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Let X be a non-singular complex projective variety. The virtual Tevelev degree of X associated to (g,d,n) is the (virtual) degree of the forgetful map from the Kontsevich moduli space M_g,n (X,d) of n-pointed stable maps to X of genus g and degree d, to the product M_g,n × X^n .

Recent work of Lian and Pandharipande shows that this invariant is enumerative in many cases, that is, it is the number of degree-d maps from a fixed genus-g curve to X, that send n fixed points in the curve to n fixed points in X. I will speak about a simple formula for this degree in terms of the (small) quantum cohomology ring of X. If X is a Grassmann variety (or more generally, a cominuscule flag variety) then the virtual Tevelev degrees of X can be expressed in terms of the (real) eigenvalues of a symmetric endomorphism of the quantum cohomology ring.
If X is a complete intersection of low degree compared to its dimension, then the virtual Tevelev degrees of X are given by an explicit product formula. I will do my best to keep this talk student-friendly, so the most of it will be about explaining the ingredients of this abstract.
The results are joint work with Rahul Pandharipande.