Weighted Seshadri constants and ellipsoid embeddings

We explore Seshadri constants associated to weighted blow-ups of complex projective varieties and demonstrate how to use this notion to construct symplectic embeddings of ellipsoids. We illustrate the utility of this point of view by providing constructions of full fillings of $\mathbb{CP}^2$ by ellipsoids corresponding to all of the exceptional (post-Fibonacci) steps of the McDuff--Schlenk staircase and some non-obvious embeddings of ellipsoids in ellipsoids.