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Mori’s program for the Kontsevich moduli space \(\overline{\mathcal M}_{0,0}(\mathbb P^3, 3)\). (English) Zbl 1147.14009
Let \(\overline {\mathcal {M}} _{0,0}(\mathbb P ^3,3)\) be the Kontsevich space of stable maps of degree \(3\) rational curves in \(\mathbb P ^3\) i.e. a compactification of the space of twisted cubics.
It is known that \(\overline {\mathcal {M}} _{0,0}(\mathbb P ^3,3)\) is Fano and hence it is a Mori dream space. Therefore, for any effective divisor \(D\) on \(\overline {\mathcal {M}} _{0,0}(\mathbb P ^3,3)\), the ring \(\bigoplus _{m\geq 0}H^0(\overline {\mathcal {M}} _{0,0}(\mathbb P ^3,3),mD)\) is finitely generated and there are finitely many spaces \(\bar{M}(D)=\text{Proj}(\bigoplus _{m\geq 0}H^0(\overline {\mathcal {M}} _{0,0}(\mathbb P ^3,3),mD))\) which are related by operations of the minimal model program.
In the paper under review, all the spaces \(\bar{M}(D)\) and the rational maps between them are explicitely described in a geometrically meaningful way. It turns out that the only possibilities for \(\bar{M}(D)\) are \({\mathcal {M}} _{0,0}(\mathbb P ^3,3)\);
\(\overline {\mathcal {M}} _{0,0}(\mathbb P ^3,3,2)\) the space of \(2\)-stable maps;
\(\mathcal {C}^\nu\) the normalization of the Chow compactification of the space of twisted cubics;
\(\mathcal {H}\) the closure of the locus of twisted cubics in the Hilbert scheme of curves of degree \(3\) and arithmetic genus \(0\);
\(\mathcal {H}(2)\) the closure of the space of twisted cubics in the Grassmannian \(\mathbb {G}(2,9)\); and \(\text{Spec}(\mathbb C )\). These spaces are related by a divisorial contraction \(\theta : \overline {\mathcal {M}} _{0,0}(\mathbb P ^3,3)\to \overline {\mathcal {M}} _{0,0}(\mathbb P ^3,3,2)\); a flipping contraction \(f: \overline {\mathcal {M}} _{0,0}(\mathbb P ^3,3)\to \mathcal {C}^\nu\); its flip \(g : \mathcal {H}\to \mathcal {C}^\nu\); and a divisorial contraction \(h:\mathcal {H}\to \mathcal {H}(2)\).

MSC:
14H10 Families, moduli of curves (algebraic)
14E30 Minimal model program (Mori theory, extremal rays)
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