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On quadratically enriched excess and residual intersections

  • Tom Bachmann and Kirsten Wickelgren EMAIL logo


We use recent duality results of Eisenbud and Ulrich to give tools to study quadratically enriched residual intersections when there is no excess bundle. We use this to prove a formula for the Witt-valued Euler number of an almost complete intersection. We give example computations of quadratically enriched excess and residual intersections.

Award Identifier / Grant number: DMS 2001890

Award Identifier / Grant number: DMS 2103838

Funding statement: Kirsten Wickelgren was partially supported by NSF CAREER DMS 2001890 and NSF DMS 2103838.

A Macaulay2 code

We have implemented functions for the computer program Macaulay2 [21] to carry out the computations implicit in Proposition 2.3 for concrete examples, in the case n = 5 . Specifically, we have a function Bprime which can be used to obtain the matrix of a symmetric bilinear form B such that B B 5 0 , and we have a function diagonalize which can be used to diagonalize a symmetric bilinear form over a field of characteristic not 2. The function Bprime uses a supporting function CDTr which is based on a function of S. Pauli [39]. The code is included at the end of this appendix.

Example A.1

Below, is a sample session computing a diagonal representative of B for a random set of five quadrics on P 5 vanishing on P 2 , over the finite field F 61 .

We thus find that

[ B ] = 1 + 11 + 29 + 12 W ( F 61 ) .

Listing 1

Functions Bprime and CDTr from form.m2

Listing 2

Function diagonalize from diagonalization.m2


We wish to thank Claudia Polini for introducing us to the paper [15], David Eisenbud for useful discussions (he owes us money as he promised payment for asking questions he can answer), Stephen McKean for useful discussions on [45], and Sabrina Pauli for useful discussions on quadratically enriched excess intersections. Kirsten Wickelgren gratefully thanks Joseph Rabinoff for providing “a room of one’s own” to work on this paper.


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Received: 2021-12-17
Revised: 2023-04-26
Published Online: 2023-07-25
Published in Print: 2023-09-01

© 2023 Walter de Gruyter GmbH, Berlin/Boston

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