Moduli Spaces of $J$-holomorphic Curves with General Jet Constraints

In this paper, we prove that the tagent map of the holomorphic $k$- jet evaluation $j^k_{hol}$ from the mapping space to holomorphic $k$-jet bundle, when restricted on the universal moduli space of simple J-holomorphic curves with one marked point, is surjective. From this we derive that for generic $J$, the moduli space of simple $J$-holomorphic curves in class $\beta\in H_2(M)$ with general jet constraints at marked points is a smooth manifold of expected dimension.