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I refer to this paper on Moduli Spaces by Ravi Vakil.

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What can possibly be a universal family over $G(k,n)$? For example, let us take the set of all linear subspaces of $\Bbb{C}^3$. What would the universal family or $U$ be here?

Matt Samuel
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As written in the text, $U$ is called the tautological vector bundle over $G(k,n)$. Each element in $G(k,n)$ is a $k$-plane in $\mathbb{C}^n$. $U$ is the vector bundle on $G(k,n)$ consisting of all pairs $(P,x)$ where $P\in G(k,n)$ is a $k$-plane and $x\in P$ is a vector in that plane. The horizontal map is the inclusion and the vertical map maps $(P,x)\mapsto P$.

See also this Wikipedia article.

Matt Samuel
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  • Thanks a lot! Could you also look at this question- http://math.stackexchange.com/questions/1060357/does-every-pair-of-varieties-have-a-morphism-between-them –  Dec 10 '14 at 04:56