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A degree one del Pezzo surface is the blowup of P^2 at 8 general points. Bythe classical Cayley-Bacharach Theorem, there is a unique 9th point whoseblowup produces a rational elliptic surface with a section. Via thisrelationship, we construct a stable pair compactification of the moduli spaceof anti-canonically polarized degree one del Pezzo surfaces.

Stable pair compactifications of the moduli space of degree one del pezzo surfaces via elliptic fibrations