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On the EKL-degree of a Weyl cover. (English) Zbl 1505.14105

Summary: More than four decades ago, Eisenbud, Khimshiashvili, and Levine [D. Eisenbud and H. I. Levine, Ann. Math. (2) 106, 19–44 (1977; Zbl 0398.57020); G. N. Khimshiashvili, Soobshch. Akad. Nauk Gruz. SSR 85, 309–312 (1977; Zbl 0346.55008)] introduced an analogue in the algebro-geometric setting of the notion of local degree from differential topology. Their notion of degree, which we call the EKL-degree, can be thought of as a refinement of the usual notion of local degree in algebraic geometry that works over non-algebraically closed base fields, taking values in the Grothendieck-Witt ring. In this note, we compute the EKL-degree at the origin of certain finite covers \(f : \mathbb{A}^n \to \mathbb{A}^n\) induced by quotients under actions of Weyl groups. We use knowledge of the cohomology ring of partial flag varieties as a key input in our proofs, and our computations give interesting explicit examples in the field of \(\mathbb{A}^1\)-enumerative geometry.

MSC:

14M15 Grassmannians, Schubert varieties, flag manifolds
55M25 Degree, winding number
14F42 Motivic cohomology; motivic homotopy theory
14G27 Other nonalgebraically closed ground fields in algebraic geometry
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