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2. Moduli


    1. Add remark

      Problem 2.1.

      a) Is there an explicit cycle C_k \subset \overline{\mathcal{M}}_{g,1}(\mathbb{P}^1,d) \cap {\rm ev}_1^*(0)
      representing the class \left[\overline{\mathcal{M}}_{g,1}(\mathbb{P}^1,d) \right]^{{\rm vir}}\cap {\rm ev}_1^*(0)\; \cdot \; \psi_1^k
      in \overline{\mathcal{M}}_{g,1}(\mathbb{P}^1,d)?

      Partial answer (Okounkov, Pandharipande): if k=\text{virdim} \,\overline{\mathcal{M}}_{g}(\mathbb{P}^1,d), then the answer is in terms of completed cycles that appear in the representation theory of symmetric groups.

      b) Construct a virtual fundamental class of maps f: (C,x) \rightarrow (\mathbb{P}^1,0) such that df admits an r-th root locally at x. Show that this virtual fundamental class is equal to \left[\overline{\mathcal{M}}_{g,1}(\mathbb{P}^1,d) \right]^{{\rm vir}}\cap {\rm ev}_1^*(0)\; \cdot \; \psi_1^k,
      ideally that it is represented by the cycle C_k.

      Comforting evidence: in the smooth case, the local equation of f is f(z)=z^{k+1}\Longleftrightarrow df = (k+1)z^k dz has a k-th root.


        • Add remark

          Problem 2.2.

          Consider \begin{align*} (1) &\,\, \text{stab}_*([\overline{\mathcal{M}}^{\,\sim}_{g,n} (\mathbb{P}^1,a_i p_i)]^{\text{vir}});\\ (2) &\, \,\text{stab}_*(\text{Adm}_{g,n}(\mathbb{P}^1,a_ip_i)), \text{where Adm is the space of admissible covers;}\\ (3) &\, \,\sigma^*(0), \text{where}\,\, \sigma: \mathcal{M}_{g,n}^{ct}\rightarrow Pic^0_g \,\,\text{is given by}\,\,(C,p_1,..,p_n)\mapsto (C, \mathcal{O}(\Sigma a_ip_i)). \end{align*}
          It is known that (1)=(3) on \mathcal{M}_{g,n}^{\rm ct}.

          a) Find an expression for (1) and (2) in the tautological ring of \overline{\mathcal{M}}_{g,n}.

          b) Does (3) extend to \overline{\mathcal{M}}_{g,n}?

              Cite this as: AimPL: Integrable systems in Gromov-Witten and symplectic field theory, available at http://aimpl.org/gwsymplectic.