Abstract
Two 3-fold flops are exhibited, both of which have precisely one flopping curve. One of the two flops is new and is distinct from all known algebraic D4-flops. It is shown that the two flops are neither algebraically nor analytically isomorphic, yet their curve-counting Gopakumar–Vafa invariants are the same. We further show that the contraction algebras associated to both are not isomorphic, so the flops are distinguished at this level. This shows that the contraction algebra is a finer invariant than various curve-counting theories, and it also provides more evidence for the proposed analytic classification of 3-fold flops via contraction algebras.
Highlights
Flopping neighbourhoods are one of the most elementary building blocks of higher dimensional algebraic geometry, and even in dimension three they exhibit a very rich structure
The produced invariant is strictly finer than the last, with the GV invariants linking to Donaldson–Thomas theory and all other modern curve counting notions
Contraction algebras were introduced in [DW1], partially to provide a new curve invariant, but mainly to unify the homological approaches to derived symmetries and twists [B02,C02,T07]. With their roots in homological algebra, and because they are an algebra as opposed to a number, this additional structure allows us to use contraction algebras to establish and control many geometric processes [DW2,W14], whilst at the same time recover the GV and other invariants [DW1,T14,HT] in a variety of natural ways
Summary
Flopping neighbourhoods are one of the most elementary building blocks of higher dimensional algebraic geometry, and even in dimension three they exhibit a very rich structure. We use the algebra structure to show that the contraction algebra is a strictly finer invariant than that of Gopakumar–Vafa. This is in some ways surprising: the GV invariants are enough to classify Type A flops [R83]. We remark that there are tables of data that numerically suggest, but do not quite yet prove, that different flops having the same GV invariants is quite typical behaviour It is perhaps worth explaining the heuristic reason as to why the noncommutativity of the contraction algebra helps, rather than hinders, distinguishing the two flops above. The proof of (1.B) is somewhat more involved than this heuristic argument; we give a direct proof in 4.7, but it is possible to give a computer algebra verification by adapting the Shirayanagi algorithm [S]
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