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ConformalBlocks :: F curve

F curve -- F curves in the moduli space of stable n-pointed genus zero curves

Let $P={P_0,P_1,P_2,P_3}$ be a partition of $\{1,...,n\}$ into four nonempty subsets. Fix four (arithmetic) genus zero at worst nodal curves $C_j$ for $j=0,1,2,3$, and $\#(P_j)$ marked points on each curve. We call the curves $C_j$ the tails. Mark one additional point $x_j$ on each tail. Next, consider $\mathbb{P}^1$ with four marked points, $y_0,...,y_3$; we call this the spine. Glue the four tails to the spine by identifying $x_j$ and $y_j$. Then, as the cross ratio of $y_0,...,y_3$ varies, we sweep out a curve $F_{P}$ in $\bar{M}_{0,n}$.

The homology class of $F_{P}$ only depends on the partition $P$, and not on the choice of the tails $C_j$ or the choices of marked points. The classes of the F-curves span $H_2(\bar{M}_{0,n},Q)$.

If we only consider F-curves up to $S_n$ symmetry, then it is enough to keep track of the four integers $\#(P_0)$, $\#(P_1)$, $\#(P_2)$, $\#(P_3)$.