# Zariski open sets in the product of two varieties

Let $$X \subseteq \mathbb{P}^n$$, $$Y \subseteq \mathbb{P}^m$$ be quasi-projective varieties (intersections of Zariski-closed and Zariski-open subsets of $$\mathbb{P}^n$$ and $$\mathbb{P}^m$$, respectively) over an algebraically closed field. We can view $$X \times Y$$ as a quasi-projective variety via the Segre embedding. Let $$U \subseteq X$$ be a nonempty open set and let $$V_u \subseteq Y$$, be a nonempty open set for each $$u \in U$$. Is the set $$\bigcup_{u \in U} \{u\}\times V_u$$ open in $$X \times Y$$? We can assume X and Y are irreducible if it helps.

I think that if this property holds for $$X, Y$$ affine, then it will hold in general, because then the set in question would be the union of open sets.

Is there a good reference that describes what the open subsets of $$X \times Y$$ look like, and finds a base for this (Zariski) topology?

• $\{u\}\times V_u$ is not open in $X\times Y$. What do you mean exactly with a family of open sets (say in the case $X,Y$ affine) ? Your set is dense. Commented Feb 15, 2020 at 21:43
• @reuns Sorry, that was vague. I have edited the question. Is it clear now?
– Ben
Commented Feb 16, 2020 at 15:49

A family of open sets may not lead to a open set in the product. Take $$k=\mathbb{C}$$ and let $$X=Y=\mathbb{A}^1_\mathbb{C}$$ and let $$V_u = Y\setminus \{e^u\}$$ for each $$u\in U=X$$. Clearly $$V_u$$ is open in $$Y$$.
However $$P = \bigcup_{u\in X} \{u\} \times V_u$$ is the complement in $$\mathbb{A}^2_\mathbb{C}$$ of the graph $$\Gamma$$ of the exponential function. Hence $$P$$ is open iff $$\Gamma$$ is closed (in the Zariski topology) and the later is not true because the exponential function is not algebraic.
The, maybe only, reasonable way to define the Zariski topology of a quasiprojective variety is through an embedding. For instance if $$X,Y$$ are affine, $$X\subset \mathbb{A}^n$$, $$Y\subset \mathbb{A}^m$$, then the Zariski topology of $$X\times Y$$ is generated by $$U\cap X\times Y$$ where $$U\subset \mathbb{A}^{n+m}$$ is open.
• Thank you! I suspect that $P$ does not even contain a Zariski open dense subset of $X\times Y$, correct?
• Correct. $\Gamma$ is Zariski dense. Commented Feb 16, 2020 at 20:10