On the complexity of the projective classification of surfaces (Q1895153)

Let \(H(g)\) be the number of all possible Hilbert polynomials of polarized pairs \((X,L)\) with \(X\) smooth complex projective surface, \(L \in \text{Pic} (X)\), \(L\) ample and such that \((X,L)\) has sectional genus \(g\). Here the authors prove the existence of a positive constant \(c > 0\) such that for large \(c\) we have \(cg^2 \leq H (g) \leq g^2 \ln (g)\). They also show that for high \(g\) the number of values of other invariants used often for the classification of polarized pairs \((X,L)\) (e.g. the Hodge numbers, degree \(L^2\) and the dimension of the cohomology groups \(h^i (X,L))\) grows at most as \(g^6\) and at least as \(c'g^3\).