Polynomial Lower Bounds for Anticanonical Volumes of Weak Fano Surfaces

Abstract

We prove an explicit lower bound of order ${\epsilon }^3/(1+\log (1/\epsilon ))$ for the anticanonical volume of every complex $\epsilon$-log canonical weak del Pezzo surface. This general estimate is close to being sharp, as it differs from the conjectured optimal quadratic order only by one power of $\epsilon$ and a logarithmic factor.

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