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Tangent and obstruction spaces for hypersurface deformations

Statement

Assume the Axiom of Choice, inherited from the projective-space cohomology suppliers (The Axiom of Choice). Let $k$ be a field, $n \geq 2$ , $d \geq 1$ , let $f \in S = k [x_{0}, \dots, x_{n}]$ be homogeneous of degree $d$ , let $X = Z (f) \subseteq P_{k}^{n}$ , and assume $X$ smooth over $k$ of pure dimension $n - 1$ . Then the embedded deformation functor of $X$ in the fixed $P_{k}^{n}$ (Embedded deformations of a closed subscheme) has:

tangent space $H^{0} (X, N_{X / P^{n}})$ at the trivial deformation, and $N_{X / P^{n}} ≅ O_{X} (d)$ , so the tangent space is $H^{0} (X, N_{X / P^{n}}) ≅ H^{0} (X, O_{X} (d)) ≅ (S / (f))_{d}, \dim_{k} = (\binom{n + d}{n}) - 1;$
vanishing obstruction space $H^{1} (X, N_{X / P^{n}}) = H^{1} (X, O_{X} (d)) = 0$ , with the embedded deformation functor formally smooth and unobstructed: every embedded deformation over a small extension extends;
for the abstract deformation functor of $X$ as a $k$ -scheme instead the tangent space is ${Ext}_{O_{X}}^{1} (L_{X / k}, O_{X}) ≅ H^{1} (X, T_{X / k})$ and the obstruction group is $H^{2} (X, T_{X / k})$ (Deformation cohomology of a smooth scheme: tangent, obstruction and automorphism spaces); the two problems are related by the normal bundle sequence $0 \to T_{X} \to i^{*} T_{P^{n}} \to N_{X / P^{n}} \to 0$ (Conormal sequence for a closed immersion), which need not have vanishing maps, so the embedded and abstract deformation spaces are different in general.

Facts & Assumptions

Given: a field $k$ , $n \geq 2$ , $d \geq 1$ , a homogeneous form $f \in S$ of degree $d$ with $X = Z (f) \subseteq P_{k}^{n}$ smooth of pure dimension $n - 1$ , and the Axiom of Choice.

[F1]

The embedded deformation functor of $X$ in $P^{n}$ is isomorphic to the equation-deformation functor $R \mapsto {F \in (S \otimes_{k} R)_{d} : F \equiv f m o d m_{R}} / (R^{\times})$ , it is formally smooth and unobstructed, and its tangent space at the trivial deformation is $(S / (f))_{d} ≅ H^{0} (X, O_{X} (d))$ of dimension $(\binom{n + d}{n}) - 1$ . (Embedded flat deformations of a smooth hypersurface are deformations of its equation)

[F2]

$N_{X / P^{n}} = H o m_{O_{X}} (I / I^{2}, O_{X}) ≅ O_{X} (d)$ with $I / I^{2} ≅ O_{X} (- d)$ invertible, $H^{0} (X, O_{X} (d)) ≅ (S / (f))_{d}$ of the stated dimension, and $H^{1} (X, O_{X} (d)) = 0$ . (Cohomology of twists on a smooth hypersurface, Internal Hom of module sheaves, Locally free sheaves of finite rank, degree projective hypersurface)

[F3]

For the smooth $k$ -scheme $X$ the abstract deformation tangent space is ${Ext}_{O_{X}}^{1} (L_{X / k}, O_{X}) ≅ H^{1} (X, T_{X / k})$ and the obstruction group is $H^{2} (X, T_{X / k})$ . (Deformation cohomology of a smooth scheme: tangent, obstruction and automorphism spaces)

[F4]

For the smooth closed immersion $i : X ↪ P_{k}^{n}$ the normal bundle sequence $0 \to T_{X} \to i^{*} T_{P^{n}} \to N_{X / P^{n}} \to 0$ is exact. (Conormal sequence for a closed immersion, Smooth closed immersion is regular with exact conormal sequence, Smooth morphism of schemes)

[F5]

Projective space is smooth by its polynomial charts and the Jacobian criterion with no relations (Relative Jacobian criterion with its presentation hypothesis). For $P_{k}^{1}$ , $H^{1} (P_{k}^{1}, O (2)) = 0$ . (Cohomology of O(d) on projective space)

Proof

technique · read the tangent and obstruction spaces of the embedded functor from the equation functor and the normal-sheaf computation, and compare with the abstract deformation spaces through the normal bundle sequence

1.1F1F2given

Tangent space and normal sheaf. By [F1] the tangent space of the embedded deformation functor at the trivial deformation is $(S / (f))_{d} ≅ H^{0} (X, O_{X} (d))$ , of dimension $(\binom{n + d}{n}) - 1$ ; by [F2] the normal sheaf is $N_{X / P^{n}} ≅ O_{X} (d)$ , so $H^{0} (X, N_{X / P^{n}}) ≅ H^{0} (X, O_{X} (d))$ has the same dimension. This proves part (1).

1.2F1F2given

Obstruction space and formal smoothness. By [F2], $H^{1} (X, N_{X / P^{n}}) = H^{1} (X, O_{X} (d)) = 0$ ; independently, [F1] proves that the equation functor is formally smooth and unobstructed, so every embedded deformation over a small extension extends. This proves part (2).

2.1F2F3F4step 1.1

Comparison with abstract deformations. By [F3] the abstract deformation functor has tangent space $H^{1} (X, T_{X / k})$ and obstruction group $H^{2} (X, T_{X / k})$ ; [F4] provides the normal bundle sequence relating $T_{X}$ , $i^{*} T_{P^{n}}$ and $N_{X / P^{n}}$ . No injectivity or surjectivity of the comparison map between embedded and abstract deformations is asserted, since the maps in the normal bundle sequence need not vanish; the two deformation problems are therefore different in general, as claimed in part (3).

3.1F3F5step 1.1algebra∎

For an explicit difference, take the line $X = Z (x_{0}) \subset P_{k}^{2}$ , which is $P_{k}^{1}$ . Its embedded tangent space has dimension $(\binom{3}{2}) - 1 = 2$ by step 1.1. On the two standard charts of $P^{1}$ , use coordinates $u$ and $v = u^{- 1}$ . The tangent frames satisfy $\partial_{v} = - u^{2} \partial_{u}$ , so after changing one frame by $- 1$ the tangent line bundle is $O (2)$ (the standard twisting transition is $u^{2}$ ). Thus [F5] gives $H^{1} (X, T_{X}) = 0$ , while the embedded tangent space has dimension two. This proves the claimed difference in general without identifying the two functors. Choice is inherited from the cited suppliers.

Depends on

Used by

First-order deformations of a plane conic and of a quadric surface Example

Dependency tree · two levels

110 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Robin Hartshorne, Lectures on Deformation Theory (Berkeley Math 274 draft, 2004/2005) (standard reference, not scraped)
The Stacks Project, The Cotangent Complex, complete chapter (Chapter 92) (standard reference, not scraped)