A lower bound for \(K^2_S\) (Q1676505)

A fundamental problem in algebraic geometry consists of finding relations between the geometry of projectively embedded varieties and their intrinsic invariants. In this setting, the authors analyze the possible values of the self intersection \(K_X^2\) of the canonical class of smooth projective surfaces \(S\), embedded with a complete linear system of high degree. The main result of the paper under review shows that a smooth surface \(S\), embedded with a very ample line bundle \(L\) of degree \(d>35\), satisfies the lower bound \(K_S^2\geq -d(d-6)\). The authors also provide a classification of surfaces for which the lower bound is attained.