ECH embedding obstructions for rational surfaces

Let $(Y,A)$ be a smooth rational surface or a possibly singular toric surface with ample divisor $A$. We show that a family of ECH-based, algebro-geometric invariants $c^{\text{alg}}_k(Y,A)$ proposed by Wormleighton obstruct symplectic embeddings into $Y$. Precisely, if $(X,\omega_X)$ is a $4$-dimensional star-shaped domain and $\omega_Y$ is a symplectic form Poincar\'e dual to $[A]$ then $$(X,\omega_X)\text{ embeds into }(Y,\omega_Y)\text{ symplectically } \implies c^{\text{ECH}}_k(X,\omega_X) \le c^{\text{alg}}_k(Y,A)$$ We give three applications to toric embedding problems: (1) these obstructions are sharp for embeddings of concave toric domains into toric surfaces; (2) the Gromov width and several generalizations are monotonic with respect to inclusion of moment polygons of smooth (and many singular) toric surfaces; and (3) the Gromov width of such a toric surface is bounded by the lattice width of its moment polygon, addressing a conjecture of Averkov--Hofscheier--Nill.