Geography of Gorenstein stable log surfaces (Q2787981)

The modular compactification of the Gieseker's moduli space of (canonical models of) surfaces of general type with fixed numerical invariants \(K^2_X=a\) and \(\chi(K_X)=b\) is given by the moduli space of \textit{stable surfaces} with the same invariants. Stable surfaces are the 2-dimensional analog of stable surfaces: they are projective surfaces that are normal crossings in codimension 1, have ``not too bad'' (i.e., \textit{slc}) singularities in codimension 2 and such that a multiple of the canonical class \(K_X\) is an \textbf{ample} Cartier divisor. As in the curve case, where one considers curves with marked points, it is natural to extend this definition to the relative case of \(2\)-dimensional stable pairs \((X,\Delta)\), where \(\Delta\) is a divisor of \(X\) that does not contain any \(1\)-dimensional component of the singular locus of \(X\) and \(K_X+\Delta\) satistfies properties that generalize the definition of stable surface.NEWLINENEWLINEWhile the moduli space of smooth curves of fixed genus is a dense open set in the moduli space of stable curves, the moduli space of stable surfaces has components that do not intersect the locus of surfaces with at most canonical singularities. This raises the so-called geographical question: what are the pairs \((a,b)\) such that there exists a stable surface with \(K^2=a\) and \(\chi(K_X)=b\)?NEWLINENEWLINEIn this paper, the authors take up the geographical question for Gorenstein stable pairs. They first prove two fundamental numerical inequalities for the numerical invariants \(\chi\) and \(K^2\) of a Gorenstein stable pair of dimension 2, that they call ``\(P_2\)-inequality'' and ``stable log Noether inequality''. Then they give examples with \(K^2=a\), \(\chi=b\) for almost all the pairs of integers in the region of the plane described by these inequalities. It is an open question whether the values for which there are no examples actually occur.