The class of Eisenbud–Khimshiashvili–Levine is the local A1-Brouwer degree

Kass, Jesse Leo; Wickelgren, Kirsten

doi:10.1215/00127094-2018-0046

Cited by 43 publications

(73 citation statements)

References 22 publications

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“…Additionally, Kass-Wickelgren [21] have shown how to compute explicitly the local contribution e x (V ; s) to the global Euler class e(V ) of an isolated zero of a section s of a vector bundle V → Y of rank equal to the dimension d of Y . In case s has only isolated zeros, this gives the formula e(V ) = s(x)=0 e x (V ; s).…”

Section: Characteristic Classes Of Symmetric Powersmentioning

confidence: 99%

“…Works by Hoyois [17] and Kass-Wickelgren [21,22], and well as our papers [23,24], have initiated a study of Euler characteristics and Euler classes with values in the Grothendieck-Witt ring or in the Chow-Witt groups. In this paper we continue this study, looking at Euler characteristics and characteristic classes with values in Witt cohomology and other SL-oriented η-invertible theories.…”

Section: Introductionmentioning

confidence: 99%

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Motivic Euler Characteristics and Witt-Valued Characteristic Classes

Levine¹

2019

Nagoya Math. J.

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This paper examines Euler characteristics and characteristic classes in the motivic setting. We establish a motivic version of the Becker-Gottlieb transfer, generalizing a construction of Hoyois. Making calculations of the Euler characteristic of the scheme of maximal tori in a reductive group, we prove a generalized splitting principle for the reduction from GLn or SLn to the normalizer of a maximal torus (in characteristic zero). Ananyevskiy's splitting principle reduces questions about characteristic classes of vector bundles in SL-oriented, η-invertible theories to the case of rank two bundles. We refine the torus-normalizer splitting principle for SL 2 to help compute the characteristic classes in Witt cohomology of symmetric powers of a rank two bundle, and then generalize this to develop a general calculus of characteristic classes with values in Witt cohomology.2010 Mathematics Subject Classification. 14F42, 55N20, 55N35.

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Section: Characteristic Classes Of Symmetric Powersmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Motivic Euler Characteristics and Witt-Valued Characteristic Classes

Levine¹

2019

Nagoya Math. J.

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“…The Wronskian of a rank-r linear series (L, V ) on an arbitrary algebraic curve C naturally defines an Euler class, associated to (the determinant of) the jet bundle J r+1 (L) on C whose fiber in a point p is H 0 (L/L(−(r + 1)p)). Moreover, when C is (hyper)elliptic and L arises via pullback from P 1 , the jet bundle in question is (relatively) orientable, which ensures that the Wronskian defines an arithmetic Euler class in the sense of A 1 -homotopy theory [4,5]. In [3] we calculate a global arithmetic inflection formula over an arbitrary field of characteristic p = 2, 3, and we analyze the geometric meaning of its constituent local arithmetic inflection indices over F q .…”

Section: Remarksmentioning

confidence: 99%

Inflection divisors of linear series on an elliptic curve

Cotterill¹,

Garay-López²

2020

RMC

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In this largely-expository note, we describe a class of divisors on elliptic curves that index the inflection points of linear series arising (as subspaces of holomorphic sections) from line bundles on P 1 via pullback along the canonical 2-to-1 projection. Associated to each inflection divisor on an elliptic curve E λ : y 2 = x(x − 1)(x − λ), there is an associated inflectionary curve in (the projective compactification of) the affine plane in coordinates x and λ. These inflectionary curves have remarkable features; among other things, they lead directly to an explicit conjecture for the number of real inflection points of linear series on E λ whenever the Legendre parameter λ is real.

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“…In this paper, we explain in more detail how to use the results [KW16] to establish an enriched Riemann-Hurwitz formula with explicitly computable branch indices when f is allowed to have wild ramification. Rather than using the formalism developed in [Lev18], we establish an enriched Riemann-Hurwitz formula using the Euler class formalism in [KW17].…”

Section: Introductionmentioning

confidence: 99%

“…The notation e(Y, Hom(f * T * X, T * Y), df) is defined in Section 2. As we explain, when f : Y → X is described explicitly, the local indices of df can be effectively computed using the main results of [Caz12,Caz08,KW16].…”

Section: Introductionmentioning

confidence: 99%

Examples of wild ramification in an enriched Riemann–Hurwitz formula

Bethea¹,

Kass²,

Wickelgren³

2020

Motivic Homotopy Theory and Refined Enumerative Geometry

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M. Levine proved an enrichment of the classical Riemann-Hurwitz formula to an equality in the Grothendieck-Witt group of quadratic forms. In its strongest form, Levine's theorem includes a technical hypothesis on ramification relevant in positive characteristic. We consider wild ramification at points whose residue fields are non-separable extensions of the ground field k. We show an analogous Riemann-Hurwitz formula, and consider an example suggested by S. Saito.Date: December 11, 2018.

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The class of Eisenbud–Khimshiashvili–Levine is the local A1-Brouwer degree

Cited by 43 publications

References 22 publications

Motivic Euler Characteristics and Witt-Valued Characteristic Classes

Motivic Euler Characteristics and Witt-Valued Characteristic Classes

Inflection divisors of linear series on an elliptic curve

Examples of wild ramification in an enriched Riemann–Hurwitz formula

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