Let $\overline{M}_{1,1}(\mathbb{P}^2, d) $ be the moduli space of degree $d$ genus one curves on $\mathbb{P}^2$ with one marked point. Let $L\longrightarrow \overline{M}_{1,1}(\mathbb{P}^2, d) $ denote the universal tangent bundle (i.e. the fibre at each point is the tangent space at the marked point). Let $\mathcal{H}$ be the divisor inside $\overline{M}_{1,1}(\mathbb{P}^2, d) $ that corresponds to the curve passing through some fixed point. Let $a$ denote the divisor that corresponds to the marked point lying on a given line. Let $B_{d_1, d_2}$ denote the boundary divisors where the marked point lies on the $d_1$ component.

$\textbf{Question:}$ Is there a formula for the Chern class $c_1(L)$ in terms of $\mathcal{H}$, $a$ and $B_{d_1, d_2}$? Moreover, is it possible to compute the intersections numbers $c_1(L)^{n_1}\cdot \mathcal{H}^{n_2} \cdot a^{n_3}$ evaluated on $[\overline{M}_{1,1}(\mathbb{P}^2, d)]$?

$\textbf{Remark:}$ Suppose I asked the same question about $\overline{M}_{0,1}(\mathbb{P}^2, d) $, the moduli space of genus zero curves. Then what I am asking is very well understood as shown in the paper by Pandharipande


One of the ways one can compute $c_1(L)$ is as follows: one goes to a covering space of $\overline{M}_{0,1}(\mathbb{P}^2, d) $ where one throws in two extra marked points $y_1$ and $y_2$ and we require the marked points to lie on two generic lines. It is then easy to construct a section of the pullback of $L$, given by $$ \frac{(y_1-y_2) dy}{(y-y-1)(y-y_2)}. $$ Here $y$ is the original marked point and $y_1$ and $y_2$ are the two new marked points. This section is well defined (invariant under mobius transformations) and hence the zeros minus the poles give us the Chern class. Finally we get a formula for the Chern class and then one can compute the relevant intersection numbers. This is what is done in this paper by Eleny Ionel (Lemma 2.7, page 28).


I am wondering if there is a similar way one can construct such a section (either by going to some appropriate cover or by taking an appropriate power of the bundle) in the case of $\overline{M}_{1,1}(\mathbb{P}^2, d)$. If so, one can just look at the zeros and poles of that section and compute the Chern class.

  • $\begingroup$ $L$ is pulled back from $\overline{\mathcal M}_{1,1}$ and is something like $-1/12$ times the pullback of the boundary divisor from $\overline{\mathcal M}_{1,1}$. Perhaps there is an explicit formula for this, like $\sum_{d_1=0}^d B_{d_1,d-d_1}$? $\endgroup$ – Will Sawin Sep 11 '15 at 21:50
  • $\begingroup$ @Will: But there should also be terms of the form $\alpha_1 \mathcal{H} + \alpha_2 a + $ the term you are saying (boundary terms). Is there any reason $\alpha_1$ and $\alpha_2$ should be zero? $\endgroup$ – Ritwik Sep 11 '15 at 21:53
  • $\begingroup$ Yes, because (I think) there is a section of the $12$th inverse power of the bundle that vanishes only on the boundary, so there should be only boundary terms. $\endgroup$ – Will Sawin Sep 11 '15 at 21:58
  • $\begingroup$ @Will: I see, I didn't know this. Hopefully someone will give a reference for this; this is precisely what I was looking for, an explicit section of the line bundle (rather a power of the line bundle). $\endgroup$ – Ritwik Sep 11 '15 at 22:01
  • $\begingroup$ The intersection numbers you're asking about are the elliptic GW invariants of $\mathbf P^2$ with gravitational descendents. These are for sure known for a long time. I am not sure what is the right reference but you could start by looking through papers of Zinger. $\endgroup$ – Dan Petersen Sep 12 '15 at 8:07

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