---
tag: 3MGF59XBD3
kind: definition
name: Complete intersection
msc: [14M10, 13C40]
defines: 43DJVJYRBN
uses: [A4H4CZ2V35, JERVQT4C12, 36X2WTHYCB, HH6VEVM6Z0, 6ZBNGS0P98, 2C3PAQ7H50, SJZW2HN1SC, 69R5R2FCS5, EPM2HKGMZS, DQA60Z3FD7]
sources:
  - url: https://en.wikipedia.org/wiki/Complete_intersection
---

## Definition

Let $k$ be an algebraically closed field.
Let $X \subseteq \mathbb{P}^n$ be a [[projective variety|A4H4CZ2V35]] of [[dimension|36X2WTHYCB]] $d$, and let $r = n - d$ be its codimension.
$X$ is a *complete intersection* if its [[homogeneous ideal|6ZBNGS0P98]] $I(X) \subseteq k[x_0, \dots, x_n]$ is generated by $r$ [[homogeneous polynomials|2C3PAQ7H50]].
In the same way, an [[affine variety|JERVQT4C12]] $X \subseteq k^n$ of dimension $d$ is a complete intersection if its [[ideal|HH6VEVM6Z0]] $I(X)$ is generated by $n - d$ polynomials.

## Remarks

By [[Krull's principal ideal theorem|SJZW2HN1SC]], every irreducible component of a nonempty set $V(f_1, \dots, f_s) \subseteq k^n$ has dimension at least $n - s$.
So $n - d$ is the least number of generators that $I(X)$ can have.
If $X \subseteq \mathbb{P}^n$ and $I(X)$ is generated by $r = n - d$ homogeneous polynomials $f_1, \dots, f_r$, then $f_1, \dots, f_r$ is a [[regular sequence|69R5R2FCS5]] in $k[x_0, \dots, x_n]$.
In both the affine and the projective case, the local rings of a complete intersection are [[complete intersection rings|EPM2HKGMZS]].
A weaker notion is a set-theoretic complete intersection: $X = V(f_1, \dots, f_r)$ as sets, without asking that the $f_i$ generate $I(X)$.

Example: if $f$ is an irreducible homogeneous polynomial of positive degree, the hypersurface $V(f) \subseteq \mathbb{P}^n$ has dimension $n - 1$ and ideal $(f)$, so it is a complete intersection.
A linear subspace defined by $r$ independent linear forms is a complete intersection of codimension $r$.

Non-example: the twisted cubic $C \subseteq \mathbb{P}^3$, the image of the [[Veronese embedding|DQA60Z3FD7]] $[s : t] \mapsto [s^3 : s^2 t : s t^2 : t^3]$, has dimension $1$ and codimension $2$.
Its ideal contains no nonzero linear form, because $s^3, s^2 t, s t^2, t^3$ are linearly independent.
Its quadrics form the $3$-dimensional space spanned by $xz - y^2$, $yw - z^2$ and $xw - yz$.
Indeed, this space is the kernel of the surjective map from the $10$ quadrics in $x, y, z, w$ to the $7$ forms of degree $6$ in $s, t$.
A set of homogeneous generators must therefore contain at least $3$ quadrics, so $C$ is not a complete intersection.