Towards Mori’s program for the moduli space of stable maps.

*(English)*Zbl 1256.14013This article studies the birational geometry of the moduli space of genus zero stable maps to projective spaces without no marked points. This fits into the study of the birational geometry of moduli spaces initiated by Hassett and intensely studied in the past years. The authors introduce several natural divisors on the moduli space of genus zero maps to \(\mathbb{P}^r\): divisors corresponding to curves with one node, the divisor of maps whose images meet a fixed codimension two linear subspace in \(\mathbb{P}^r\), the divisor consisting of maps whose images have a node on a fixed line, the divisor of the locus corresponding to maps whose images have a tacnode, the divisor class parameterizing maps whose images have a triple point. Theorem 1.1 gives relations between all these divisor classes. Theorem 1.2 provides generators of the nef cone of the moduli space of stable maps of degree 4 to \(\mathbb{P}^r\) for \(r\) equal to 2,3,4. Moreover for \(r=4\) there is an explicit chamber decomposition. In Section 3 birational models corresponding to certain divisors described above are being studied. One very nice application is that these birational models provide normalizations of the Chow scheme and the scheme of branch curves with fixed Hilbert polynomial defined by Alexeev and Knutson. The authors also study contractions of divisors corresponding to reducible curves. These contractions lead to the moduli spaces of \(k\)-stable maps constructed by Mustată and Mustată.

Reviewer: Cristina Manolache (London)