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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)
##### Keywords:
Mori theory; Kontsevich moduli space
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