Oriented cohomology theories provide a general framework for refined topological invariants of schemes, which admits intersection-theory-type calculus. The Chow ring, the \$K_0\$-ring, and algebraic cobordism of Levine--Morel are all instances of such theories. In this talk, we give a presentation of the oriented cohomology ring of the blowup of a smooth scheme along a smooth center. We compute explicit examples of such presentations for the cases of del Pezzo surfaces and the blowup of \$\mathbb{P}^5\$ along the Veronese surface, the latter of which can be identified with the moduli space of complete conics. We demonstrate that one can recover solutions to enumerative problems such as Steiner's \$3264\$ conics in an arbitrary oriented cohomology theory. Finally, we give a presentation to the algebraic cobordism ring of \$\overline M_{0,n}\$, which generalizes Keel's presentation of the Chow ring.