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We formulate conjectural blowup formulas for Segre and Verlinde numbers onmoduli spaces of sheaves on projective surfaces $S$ with $p_g(S)>0$ and$b_1(S)=0$. As applications we give a give a conjectural formula for theDonaldson invariants of $S$ in arbitrary rank, as well as for the $K$-theoreticDonaldson invariants, and some Donaldson invariants with fundamental matters.

Blowup formulas for Segre and Verlinde numbers of surfaces and higher rank Donaldson invariants