Polynomial Lower Bounds for Anticanonical Volumes of Weak Fano Surfaces

RESUMO

We prove an explicit lower bound of order $\varepsilon^3/(1+\log(1/\varepsilon))$ for the anticanonical volume of every complex $\varepsilon$-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 $\varepsilon$ and a logarithmic factor.