Gromov–Witten Invariants with Naive Tangency Conditions

We introduce Gromov–Witten invariants with naive tangency conditions at the marked points of the source curve. We then establish an explicit formula which expresses Gromov–Witten invariants with naive tangency conditions in terms of descendent Gromov–Witten invariants. Several examples of genus zero Gromov–Witten invariants with naive tangencies are computed in the case of curves and surfaces. In particular, the counts of rational curves naively tangent to an anticanonical divisor on a del Pezzo surface are studied, and via mirror symmetry, we obtain a relation to the local Gromov–Witten invariants.