Mathematical Instantons with Maximal Order Jumping Lines

Prabhakar Rao

doi:10.2140/pjm.1997.178.331

Mathematical Instantons with Maximal Order Jumping Lines

Prabhakar Rao

University of Missouri-St. Louis

Research output: Contribution to journal › Article › peer-review

Abstract

<div class="line" id="line-7"> A mathematical instanton bundle on P 3  (over an algebraically closed field) is a rank two vector bundle  &expectation;  on P 3  with c 1  = 0 and with H 0 ( &expectation; ) = H 1 ( &expectation; ( − 2)) = 0. Let c 2 ( &expectation; ) = n. Then n > 0. A jumping line of  &expectation;  of order a, (a > 0), is a line &ell; in P 3  on which  &expectation; splits as  &Oscr; &ell; ( − a)  &oplus;&Oscr; &ell; (a). It is easy to see that the jumping lines of  &expectation;  all have order  ≤  n. We will say that  &expectation;  has a maximal order jumping line if it has a jumping line of order n. Our goal is to show that such an  &expectation;  is unobstructed in the moduli space of stable rank two bundles, i.e., H 2 ( &expectation;&otimes;&expectation; ) = 0. The technique can be slightly extended. We show that when c 2  = 5, any  &expectation;  with a jumping line of order 4 is unobstructed. We describe at the end how mathematical instantons with maximal order jumping lines arise and estimate the dimension of this particular smooth locus of bundles. </div>

Original language	American English
Journal	Pacific Journal of Mathematics
Volume	178
DOIs	https://doi.org/10.2140/pjm.1997.178.331
State	Published - 1997

Disciplines

Physical Sciences and Mathematics
Mathematics

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Cite this

@article{715e44c66e1349cca3899294c0ef5b2f,

title = "Mathematical Instantons with Maximal Order Jumping Lines",

abstract = " A mathematical instanton bundle on P 3  (over an algebraically closed field) is a rank two vector bundle  &expectation;  on P 3  with c 1  = 0 and with H 0 ( &expectation; ) = H 1 ( &expectation; ( − 2)) = 0. Let c 2 ( &expectation; ) = n. Then n > 0. A jumping line of  &expectation;  of order a, (a > 0), is a line &ell; in P 3  on which  &expectation; splits as  &Oscr; &ell; ( − a)  &oplus;&Oscr; &ell; (a). It is easy to see that the jumping lines of  &expectation;  all have order  ≤  n. We will say that  &expectation;  has a maximal order jumping line if it has a jumping line of order n. Our goal is to show that such an  &expectation;  is unobstructed in the moduli space of stable rank two bundles, i.e., H 2 ( &expectation;&otimes;&expectation; ) = 0. The technique can be slightly extended. We show that when c 2  = 5, any  &expectation;  with a jumping line of order 4 is unobstructed. We describe at the end how mathematical instantons with maximal order jumping lines arise and estimate the dimension of this particular smooth locus of bundles. ",

author = "Prabhakar Rao",

year = "1997",

doi = "10.2140/pjm.1997.178.331",

language = "American English",

volume = "178",

journal = "Pacific Journal of Mathematics",

}

TY - JOUR

T1 - Mathematical Instantons with Maximal Order Jumping Lines

AU - Rao, Prabhakar

PY - 1997

Y1 - 1997

N2 - A mathematical instanton bundle on P 3  (over an algebraically closed field) is a rank two vector bundle  &expectation;  on P 3  with c 1  = 0 and with H 0 ( &expectation; ) = H 1 ( &expectation; ( − 2)) = 0. Let c 2 ( &expectation; ) = n. Then n > 0. A jumping line of  &expectation;  of order a, (a > 0), is a line &ell; in P 3  on which  &expectation; splits as  &Oscr; &ell; ( − a)  &oplus;&Oscr; &ell; (a). It is easy to see that the jumping lines of  &expectation;  all have order  ≤  n. We will say that  &expectation;  has a maximal order jumping line if it has a jumping line of order n. Our goal is to show that such an  &expectation;  is unobstructed in the moduli space of stable rank two bundles, i.e., H 2 ( &expectation;&otimes;&expectation; ) = 0. The technique can be slightly extended. We show that when c 2  = 5, any  &expectation;  with a jumping line of order 4 is unobstructed. We describe at the end how mathematical instantons with maximal order jumping lines arise and estimate the dimension of this particular smooth locus of bundles.

AB - A mathematical instanton bundle on P 3  (over an algebraically closed field) is a rank two vector bundle  &expectation;  on P 3  with c 1  = 0 and with H 0 ( &expectation; ) = H 1 ( &expectation; ( − 2)) = 0. Let c 2 ( &expectation; ) = n. Then n > 0. A jumping line of  &expectation;  of order a, (a > 0), is a line &ell; in P 3  on which  &expectation; splits as  &Oscr; &ell; ( − a)  &oplus;&Oscr; &ell; (a). It is easy to see that the jumping lines of  &expectation;  all have order  ≤  n. We will say that  &expectation;  has a maximal order jumping line if it has a jumping line of order n. Our goal is to show that such an  &expectation;  is unobstructed in the moduli space of stable rank two bundles, i.e., H 2 ( &expectation;&otimes;&expectation; ) = 0. The technique can be slightly extended. We show that when c 2  = 5, any  &expectation;  with a jumping line of order 4 is unobstructed. We describe at the end how mathematical instantons with maximal order jumping lines arise and estimate the dimension of this particular smooth locus of bundles.

U2 - 10.2140/pjm.1997.178.331

DO - 10.2140/pjm.1997.178.331

M3 - Article

VL - 178

JO - Pacific Journal of Mathematics

JF - Pacific Journal of Mathematics

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