# Virtual fundamental class of Moduli space of stable maps in genus 1

What is the virtual fundamental class of $\overline{M}_{1,n}(\mathbb{P}^2,d)$? In general the virtual fundamental class is difficult to compute I guess. But if you look at Proposition 2.5 of https://arxiv.org/abs/alg-geom/9612004 an explicit formula is given.

In particular if $f: \overline{M}_{1,n+1}(\mathbb{P}^2,d) \to \mathbb{P}^2$ is the evaluation map at the last marked point and $\pi: \overline{M}_{1,n+1}(\mathbb{P}^2,d) \to \overline{M}_{1,n}(\mathbb{P}^2,d)$ is the universal curve, then the virtual fundamental class should be $c_e(R^1\pi_*f^* T\mathbb{P}^2) \cap [\overline{M}_{1,n}(\mathbb{P}^2,d)]$ where $R^1\pi_*f^* T\mathbb{P}^2$ is a rank $e$ vector bundle and $c_e$ is the top Chern class. Can we explicitly write down what this vector bundle is?

If this is too complicated, what is the answer at least when $n=1$ and $d=0$.

• For $n=1$ and $d=0$, the moduli space is $\overline{\mathcal{M}}_{1,1}\times \mathbb{P}^2$, and the rank $e=2$ bundle is the tensor product of the pullback of the tangent bundle of $\mathbb{P}^2$ with the pullback of the dual of the Hodge bundle of $\overline{\mathcal{M}}_{1,1}$. The $2^{\text{nd}}$ Chern class of this bundle is the pullback of $3[{*}]$ from $\mathbb{P}^2$ minus the product of the pullback of $3c_1(\mathcal{O}(1))$ from $\mathbb{P}^2$ with the pullback of $\lambda$ from $\overline{\mathcal{M}}_{1,1}$. – Jason Starr Jul 19 '18 at 12:01