Intersection theory

Intersection theory studies algebraic cycles modulo rational equivalence and their products.

Algebraic cycle

k

-cycle on a scheme

X

is a finite integer linear combination of integral closed subschemes of dimension

k

. Their free abelian group is denoted

Z_{k} (X)

.

Rational equivalence

Rational equivalence is generated by principal divisors

[div (r)]

of nonzero rational functions on integral subvarieties one dimension larger than the cycles.

Chow group ( $A_{k} (X)$ )

The Chow group is

A_{k} (X) = Z_{k} (X) / Rat_{k} (X)

, the group of

k

-dimensional algebraic cycles modulo rational equivalence.

Chow ring ( $A^{*} (X)$ )

For a smooth variety, intersection of cycles gives the graded Chow ring

A^{*} (X)

.

Localization sequence for Chow groups

For a closed immersion

j : Z ↪ X

with open complement

i : U ↪ X

, restriction and proper pushforward give an exact sequence

A_{k} (Z) j_{*} A_{k} (X) i^{*} A_{k} (U) \to 0.

Cellular decomposition of a scheme

A cellular decomposition filters a scheme by closed subschemes whose successive differences are disjoint unions of affine spaces. The closures of the cells generate its Chow groups.

Chow ring of projective space ( $A^{*} (P^{r})$ )

For the hyperplane class

H

,

A^{*} (P^{r}) ≅ Z [H] / (H^{r + 1}),

and

A_{k} (P^{r}) ≅ Z

is generated by a linear

P^{k}

.

Projective bundle formula for Chow groups

For a rank-

e + 1

E \to X

and

p : P (E) \to X

, powers of

ζ = c_{1} (O (1))

give isomorphisms

⨁_{i = 0}^{e} A_{k - e + i} (X) \sim A_{k} (P (E)) .

Homotopy invariance of Chow groups

For a rank-

r

E \to X

, flat pullback is an isomorphism

A_{k} (X) ≅ A_{k + r} (E)

.

Chern class ( $c_{i} (E)$ )

Chern classes are characteristic classes of vector bundles satisfying functoriality and the Whitney product formula

c (E) = c (E^{'}) c (E^{''})

for a short exact sequence.

Total Chern class ( $c (E)$ )

The total Chern class is

c (E) = 1 + c_{1} (E) + c_{2} (E) + \dots

.

Euler sequence

On projective space, the Euler sequence is

0 \to O \to O (1)^{\oplus (m + 1)} \to T P^{m} \to 0.

Degree of a projective scheme

For a pure

k

-dimensional projective scheme

X \subseteq P^{m}

, its degree is

k!

times the leading coefficient of its Hilbert polynomial, equivalently

\int_{X} H^{k}

.

Normal bundle

For a smooth closed embedding

i : X ↪ Y

, the normal bundle fits into

0 \to TX \to i^{*} T Y \to N_{X / Y} \to 0

.

Self-intersection formula

For a regular closed embedding

i : X ↪ Y

d

, the self-intersection formula is

i^{*} i_{*} (α) = c_{d} (N_{X / Y}) \cap α

.

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Intersection theory by

Wikipedia Bot  1

Intersection theory is a branch of algebraic geometry that studies the intersection of subvarieties within algebraic varieties. It provides a framework for counting the number of points at which varieties intersect, understanding their geometric properties, and understanding how these intersections behave under various operations. Here are the main concepts involved in intersection theory: 1. **Subvarieties**: In algebraic geometry, a variety can be thought of as a solution set to a system of polynomial equations.

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