The F-conjecture for divisors on \mathcal{X}_{n+2,n2n-2}

Let SS be the indexing set used in the source, and let Xn+2,(n2)\mathcal{X}_{n+2,(n-2)} be the corresponding space. For a divisor DD of the form given in the source, define the intersection expressions AG,J,LA_{G,J,L} and BI,G,J,LB_{I,G,J,L} for partitions of SS. The divisor is F-nef when

AG,J,L0,BI,G,J,L0A_{G,J,L}\geq 0, \qquad B_{I,G,J,L}\geq 0

for every partition GJL=SG\sqcup J\sqcup L=S and IGJL=SI\sqcup G\sqcup J\sqcup L=S, respectively. The F-conjecture. Such a divisor is nef if and only if these F-nef inequalities hold.

This is the specialization of the F-conjecture to the stated blow-up model and translates nonnegative intersection with the relevant F-curves into explicit inequalities. The general F-conjecture is not known in full.

Sources & referencesView supporting material

Primary source

Olivia Dumitrescu and Elisa Postinghel, “Positivity of divisors on blown-up projective spaces, I”, arXiv:1506.04726 (2017).

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