Complete intersection

Remarks

By Krull's principal ideal theorem, 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 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. 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 $[s : t] \mapsto [s^3 : s^2 t : s t^2 : t^3]$, 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.