I wan to know if Grassmannians are rational connected? Any reference describe how to tell if a variety is rational connected or not?
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4$\begingroup$ Rational varieties are rationally connected. $\endgroup$– Dan PetersenJul 12, 2014 at 9:59
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$\begingroup$ This is not exactly what you expect, but still Grassmanians are Fano (by the Euler-sequence-like characterization of its tangent bundle) and therefore by Mori's Theory they are rationally chain connected. $\endgroup$– MaciekJul 12, 2014 at 21:14
1 Answer
Let $G(k,n)$ be the Grassmannian parametrizing $k$-linear subspaces of $\mathbb{P}^n$. By taking local coordinates on $G(k,n)$ one can prove that $G(k,n)$ is a smooth rational variety of dimension $(k+1)(n-k)$. This is a standard fact. You may look for instance at pag $3$ of these notes:
http://math.rice.edu/~evanmb/math465spring11/math465Grassmannians.pdf
Now, any rational variety (indeed any unirational variety) is rationally connected. Let $X$ be a proper unirational variety of dimension $n$. Then there is a dominant rational map $f:\mathbb{P}^n\dashrightarrow X$. Now, take two general points $x_1,x_2\in X$, and consider two points $y_1\in f^{-1}(x_1)$ and $y_2\in f^{-1}(x_2)$. Consider the line $L\subseteq\mathbb{P}^n$ through $y_1,y_2$. Then $f(L)$ is an irreducible rational curve in $X$ through $x_1,x_2$. Hence $X$ is rationally connected.