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Let $\overline{M}_{0,2}(G(3,7),4)$ be the moduli space of $2$-pointed degree $4$ stable maps to the Grassmannian of $3$-planes in $\mathbb{P}^7$. Consider the divisor $\Delta$ whose general point is a stable map with reducible domain $C\cup \Gamma$, the map has degree $1$ on $C$ and degree $3$ on $\Gamma$, $x_1\in C$, $x_2\in \Gamma$.

Similarly, let $\Delta'$ be the divisor whose general point is a stable map with reducible domain $C\cup \Gamma$, the map has degree $3$ on $C$ and degree $1$ on $\Gamma$, $x_1\in C$, $x_2\in \Gamma$.

There are two birational morphism $f:\overline{M}_{0,2}(G(3,7),4)\rightarrow X$, $f':\overline{M}_{0,2}(G(3,7),4)\rightarrow X'$ contracting respectively $\Delta$ and $\Delta'$.

The morphism $f\times f':\overline{M}_{0,2}(G(3,7),4)\rightarrow X\times X'$ is a small contraction contracting he codimesnion two locus $\Delta\cap\Delta'$ whose general point correspond to a stable map from a curve with three components $C\cup \Gamma\cup F$ with degree $1,2,1$ respectively, $x_1\in C,x_2\in F$.

Is it know if there exists a moduli spaces of some kind of geometric objects realizing the flip of $f\times f'$ ?

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  • $\begingroup$ The method from Coskun-Harris-Starr produces a basepoint free divisor class giving the contraction $f\times f'$ from a basepoint free divisor class on $\overline{M}_{0,1+4+1}$ contracting the codimension $2$ locus parameterizing $6$-pointed, genus $0$, stable curves with three components $C\cup \Gamma \cup F$ each containing two marked points where the first marked point, resp. final marked point, is contained in $C$, resp. $F$. If you find a moduli interpretation of the corresponding flip of $\overline{M}_{0,6}$, that should give a moduli interpretation of your flip as well. $\endgroup$ – Jason Starr Feb 6 '18 at 20:46

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