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A^1-Euler classes: six functors formalisms, dualities, integrality and linear subspaces of complete intersections
TL;DR: In this paper, the Euler classes associated to arithmetic counts of d-planes on complete intersections in P^n in terms of topological Euler numbers over R and C are compared.
Abstract: We equate various Euler classes of algebraic vector bundles, including those of [BM, KW, DJK], and one suggested by M.J. Hopkins, A. Raksit, and J.-P. Serre. We establish integrality results for this Euler class, and give formulas for local indices at isolated zeros, both in terms of 6-functor formalism of coherent sheaves and as an explicit recipe in commutative algebra of Scheja and Storch. As an application, we compute the Euler classes associated to arithmetic counts of d-planes on complete intersections in P^n in terms of topological Euler numbers over R and C.
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TL;DR: In this article, an arithmetic count of the lines on a smooth cubic surface over an arbitrary field was given and generalized to complex and real algebraic geometry can be obtained with similar methods.
Abstract: We give an arithmetic count of the lines on a smooth cubic surface over an arbitrary field ,
\[ \sum_{\text{lines}} \operatorname{Tr}_{L/k} \langle \alpha \rangle = 15 \cdot \langle 1 \rangle + 12 \cdot \langle -1 \rangle, \]
where -homotopy theory for algebraic vector bundles. We expect that further arithmetic counts generalizing enumerative results in complex and real algebraic geometry can be obtained with similar methods.
30 citations
Abstract: The apparatus of motivic stable homotopy theory provides a notion of Euler characteristic for smooth projective varieties, valued in the Grothendieck–Witt ring of the base field. Previous work of the first author and recent work of Deglise, Jin and Khan established a motivic Gauss–Bonnet formula relating this Euler characteristic to pushforwards of Euler classes in motivic cohomology theories. We apply this formula to SL-oriented motivic cohomology theories to obtain explicit characterizations of this Euler characteristic. The main new input is a uniqueness result for pushforward maps in SL-oriented theories, identifying these maps concretely in examples of interest.
19 citations
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TL;DR: In this article, a 2-term resolution of the 2-local Witt K-theory spectrum was established over any field of characteristic not 2, which is curiously similar to the resolution of K(1)-local sphere in classical stable homotopy theory.
Abstract: Over any field of characteristic not 2, we establish a 2-term resolution of the $\eta$-periodic, 2-local motivic sphere spectrum by shifts of the connective 2-local Witt K-theory spectrum. This is curiously similar to the resolution of the K(1)-local sphere in classical stable homotopy theory. As applications we determine the $\eta$-periodized motivic stable stems and the $\eta$-periodized algebraic symplectic and SL-cobordism groups. Along the way we construct Adams operations on the motivic spectrum representing Hermitian K-theory and establish new completeness results for certain motivic spectra over fields of finite virtual 2-cohomological dimension. In an appendix, we supply a new proof of the homotopy fixed point theorem for the Hermitian K-theory of fields.
14 citations
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TL;DR: In this paper, a geometric interpretation of the local contribution of a line to the count of lines on a quintic threefold over a field k of characteristic not equal to 2 is provided.
Abstract: We provide a geometric interpretation of the local contribution of a line to the count of lines on a quintic threefold over a field k of characteristic not equal to 2, that is, we define the type of a line on a quintic threefold and show that it coincides with the local index at the corresponding zero of the section of Sym^5 S^* -> Gr(2, 5) defined by the threefold. Furthermore, we define the dynamic Euler number which allows us to compute the A^1-Euler number as the sum of local contributions of zeros of a section with non-isolated zeros which deform with a general deformation. As an example we provide a quadratic count of 2875 distinguished lines on the Fermat quintic threefold which computes the dynamic Euler number of Sym^5 S^* -> Gr(2, 5). Combining those two results we get that the sum of the types of lines on a general quintic threefold is 1445 + 1430 in GW(k) when k is a field of characteristic not equal to 2 or 5.
12 citations
TL;DR: In this article, a bilinear form-valued count of the intersection points of hypersurfaces in projective space over non-algebraically closed fields has been obtained for the reals, rationals and finite fields of odd characteristic.
Abstract: The classical version of Bezout’s Theorem gives an integer-valued count of the intersection points of hypersurfaces in projective space over an algebraically closed field. Using work of Kass and Wickelgren, we prove a version of Bezout’s Theorem over any perfect field by giving a bilinear form-valued count of the intersection points of hypersurfaces in projective space. Over non-algebraically closed fields, this enriched Bezout’s Theorem imposes a relation on the gradients of the hypersurfaces at their intersection points. As corollaries, we obtain arithmetic–geometric versions of Bezout’s Theorem over the reals, rationals, and finite fields of odd characteristic.
10 citations
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Book•
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01 Jan 1993
TL;DR: In this article, the authors present a self-contained introduction to the homological and combinatorial aspects of the theory of Cohen-Macaulay rings, Gorenstein rings, local cohomology, and canonical modules.
Abstract: In the last two decades Cohen-Macaulay rings and modules have been central topics in commutative algebra. This book meets the need for a thorough, self-contained introduction to the homological and combinatorial aspects of the theory of Cohen-Macaulay rings, Gorenstein rings, local cohomology, and canonical modules. A separate chapter is devoted to Hilbert functions (including Macaulay's theorem) and numerical invariants derived from them. The authors emphasize the study of explicit, specific rings, making the presentation as concrete as possible. So the general theory is applied to Stanley-Reisner rings, semigroup rings, determinantal rings, and rings of invariants. Their connections with combinatorics are highlighted, e.g. Stanley's upper bound theorem or Ehrhart's reciprocity law for rational polytopes. The final chapters are devoted to Hochster's theorem on big Cohen-Macaulay modules and its applications, including Peskine-Szpiro's intersection theorem, the Evans-Griffith syzygy theorem, bounds for Bass numbers, and tight closure. Throughout each chapter the authors have supplied many examples and exercises which, combined with the expository style, will make the book very useful for graduate courses in algebra. As the only modern, broad account of the subject it will be essential reading for researchers in commutative algebra.
2,760 citations
TL;DR: In this article, the authors propose a solution to solve the problem of spamming, which is called spamming-based spamming.$$$/$/$/$/$$
1,294 citations
TL;DR: In this paper, the authors prove a localization formula for virtual fundamental classes in the context of torus equivariant perfect obstruction theories, where the higher genus Gromov-Witten invariants of projective space are expressed as graph sums of tautological integrals over moduli spaces of stable pointed curves.
Abstract: We prove a localization formula for virtual fundamental classes in the context of torus equivariant perfect obstruction theories. As an application, the higher genus Gromov-Witten invariants of projective space are expressed as graph sums of tautological integrals over moduli spaces of stable pointed curves (generalizing Kontsevich's genus 0 formulas). Also, excess integrals over spaces of higher genus multiple covers are computed.
808 citations
Book•
01 Nov 2004TL;DR: In this article, the Brauer-Wall group of quadratic algebras and their norm forms are studied. But the focus is on the Witt rings of the quadratics and not on the norm forms.
Abstract: Foundations Introduction to Witt rings Quaternion algebras and their norm forms The Brauer-Wall group Clifford algebras Local fields and global fields Quadratic forms under algebraic extensions Formally real fields, real-closed fields, and pythagorean fields Quadratic forms under transcendental extensions Pfister forms and function fields Field invariants Special topics in quadratic forms Special topics on invariants Bibliography Index.
767 citations