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# The Ample Cone of the Kontsevich Moduli Space

Published:2009-02-01
Printed: Feb 2009
• Izzet Coskun
• Joe Harris
• Jason Starr
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## Abstract

We produce ample (resp.\ NEF, eventually free) divisors in the Kontsevich space $\Kgnb{0,n} (\mathbb P^r, d)$ of $n$-pointed, genus $0$, stable maps to $\mathbb P^r$, given such divisors in $\Kgnb{0,n+d}$. We prove that this produces all ample (resp.\ NEF, eventually free) divisors in $\Kgnb{0,n}(\mathbb P^r,d)$. As a consequence, we construct a contraction of the boundary $\bigcup_{k=1}^{\lfloor d/2 \rfloor} \Delta_{k,d-k}$ in $\Kgnb{0,0}(\mathbb P^r,d)$, analogous to a contraction of the boundary $\bigcup_{k=3}^{\lfloor n/2 \rfloor} \tilde{\Delta}_{k,n-k}$ in $\kgnb{0,n}$ first constructed by Keel and McKernan.
 MSC Classifications: 14D20 - Algebraic moduli problems, moduli of vector bundles {For analytic moduli problems, see 32G13} 14E99 - None of the above, but in this section 14H10 - Families, moduli (algebraic)

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