Past exam of the algebraic Geometry course of the University of Oxford / 2011 / 2

A projective variety is a Zariski-closed subset of projective space ${\mathbb{P}}_k^n$, equivalently the common zero set of homogeneous polynomials. Homogeneity makes their vanishing independent of the nonzero scalar used to represent a projective point.

A morphism is a map locally given by regular coordinate functions in affine charts. Equivalently, near each point of a projective source embedded in ${\mathbb{P}}^n$, a morphism to ${\mathbb{P}}^m$ can be writtenusing homogeneous polynomials of a common degree that are not all zero on that neighbourhood. The expressions must agree on overlaps, and for a specified target variety their values must lie in it. A single global tuple is a sufficient description when available, but is not part of the general definition.

The Segre embedding isRescaling the two representatives multiplies every target coordinate by the same scalar, and at least one product is nonzero, so the map is well defined. In affine source charts its coordinate ratios are regular products, so it is a morphism.

Arrange target coordinates into a matrix $Z=(z_{ij})$. Every image matrix has rank one and satisfies all homogeneous equationsConversely, a nonzero matrix satisfying these equations has rank one. Choose $z_{i_0{j}_0}\ne 0$ and set $x_i=z_{i{j}_0}$, $y_j=z_{i_{0}j}/z_{i_0{j}_0}$. The minors imply $z_{ij}=x_i{y}_j$. Thereforea projective variety. The two projective factors are uniquely recoverable from a nonzero column and row. This also shows injectivity and supplies the local inverse formulas used in the next part.

Choose projective embeddings $X\subseteq {\mathbb{P}}^n$ and $Y\subseteq {\mathbb{P}}^m$. Define their product as the Segre image of the set of pairs $(x,y)$, with its induced variety structure.

It is closed in the Segre variety. To see this directly, for every homogeneous $f\in I(X)$ impose the equations $f(z_{0j},\dots ,z_{nj})=0$ on every column, and for every homogeneous $g\in I(Y)$ impose $g(z_{i0},\dots ,z_{im})=0$ on every row. For a rank-one matrix $z_{ij}=x_i{y}_j$, the column equations are $y_j^{\mathrm{deg}f}f(x)=0$. A nonzero $y_j$ therefore forces $x\in X$. The row equations similarly force $y\in Y$. Thus these homogeneous equations together with the minors cut out exactly $X\times Y$, proving it is projective.

On a chart where $z_{i_0{j}_0}\ne 0$, the projections have formulasThese homogeneous coordinate tuples are nonzero on the chart and agree on overlaps, so both projections are morphisms.

Given morphisms $\alpha :Z\to X$ and $\beta :Z\to Y$, define $\gamma (z)={\sigma }_{n,m}(\alpha (z),\beta (z))$. Locally write their coordinates as $[a_0:\cdots :a_n]$ and $[b_0:\cdots :b_m]$. Then $\gamma$ has coordinates $[a_i{b}_j]$, which are regular with a nonzero coordinate near each point. Hence $\gamma$ is a morphism andThe two projections determine the underlying pair uniquely, so this morphism is unique. This is the product universal property, and it makes the construction independent up to canonical isomorphism of the chosen embeddings.

Let $S=k[x_0,\dots ,x_n]$ and let $B=S/I(X)$ be the homogeneous coordinate ring of a nonempty projective variety. Its Hilbert function is $h_X(t)={\dim }_k{B}_t$. For all sufficiently large integers $t$, this function agrees with a polynomial $P_X(t)$, the Hilbert polynomial.

The dimension $r$ is the largest length of a strict chain of nonempty irreducible closed subvarieties of $X$. Equivalently $r=\dim B-1=\mathrm{deg}{P}_X$. The degree of the embedded variety is the integer specified by the leading termGeometrically it is the number of intersection points with a general linear subspace of codimension $r$, counted with intersection multiplicities. The polynomial's degree is the dimension $r$; the variety's degree is the leading coefficient multiplied by $r!$. Keeping those two meanings separate is important in the next computation.

The homogeneous coordinate ring of the Segre image is the subalgebra $k[x_i{y}_j]\subseteq k[x_0,\dots ,x_n,y_0,\dots ,y_m]$, grading each product $x_i{y}_j$ as degree one. Its degree-$t$ part is precisely the space of polynomials bihomogeneous of bidegree $(t,t)$.

Indeed, every product of $t$ generators has those two degrees. Conversely, any monomial of bidegree $(t,t)$ can pair its $t$ individual $x$ factors with its $t$ individual $y$ factors, expressing it as a product of generators. The distinct resulting monomials are linearly independent. Counting them givesIn fact this agrees with the Hilbert function for every integer $t\ge 0$. Its leading coefficient is $1/(n!m!)$, so the two degree quantities areThus the Segre variety has dimension $n+m$ and embedded projective degree $\left(\genfrac{}{}{0ex}{}{n+m}{n}\right)$.

This gives the Segre Hilbert polynomial and degree directly from its graded monomials.