Date of Award
Open Access Dissertation
Jesse Leo Kass
Enumerative geometry studies the number of geometric objects in a given class satisfying specific geometric conditions. For example, we can ask how many conics in the plane pass through four points. One such example is a specific case of Göttsche’s conjecture, stated in (Göttsche 1998): given a pencil of conics in P2 C, how many curves in the pencil have nodal singularities? The answer is three as long as the defining conics of the pencil are general. The conjecture was proved in full generality by Y. Tzeng in 2010 in (Tzeng 2012), and another proof exists due to Kool, Shende, and Thomas in 2011 in (Kool, Shende, and Thomas 2011).
Recent developments in motivic homotopy theory have led to enrichments of many enumerative results over non-algebraically closed fields, and examples can be found in (Hoyois 2014), (Kass and Wickelgren 2019), and (Levine 2017). However, the question of replicating enumerative results in the presence of a group action is unstudied.
This work takes a classical enumerative problem in the presence of a group action and enriches it using equivariant topology. Specifically, let X be a pencil of general conics in P2 C which is invariant under the linear action of a finite group. The main result is that for finite groups not isomorphic to Z/2◊Z/2 or D8, there is a weighted sum valued in the Burnside ring of G-sets of the orbits of nodal conics in X in terms of the base locus of the defining equations. Counterexamples for Z/2 ◊ Z/2 and D8 are also given. This is a direct generalization of the specific aforementioned case of Göttsche’s conjecture as well as its real analogue, as the classical case is obtained by taking a trivial group action and the case over R is obtained by taking G ≥= Z/2 and the action on P2 C to be coordinate-wise conjugation.
Bethea, C.(2020). An Equivariant Count of Nodal Orbits in an Invariant Pencil of Conics. (Doctoral dissertation). Retrieved from https://scholarcommons.sc.edu/etd/5937