Orbifold Degenerations of Hirzebruch Surfaces

We study orbifold degenerations of Hirzebruch surfaces. Our main theorem shows that every such degeneration $X$ arises as a partial smoothing of a toric surface. We then give a combinatorial description of the singularities that arise when $-K_X$ is not nef. This complements previous joint work with G. Urzúa, which treated the case in which $-K_X$ is ample. For Hirzebruch surfaces $\mathbb{F}_k$ with $k\geq 3$, we obtain an explicit description of all possible central fibers. For $k\leq 1$, we use the threefold minimal model program to reduce the problem to the case of central fibers whose anticanonical divisor is nef. The remaining surfaces are toric del Pezzo surfaces of degree $8$ with T-singularities. We classify these by studying the birational geometry arising from mutations of Fano polygons.