Abstract
A recent paper of Totaro developed a theory of q-ample bundles in characteristic 0. Specifically, a line bundle L on X is q-ample if for every coherent sheaf F on X, there exists an integer m 0 such that m ≥ m 0 implies H i (X, F ⊗ O(mL))=0 for i>q. We show that a line bundle L on a complex projective scheme X is q-ample if and only if the restriction of L to its augmented base locus is q-ample. In particular, when X is a variety and L is big but fails to be q-ample, then there exists a codimension-one subscheme D of X such that the restriction of L to D is not q-ample.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 790-798 |
| Number of pages | 9 |
| Journal | Compositio Mathematica |
| Volume | 148 |
| Issue number | 3 |
| DOIs | |
| State | Published - May 2012 |
| Externally published | Yes |
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Keywords
- augmented base loci
- linear series
- partial amplitude
ASJC Scopus subject areas
- Algebra and Number Theory
Cite this
Big q-ample line bundles. / Brown, Morgan.
In: Compositio Mathematica, Vol. 148, No. 3, 05.2012, p. 790-798.Research output: Contribution to journal › Article
}
TY - JOUR
T1 - Big q-ample line bundles
AU - Brown, Morgan
PY - 2012/5
Y1 - 2012/5
N2 - A recent paper of Totaro developed a theory of q-ample bundles in characteristic 0. Specifically, a line bundle L on X is q-ample if for every coherent sheaf F on X, there exists an integer m 0 such that m ≥ m 0 implies H i (X, F ⊗ O(mL))=0 for i>q. We show that a line bundle L on a complex projective scheme X is q-ample if and only if the restriction of L to its augmented base locus is q-ample. In particular, when X is a variety and L is big but fails to be q-ample, then there exists a codimension-one subscheme D of X such that the restriction of L to D is not q-ample.
AB - A recent paper of Totaro developed a theory of q-ample bundles in characteristic 0. Specifically, a line bundle L on X is q-ample if for every coherent sheaf F on X, there exists an integer m 0 such that m ≥ m 0 implies H i (X, F ⊗ O(mL))=0 for i>q. We show that a line bundle L on a complex projective scheme X is q-ample if and only if the restriction of L to its augmented base locus is q-ample. In particular, when X is a variety and L is big but fails to be q-ample, then there exists a codimension-one subscheme D of X such that the restriction of L to D is not q-ample.
KW - augmented base loci
KW - linear series
KW - partial amplitude
UR - http://www.scopus.com/inward/record.url?scp=84861428949&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84861428949&partnerID=8YFLogxK
U2 - 10.1112/S0010437X11007457
DO - 10.1112/S0010437X11007457
M3 - Article
VL - 148
SP - 790
EP - 798
JO - Compositio Mathematica
JF - Compositio Mathematica
SN - 0010-437X
IS - 3
ER -