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Compactification of the moduli space of minimal instantons on the Fano threefold V_4

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Xuqiang Qin at University of North Carolina at Chapel Hill
Xuqiang Qin
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Abstract

We study semistable sheaves of rank 2 with Chern classes c1=0c1=0c_1=0, c2=2c2=2c_2=2 and c3=0c3=0c_3=0 on the Fano threefold V4V4V_4 of Picard number 1, degree 4 and index 2. We show that the moduli space of such sheaves is isomorphic to the moduli space of semistable rank 2, degree 0 vector bundles on a genus 2 curve. This also provides a natural smooth compactification of the moduli space of Ulrich bundles of rank 2 on V4V4V_4.
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European Journal of Mathematics (2021) 7:1502–1523
https://doi.org/10.1007/s40879-021-00486-5
RESEARCH ARTICLE
Compactification of the moduli space of minimal instantons
on the Fano threefold V4
Xuqiang Qin1
Received: 9 April 2019 / Revised: 25 December 2020 / Accepted: 16 June 2021 /
Published online: 19 July 2021
© The Author(s), under exclusive licence to Springer Nature Switzerland AG 2021
Abstract
We study semistable sheaves of rank 2 with Chern classes c1=0, c2=2 and c3=0
on the Fano threefold V4of Picard number 1, degree 4 and index 2. We show that the
moduli space of such sheaves is isomorphic to the moduli space of semistable rank
2, degree 0 vector bundles on a genus 2 curve. This also provides a natural smooth
compactification of the moduli space of Ulrich bundles of rank 2 on V4.
Keywords Moduli spaces ·Instanton bundles ·Fano threefolds ·Semiorthogonal
decomposition
Mathematics Subject Classification 14J10 ·14J30 ·14F05 ·14H60
1 Introduction
Instanton bundles first appeared in [3] as a way to describe Yang–Mills instantons on
a 4-sphere S4. They provide extremely useful links between mathematical physics and
algebraic geometry. The notion of mathematical instanton bundle was first introduced
on P3. By definition a mathematical instanton of charge nis a stable rank 2 vector
bundle with on P3with Chern classes c1(E)=0, c2(E)=n, satisfying the instantonic
vanishing condition
h1(E(−2)) =0.
Since then the irreducibility [34] and smoothness [19] of their moduli space were
heavily investigated. Faenzi [11] and Kuznetsov [21] generalized this notion to Fano
threefolds, we recall
BXuqiang Qin
qinx@iu.edu
1Department of Mathematics, Indiana University, 831 E. Third St., Bloomington, IN 47405, USA
123
Content courtesy of Springer Nature, terms of use apply. Rights reserved.
... . But E is locally free therefore: (36) for O Y and the vanishing of H 1 (N Y /X ) is due to Lemma (50)). Therefore Ext 2 (E, I Y ) = 0 which implies Ext 2 (E, E) = 0. Now, the stability of E leads to Hom(E, E) ≃ C and Ext 3 (E, E) ≃ Hom(E, E(−3)) * = 0. Since E has homological dimension one, we can apply an argument equivalent to [14,Proposition 3.4], obtaining: ...
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