Lemma 43.13.1. Let $X$ and $Y$ be varieties. Then $X \times Y$ is a variety and $\dim (X \times Y) = \dim (X) + \dim (Y)$.
43.13 Proper intersections
First a few lemmas to get dimension estimates.
Proof. The scheme $X \times Y = X \times _{\mathop{\mathrm{Spec}}(\mathbf{C})} Y$ is a variety by Varieties, Lemma 33.3.3. The statement on dimension is Varieties, Lemma 33.20.5. $\square$
Recall that a regular immersion $i : X \to Y$ of schemes is a closed immersion whose corresponding sheaf of ideals is locally generated by a regular sequence, see Divisors, Section 31.21. Moreover, the conormal sheaf $\mathcal{C}_{X/Y}$ is finite locally free of rank equal to the length of the regular sequence. Let us say $i$ is a regular immersion of codimension $c$ if $\mathcal{C}_{X/Y}$ is locally free of rank $c$.
More generally, recall (More on Morphisms, Section 37.62) that $f : X \to Y$ is a local complete intersection morphism if we can cover $X$ by opens $U$ such that we can factor $f|_ U$ as
\[ \xymatrix{ U \ar[rr]_ i \ar[rd] & & \mathbf{A}^ n_ Y \ar[ld] \\ & Y } \]
where $i$ is a Koszul regular immersion (if $Y$ is locally Noetherian this is the same as asking $i$ to be a regular immersion, see Divisors, Lemma 31.21.3). Let us say that $f$ is a local complete intersection morphism of relative dimension $r$ if for any factorization as above, the closed immersion $i$ has conormal sheaf of rank $n - r$ (in other words if $i$ is a Koszul-regular immersion of codimension $n - r$ which in the Noetherian case just means it is regular immersion of codimension $n - r$).
Lemma 43.13.2. Let $f : X \to Y$ be a morphism of varieties.
If $Z \subset Y$ is a subvariety dimension $d$ and $f$ is a regular immersion of codimension $c$, then every irreducible component of $f^{-1}(Z)$ has dimension $\geq d - c$.
If $Z \subset Y$ is a subvariety of dimension $d$ and $f$ is a local complete intersection morphism of relative dimension $r$, then every irreducible component of $f^{-1}(Z)$ has dimension $\geq d + r$.
Proof. Proof of (1). We may work locally, hence we may assume that $Y = \mathop{\mathrm{Spec}}(A)$ and $X = V(f_1, \ldots , f_ c)$ where $f_1, \ldots , f_ c$ is a regular sequence in $A$. If $Z = \mathop{\mathrm{Spec}}(A/\mathfrak p)$, then we see that $f^{-1}(Z) = \mathop{\mathrm{Spec}}(A/\mathfrak p + (f_1, \ldots , f_ c))$. If $V$ is an irreducible component of $f^{-1}(Z)$, then we can choose a closed point $v \in V$ not contained in any other irreducible component of $f^{-1}(Z)$. Then
\[ \dim (Z) = \dim \mathcal{O}_{Z, v} \quad \text{and}\quad \dim (V) = \dim \mathcal{O}_{V, v} = \dim \mathcal{O}_{Z, v}/(f_1, \ldots , f_ c) \]
The first equality for example by Algebra, Lemma 10.116.1 and the second equality by our choice of closed point. The result now follows from the fact that dividing by one element in the maximal ideal decreases the dimension by at most $1$, see Algebra, Lemma 10.60.13.
Proof of (2). Choose a factorization as in the definition of a local complete intersection and apply (1). Some details omitted. $\square$
Lemma 43.13.3. Let $X$ be a nonsingular variety. Then the diagonal $\Delta : X \to X \times X$ is a regular immersion of codimension $\dim (X)$.
Proof. In fact, any closed immersion between nonsingular projective varieties is a regular immersion, see Divisors, Lemma 31.22.11. $\square$
The following lemma demonstrates how reduction to the diagonal works.
Lemma 43.13.4. Let $X$ be a nonsingular variety and let $W,V \subset X$ be closed subvarieties with $\dim (W) = s$ and $\dim (V) = r$. Then every irreducible component $Z$ of $V \cap W$ has dimension $\geq r + s - \dim (X)$.
Proof. Since $V \cap W = \Delta ^{-1}(V \times W)$ (scheme theoretically) we conclude by Lemmas 43.13.3 and 43.13.2. $\square$
This lemma suggests the following definition.
Definition 43.13.5. Let $X$ be a nonsingular variety.
Let $W,V \subset X$ be closed subvarieties with $\dim (W) = s$ and $\dim (V) = r$. We say that $W$ and $V$ intersect properly if $\dim (V \cap W) \leq r + s - \dim (X)$.
Let $\alpha = \sum n_ i [W_ i]$ be an $s$-cycle, and $\beta = \sum _ j m_ j [V_ j]$ be an $r$-cycle on $X$. We say that $\alpha $ and $\beta $ intersect properly if $W_ i$ and $V_ j$ intersect properly for all $i$ and $j$.
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