Use the Segre embedding to identify \(\PP^n \times \PP^m\) with its image, and hence give it the structure of a projective variety. Now let \(X \subseteq \PP^n\) and \(Y \subseteq \PP^m\) be quasi-projective varieties, and consider \(X \times Y \subseteq \PP^n \times \PP^m\).
Show that \(X \times Y\) is a quasi-projective variety.
If \(X\) and \(Y\) are both projective, show that \(X \times Y\) is projective.
Show that \(X \times Y\) is a product in the category of varieties.
Solution
Let \[ S\subseteq\PP^N, \qquad N=nm+n+m, \] be the Segre variety from P-AGH214SEGRE, with homogeneous coordinates \(z_{ij}=x_i y_j\). We identify the set \(\PP^n\times\PP^m\) with \(S\).
1. The two set-theoretic projections\[ \pi_1:S\to\PP^n, \qquad \pi_2:S\to\PP^m \]are morphisms.
The opens \[ D_+(z_{ij})\cap S \] cover \(S\). On such an open, the inverse construction in P-AGH214SEGRE recovers the first factor from the \(j\)th column and the second from the \(i\)th row: \[ \pi_1([z_{ab}])=[z_{0j}:\cdots:z_{nj}], \] \[ \pi_2([z_{ab}])=[z_{i0}:\cdots:z_{im}]. \] Because \(z_{ij}\ne0\), neither displayed coordinate vector is zero. Each formula is given by homogeneous linear forms and therefore defines a morphism on the chart. The rank-one minor equations make these formulas agree on overlaps, so they glue to the stated morphisms.
2. The subset \(X\times Y\subseteq S\) is a quasi-projective variety, proving (a).
Since \(X\subseteq\PP^n\) and \(Y\subseteq\PP^m\) are locally closed and the projections are continuous, \[ X\times Y = \pi_1^{-1}(X)\cap\pi_2^{-1}(Y) \] is locally closed in the projective variety \(S\).
It remains to check irreducibility. Choose affine open covers \(\{U_\alpha\}\) of \(X\) and \(\{V_\beta\}\) of \(Y\), refined so that each \(U_\alpha\) lies in a standard affine chart of \(\PP^n\) and each \(V_\beta\) lies in a standard affine chart of \(\PP^m\). Then \[ U_\alpha\times V_\beta = \pi_1^{-1}(U_\alpha)\cap\pi_2^{-1}(V_\beta) \] is open in \(X\times Y\). On the corresponding Segre chart \(D_+(z_{ij})\), the row-and-column formulas of step 1 identify this open with the affine product of \(U_\alpha\) and \(V_\beta\). It is therefore irreducible by P-AGH315AFFPRODUCT.
Any two nonempty members \(U_\alpha\times V_\beta\) and \(U_{\alpha'}\times V_{\beta'}\) intersect: irreducibility of \(X\) gives \[ U_\alpha\cap U_{\alpha'}\ne\varnothing, \] and irreducibility of \(Y\) gives \[ V_\beta\cap V_{\beta'}\ne\varnothing. \] Moreover, their intersection is open in each product chart. Fix one nonempty chart \(U_{\alpha_0}\times V_{\beta_0}\). For every other nonempty chart \(U_\alpha\times V_\beta\), the intersection with the fixed chart is a nonempty open subset of the irreducible variety \(U_\alpha\times V_\beta\), hence is dense in that chart. Therefore every product chart is contained in the closure of the fixed irreducible chart. The fixed chart is consequently dense in \(X\times Y\), and its closure is irreducible. Hence \(X\times Y\) is irreducible. Being irreducible and locally closed in projective space, it is quasi-projective.
3. If \(X\) and \(Y\) are projective, then \(X\times Y\) is projective.
In this case \(X\) is closed in \(\PP^n\) and \(Y\) is closed in \(\PP^m\). Therefore \[ X\times Y=\pi_1^{-1}(X)\cap\pi_2^{-1}(Y) \] is closed in \(S\). The Segre variety \(S\) is projective by P-AGH214SEGRE, and step 2 gives irreducibility. Hence \(X\times Y\) is a projective variety, proving (b).
4. The restricted projections make \(X\times Y\) a categorical product of \(X\) and \(Y\).
The restrictions \[ p_X:X\times Y\to X, \qquad p_Y:X\times Y\to Y \] of the morphisms in step 1 are morphisms by P-AGH310SUBVARIETY.
1.Thus \(h|_W\) is a morphism.Since such neighborhoods cover \(Z\), \(h\) is a morphism globally.
By construction \[ p_X\circ h=f, \qquad p_Y\circ h=g, \] and uniqueness follows because these two coordinates determine every point of the Cartesian set \(X\times Y\). This proves (c).
5. Q.E.D.
Steps <1>2–<1>4 prove (a)–(c), respectively.
Source Collections
Dependencies
P-AGH315AFFPRODUCTProducts of affine varieties and the tensor product of coordinate ringsusesP-AGH214SEGREThe Segre embedding of $\PP^r \times \PP^s$uses
Backlinks
P-AGH321GROUPVARGroup varieties $\GG_a$ and $\GG_m$, and the group structure on $\Hom(X,G)$uses