The maximum cut problem on blow-ups of multiprojective spaces
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by
Mauricio Junca Mauricio Velasco
2012
Abstract
The maximum cut problem for a quintic del Pezzo surface
Bl_4(P^2) asks: Among all partitions of the 10 exceptional curves
into two disjoint sets, what is the largest possible number of pairwise
intersections? In this article we show that the answer is twelve. More
generally, we obtain bounds for the maximum cut problem for the minuscule
varieties X_a,b,c:= Bl_b+c(P^c-1)^a-1 studied by Mukai
and Castravet-Tevelev and show that these bounds are asymptotically sharp for
infinite families. We prove our results by constructing embeddings of the
classes of (-1)-divisors on these varieties which are optimal for the
semidefinite relaxation of the maximum cut problem on graphs proposed by
Goemans and Williamson. These results give a new optimality property of the
Weyl orbits of root systems of type A,D and E.
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