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In mathematics, specifically enumerative geometry, the virtual fundamental class E ∙ vir {\displaystyle _{E^{\bullet }}^{\text{vir}}} of a space X {\displaystyle X} is a replacement of the classical fundamental class ∈ A ∗ {\displaystyle \in A^{*}} in its chow ring which has better behavior with respect to the enumerative problems being considered. In this way, there exists a cycle with can be used for answering specific enumerative problems, such as the number of degree d {\displaystyle d} rational curves on a quintic threefold. For example, in Gromov–Witten theory, the Kontsevich moduli spaces
M ¯ g , n {\displaystyle {\overline {\mathcal {M}}}_{g,n}}
for X {\displaystyle X} a scheme and β {\displaystyle \beta } a class in A 1 {\displaystyle A_{1}} , their behavior can be wild at the boundary, such as having higher-dimensional components at the boundary than on the main space. One such example is in the moduli space
M ¯ 1 , n ] {\displaystyle {\overline {\mathcal {M}}}_{1,n}]}