# The virtual fundamental class as derived intersection

Say $$X$$ is a smooth projective variety and $$\beta\in H_2(X)$$ is a class. Then there is a finite-type proper scheme (or in general, stack) $$SM : = \overline{\mathcal{M}}_{g,n}(X,\beta)$$ of stable maps to $$X$$ from nodal curves of genus $$g$$ with $$n$$ marked points in $$X$$ with class $$\beta$$. Famously, this scheme has a "virtual fundamental class" which is often nowhere close to the fundamental class of $$SM$$ (which may be highly singular and of the "wrong" dimension). However, it is now an accepted fact that the scheme $$SM$$ has a derived thickening, $$\widetilde{SM},$$ which correctly captures the "intersection-theoretic" meaning of $$SM$$ and has the appropriate virtual fundamental class.

All the constructions of such a scheme that I found seem complicated: they involve obstruction theory, deformation to the normal curve, etc. However there seems to me a very straightforward construction using formal intersection theory which has probably been considered by many people, but I have not found it written down. I suspect there is some reason why it does not work or requires too much new technology.

Namely, say $$f:C\to X$$ is a stable map, and $$C = U_1\cup U_2$$ is a cover of $$C$$ by two affines. Let $$U_{12}$$ be their intersection. For any affine curve, $$Map(U, X)$$ is a smooth infinite-dimensional formal (ind-) variety. Now consider the derived intersection in a formal-geometric sense $$Map(U_1,X)\times^L_{Map(U_{12}, X)} Map(U_2, X).$$ This is a (quasi-)smooth derived scheme whose underlying classical scheme is a subscheme of $$Map(C, X)$$, and it can be seen (at least in a neighborhood of the map $$f:C\to X$$) to be locally of finite type (i.e. there is some half-infinite intersection business going on). It seems that this procedure can be performed "consistently" for every fiber of $$SM$$ (possibly involving some patching), and this should be the "correct" derived intersection-theoretic meaning for the moduli of stable maps.

Are there difficulties in using this technique? Why doesn't it seem to exist in the literature?

• Kontsevich's original construction was very similar. He worked analytically and broke the curve into compact pieces with boundary. I think it's valuable motivation that seems to be largely forgotten. But infinite dimensions are hard! I don't think anyone claimed to make it work. Working algebraically and dropping compactness sounds even more difficult. – Ben Wieland Sep 11 at 22:21
• I don’t folllow your construction but for the record, smooth derived schemes are automatically 0-truncated. Also, it is not “complicated” to define $\tilde{SM}$, it is just the obvious extension of the moduli problem from ordinary to simplicial commutative rings. See the paper by Schurg, Toën, Vezzosi in Crellle journal. – crystalline Sep 15 at 3:48
• @crystalline I meant of course "quasi-smooth". "Complicated" is in the eye of the beholder, but the definitions I knew of seemed so to me. Thank you for the Schurg, Toen, Vezzosi reference -- that seems very close to what I need (even closer, though probably non-existent, is Lurie's assertion in HAG that stable maps are a representable DG moduli problem. – Dmitry Vaintrob 2 days ago
• Well what STV show is that the derived enhancement of $SM$ is a quasi-smooth proper derived DM stack (ie representable by a derived DM stack), using Lurie’s represetabikity theorem using the fact that the 0-truncation is known to be representable by Kontsevich. – crystalline 2 days ago
• @crystalline Thanks! That's exactly what I was hoping for -- will read the paper. If you can turn that into an answer, I'll approve it. – Dmitry Vaintrob 2 days ago