# Double lines in \$mathbb{G}(k,n)\$

In the Hilbert scheme $$Hilb_2(mathbb{G}(k,n))$$ of conics in $$mathbb{G}(k,n)$$ consider a point $$(C)$$ corresponding to a conic in $$mathbb{G}(k,n)$$ supported on a line and whose linear span is a plane not contained in $$mathbb{G}(k,n)$$.

Consider the union $$bigcup_{Hin C}Hsubset mathbb{P}^n$$ the union of all the $$k$$-planes parametrized by the conic $$C$$, and let $$Lambda_Csubsetmathbb{P}^n$$ be the linear span of $$bigcup_{Hin C}H$$. Is the dimension $$k+1$$ or $$k+2$$?