Let $X$ be a smooth irreducible projective surface with generic point $\eta$ over an algebraically closed field $k$, $E$ a rank 2 vector bundle on $X$. Consider Quot scheme $Q=\mathrm{Quot}_{E/X/k}^{\Phi,H}$. We assume that the quotient sheaves parameterized by $Q$ are all torsion free of rank 1. (This is possible, e.g., the kernels of these quotients have maximal Gieseker slope.) Let $K$ be the function field of $X$.
We can map a closed point $t\in Q$ to a $K$-point of $\mathbb{P}^1_K$, i.e., $t$ gives a torsion free quotient $E\rightarrow G_t\rightarrow 0$, hence a locally free quotient on generic point $E_\eta\rightarrow G_{t,\eta}\rightarrow 0$, hence a $K$-point on $\mathbb{P}^1_K$. By Proposition 1 of Stacy G. Langton's paper, this map is injective.
My question is: can we estimate $\dim Q$ by this map, for example $\dim Q\leq \dim \mathbb{P}^1_K$? This map is no a morphism of scheme. I want to construct an injective morphism but fail.