Higher jet evaluation transversality of $J$-holomorphic curves

In this paper, we establish general stratawise higher jet evaluation transversality of $J$-holomorphic curves for a generic choice of almost complex structures $J$ tame to a given symplectic manifold $(M,\omega )$. Using this transversality result, we prove that there exists a subset $\CJ_\omega^{ram} \subset \CJ_\omega$ofsecondcategorysuchthatforevery$J \in \CJ_\omega^{ram}$,thedimensionofthemodulispaceof(somewhereinjective)$J$-holomorphiccurveswithagivenramificationprofilegoesdownby$2n$or$2(n-1)$ depending on whether the ramification degree goes up by one or a new ramification point is created. We also derive that for each $J \in \CJ_\omega^{ram}$thereareonlyafinitenumberoframificationprofilesof$J$-holomorphiccurvesinagivenhomologyclass$\beta \in H_2(M;\Z)$ and provide an explicit upper bound on the number of ramification profiles in terms of $c_1(\beta )$ and the genus $g$ of the domain surface.