Cohomology rings of moduli of point configurations on the projective line

Abstract

We describe the Chow rings of moduli spaces of ordered configurations of points on the projective line for arbitrary (sufficiently generic) stabilities. As an application, we exhibit such a moduli space admitting two small desingularizations with non-isomorphic cohomology rings.

Fano quiver moduli

“…The semistable moduli space is isomorphic to the Segre cubic by [7], which is indeed a singular projective Fano threefold with ten isolated singularities. Slightly deforming the stability Θ to a new stability Θ + as in [5] yields a small desingularization of the Segre cubic, which cannot be Fano: a Fano threefold of Picard rank six is automatically isomorphic to the product of the projective line and the degree five del Pezzo surface, thus contains a 2-nilpotent element in second cohomology. But in [5] it is shown that this does not hold for this desingularization, disproving the Fano property.…”

“…Slightly deforming the stability Θ to a new stability Θ + as in [5] yields a small desingularization of the Segre cubic, which cannot be Fano: a Fano threefold of Picard rank six is automatically isomorphic to the product of the projective line and the degree five del Pezzo surface, thus contains a 2-nilpotent element in second cohomology. But in [5] it is shown that this does not hold for this desingularization, disproving the Fano property. A theorem of Kobayashi and Ochiai [10] asserts that a smooth projective Fano variety X of dimension n and index q satisfies q ≤ n + 1 and equality holds if and only if X ≃ P n .…”

$PGL_2$-equivariant strata of point configurations in $\mathbb{P}^1$

“…In this case, the ideal given by excision is generated by the classes of the excised strata. See [14] for an approach via quiver representations. These GIT quotients are Hassett spaces with total weight 2 + ǫ [18, Section 8] and receive maps from M 0,n via reduction morphisms [18,Theorem 4.1], as induced maps between GIT quotients [19,Theorem 3.4], or by viewing M 0,n as a Chow quotient [20].…”

Fano quiver moduli

“…The semistable moduli space is isomorphic to the Segre cubic by [14], which is indeed a singular projective Fano threefold with ten isolated singularities. Slightly deforming the stability Θ to a new stability Θ + as in [9] yields a small desingularization of the Segre cubic, which cannot be Fano: a Fano threefold of Picard rank six is automatically isomorphic to the product of the projective line and the degree five del Pezzo surface, thus contains a 2-nilpotent element in second cohomology. But in [9] it is shown that this does not hold for this desingularization, disproving the Fano property.…”

“…Slightly deforming the stability Θ to a new stability Θ + as in [9] yields a small desingularization of the Segre cubic, which cannot be Fano: a Fano threefold of Picard rank six is automatically isomorphic to the product of the projective line and the degree five del Pezzo surface, thus contains a 2-nilpotent element in second cohomology. But in [9] it is shown that this does not hold for this desingularization, disproving the Fano property. A theorem of Kobayashi and Ochiai [18] asserts that a smooth projective Fano variety of dimension and index satisfies ≤ + 1 and equality holds if and only if 2020/12/18 12:33 P .…”