $X$ (Hauser, 2014).

2. Affine Charts and Local Models

For $X = \operatorname{Spec} A$ and $I = (f_1, \ldots, f_r) \subset A$, the Rees algebra $R = A[f_1 t, \ldots, f_r t]$ localizes over charts $$D_+(f_i t) = \operatorname{Spec} A[ f_1 / f_i, \ldots, f_r / f_i ]$$. Thus, each chart corresponds to localization by a generator of $I$; on overlaps, these glue to reconstruct $\operatorname{Bl}_I \operatorname{Spec} A$. In these charts, the equation $f_i = 0$ defines the exceptional divisor (Hauser, 2014).

This local nature allows computations and explicit examples. For instance, the blow-up of $\mathbb{A}^2$ at the origin with $I = (x, y)$ is built from $x$- and $y$-charts, separating the coordinates and ensuring the exceptional divisor is regular.

3. Exceptional Divisor and Transform Theory

The exceptional divisor $E$ of the blow-up is defined scheme-theoretically by the homogeneous ideal $R_+ = \bigoplus_{n \geq 1} I^n t^n$ in $\operatorname{Proj} R$; on each chart, the vanishing of a generator $f_i$ cuts out $E$. The ideal sheaf of $E$ is $I \cdot \mathcal{O}_{\operatorname{Bl}_Z X}$. Several notions of transform are defined for an ideal $J \subset \mathcal{O}_X$:

• Total transform: $J^* = J \cdot \mathcal{O}_{X'}$, whose zero locus is $\pi^{-1}(V(J))$.
• Weak (controlled) transform: If $d = \operatorname{ord}_Z J$, $J^! = I_E^{-d} J^*$.
• Strict transform: Defined as the closure of the complement of the exceptional divisor in the total transform, universally as the largest subsheaf

$J^s = \bigcup_{k \geq 0} (J^* : I_E^k).$

If $J$ is principal, $J^s = J^!$ (Hauser, 2014).

4. Functoriality, Base Change, and Composition

Blow-ups possess several robust formal properties:

• Projectivity: $\operatorname{Bl}_Z X \to X$ factors as a closed immersion followed by projection, ensuring projectivity.
• Birationality: If $\operatorname{codim}_X Z \geq 1$, then over $X\setminus Z$, the blow-up is an isomorphism.
• Functoriality: Blow-up commutes with localization and flat base change:

$$\operatorname{Bl}_{Z \times_X Y} Y \cong \operatorname{Bl}_Z X \times_X Y$$

for any flat morphism $Y \to X$.

• Iterativity: The blow-up of a blow-up is itself a blow-up of the original scheme in a suitable “total” ideal, typically the product of pullbacks of centers (Hauser, 2014).

5. Generalizations: Blow-Up Along Fractional Ideals and Conductors

The scheme-theoretic blow-up extends to coherent submodules $I \subset Q_X$ (the sheaf of total quotient rings) of rank 1, i.e., fractional ideals. The construction remains as $\operatorname{Proj}_X \bigoplus_{n \geq 0} I^n$. The universal property persists, but the morphisms considered are "fractional" (preserving associated points and inducing maps of quotient sheaves) (Birghila et al., 2016).

A principal application is the blow-up along the conductor ideal. For the normalization $\nu : X^\nu \to X$ of a reduced Nagata scheme $X$, the conductor is the coherent ideal $$\mathcal{C}_{X^\nu/X} = \mathrm{Ann}_{\mathcal{O}_X}( \nu_* \mathcal{O}_{X^\nu} / \mathcal{O}_X )$$. The blow-up of $X$ along its conductor relates to normalization:

• If $X$ is Cohen–Macaulay (CM) and $X^\nu$ admits an invertible relative canonical module, then $$\operatorname{Bl}_{\mathcal{C}_{X^\nu/X}} X \cong X^\nu$$ if and only if $X^\nu$ is Gorenstein.
• If $X^\nu$ is not Gorenstein, then the blow-up strictly dominates $X^\nu$ (Birghila et al., 2016).

This demonstrates that, under suitable conditions, blowing up the conductor is a universal solution to normalizing $X$ via a blow-up construction.

6. Families of Sections: The Blow-Up Split Sections Family

The blow-up split sections family ("blow $\mathcal{S}$ family" [Editor's term]) generalizes the scheme-theoretic blow-up to families of sections; this extends the classical iterated blow-up to settings parametrizing sections over families. Given an $S$-scheme $X$ and $Y$, let $Z \subset X_Y = X \times_S Y$ be a closed subscheme. The blow-up split sections family $(\mathfrak{B}, b)$ over $X$ is equipped so that $(b_Y)^{-1}(Z) \subset \mathfrak{B}_Y$ is a Cartier divisor, and its universal property mirrors that of the classical blow-up: for any $T \to X$, if $(g_Y)^{-1}(Z)$ is Cartier, there is a unique $h: T \to \mathfrak{B}$ compatible with $b$.

An outline of the construction involves:

1. Blowing up $X_Y$ along $Z$.
2. Taking the universal sections (Weil restriction).
3. "Constfy" construction to ensure constant behavior along fibers.
4. Final blow-up to achieve universality.

This setup recovers the usual scheme-theoretic blow-up when $Y = S$, and more generally, enables the parametrized resolution of families of sections. The family is birational when certain codimension and associated point hypotheses are satisfied. Flattening stratification plays a key role in characterizing the locus of isomorphism, reflecting where the Hilbert polynomial of fibers jumps (Moncusí, 2018).

7. Illustrative Examples and Limitations

Explicit computations in affine space confirm the concrete behavior of the blow-up:

• Blowing up $\mathbb{A}^2$ at the origin separates the coordinates, making the exceptional divisor regular and yielding a smooth strict transform of curves such as $y = x^2$.
• Blowing up along a principal ideal yields the identity morphism, as the center is already Cartier.
• For conductor blow-up, the operation coincides with normalization for reduced curves, hypersurfaces with Gorenstein normalization, and fails to do so in the absence of Cohen–Macaulay or Gorenstein conditions (Hauser, 2014, Birghila et al., 2016).

These concrete cases clarify how the scheme-theoretic blow-up, including its fractional and family-theoretic variants, underpins the resolution of singularities and the construction of refined morphisms in algebraic geometry.