Twisted cubic

One method for constructing local complete intersections is to take a projective complete intersection variety and embed it into a higher dimensional projective space. A classic example of this is the twisted cubic in ${\displaystyle \mathbb {P} _{R}^{3}}$: it is a smooth local complete intersection meaning in any chart it can be expressed as the vanishing locus of two polynomials, but globally it is expressed by the vanishing locus of more than two polynomials. We can construct it using the very ample line bundle ${\displaystyle {\mathcal {O}}(3)}$ over ${\displaystyle \mathbb {P} ^{1}}$ giving the embedding

${\displaystyle \mathbb {P} _{R}^{1}\to \mathbb {P} _{R}^{3}}$ by ${\displaystyle [s:t]\mapsto [s^{3}:s^{2}t:st^{2}:t^{3}]}$

Note that ${\displaystyle \Gamma ({\mathcal {O}}(3))={\text{Span}}_{R}\{s^{3},s^{2}t,st^{2},t^{3}\}}$. If we let ${\displaystyle \mathbb {P} _{R}^{3}={\text{Proj}}(R[x_{0},x_{1},x_{2},x_{3}])}$ the embedding gives the following relations:

${\displaystyle {\begin{aligned}f_{1}&=x_{0}x_{3}-x_{1}x_{2}\\f_{2}&=x_{1}^{2}-x_{0}x_{2}\\f_{3}&=x_{2}^{2}-x_{1}x_{3}\end{aligned}}}$

Hence the twisted cubic is the projective scheme

${\displaystyle {\text{Proj}}\left({\frac {R[x_{0},x_{1},x_{2},x_{3}]}{(f_{1},f_{2},f_{3})}}\right)}$

Union of varieties differing in dimension

Another convenient way to construct a non complete intersection, which can never be a local complete intersection, is by taking the union of two different varieties where their dimensions do not agree. For example, the union of a line and a plane intersecting at a point is a classic example of this phenomenon. It is given by the scheme

${\displaystyle {\text{Spec}}\left({\frac {\mathbb {C} [x,y,z]}{(xz,yz)}}\right)}$

Homology

Since complete intersections of dimension ${\displaystyle n}$ in ${\displaystyle \mathbb {CP} ^{n+m}}$ are the intersection of hyperplane sections, we can use the Lefschetz hyperplane theorem to deduce that

${\displaystyle H^{j}(X)=\mathbb {Z} }$

for ${\displaystyle j<n}$. In addition, it can be checked that the homology groups are always torsion-free using the universal coefficient theorem. This implies that the middle homology group is determined by the Euler characteristic of the space.

Euler characteristic

Hirzebruch gave a generating function computing the dimension of all complete intersections of multi-degree ${\displaystyle (a_{1},\ldots ,a_{r})}$. It reads

${\displaystyle \sum _{n=0}^{\infty }\chi (X_{n}(a_{1},\ldots ,a_{r}))z^{n}={\frac {a_{1}\cdots a_{r}}{(1-z)^{2}}}\prod _{i=1}^{r}{\frac {1}{(1+(a_{i}-1)z)}}}$