I hope this question is suitable for this site and not too easy. It is just about a knoledge of well known results about moduli spaces of complex algebraic surfaces.
With the notation $\mathcal{M}_{K^2,\chi}$ i mean the moduli space of a surface of general type (surface: complex algebraic ) with the invariants $K^2$ and $\chi$ fixed. I've seen that there is an inequality for that moduli space dimension due to Castelnuovo that is $$dim \mathcal{M}_{K^2, \chi}\ge 10\chi -2K^2$$.
If i want to compute directly the dimension of $\mathcal{M}_{K^2,\chi}$ How can i do it?
Maybe useful.
This question rises up from that problem: i know the value of the second term of the previous inequality (48). I know the dimension of the space $H^1(S,T_S)$ where $S$ is a canonical surface and $T$ the tangent bundle (64). Knowing that parameters can i find the dimension of the moduli space of my surface?