The nonvanishing locus of a section of an invertible sheaf #
Let M be an invertible sheaf on a scheme X and s a section of M over an open U. The
nonvanishing locus X_s is the set of points x ∈ U at which the germ of s generates the
stalk M_x, equivalently at which s does not vanish in the fibre M_x ⊗ κ(x). Over an open V
on which M is trivialized, s is a regular function times the basis section, and X_s ∩ V is
the basic open set of that function; in particular X_s is open. It generalizes the basic open
set X_f of a regular function, the case of the structure sheaf.
These open sets are the ones through which positivity of line bundles is defined: M is globally
generated when the sets X_s of its global sections cover X, and ample when X is
quasi-compact and the sets X_s that are affine, for global sections s of the positive tensor
powers of M, cover X.
The section s is a basis of M exactly over its nonvanishing locus: an open W ⊆ U lies in
X_s if and only if, on every open W' ⊆ W, every section of M is a unique regular multiple
of the restriction of s. In other words, s trivializes M over X_s.
Main definitions #
AlgebraicGeometry.Scheme.Modules.nonvanishingLocus: the open setX_sof points at which the germ ofsgenerates the stalk.
Main results #
AlgebraicGeometry.Scheme.Modules.nonvanishingLocus_eq_basicOpen: on the domain of a rank-one trivialization,X_sis the basic open set of the coordinate ofs;AlgebraicGeometry.Scheme.Modules.nonvanishingLocus_mapandAlgebraicGeometry.Scheme.Modules.nonvanishingLocus_smul: restriction intersectsX_swith the smaller open, andX_{r s} = X_r ∩ X_sfor a regular functionr;AlgebraicGeometry.Scheme.Modules.le_nonvanishingLocus_iff:sis a basis ofMexactly over open subsets ofX_s;AlgebraicGeometry.Scheme.Modules.nonvanishingLocus_hom_app: the nonvanishing locus is invariant under isomorphisms of invertible sheaves;AlgebraicGeometry.Scheme.Modules.range_nonvanishingLocus_eq_of_iso: isomorphic invertible sheaves have the same nonvanishing loci of sections over each open;AlgebraicGeometry.Scheme.Modules.nonvanishingLocus_unit: on the structure sheaf,X_ris the basic open set of the regular functionr.
References #
- The Stacks Project, Tag 01CT (the open
X_sattached to a section of an invertible sheaf) and Tag 01PS (ample invertible sheaves). - R. Hartshorne, Algebraic Geometry, Chapter II, Section 7.
On the domain of a rank-one trivialization, the germ of a section generates the stalk exactly when the germ of its coordinate is a unit of the local ring.
The nonvanishing locus X_s of a section s of an invertible sheaf over U: the open
set of points of U at which the germ of s generates the stalk.
Instances For
A point lies in the nonvanishing locus of s exactly when it lies in the domain of s and the
germ of s there generates the stalk.
A point of the domain of s lies in its nonvanishing locus exactly when the germ of s
generates the stalk there.
The nonvanishing locus of a section over U is contained in U.
On the domain of a rank-one trivialization, the nonvanishing locus of a section is the basic open set of its coordinate.
The nonvanishing locus of a restricted section is the intersection of the smaller open with the nonvanishing locus of the section.
The nonvanishing locus of a regular multiple r • s is the intersection of the basic open set
of r with the nonvanishing locus of s.
The zero section vanishes everywhere.
The basis section of a rank-one trivialization over V vanishes nowhere on V.
A section is a nonzerodivisor for the regular functions over any open contained in its nonvanishing locus.
Every section over an open contained in the nonvanishing locus of s is a regular multiple
of s.
A section is a basis exactly over its nonvanishing locus. An open W ⊆ U lies in the
nonvanishing locus of s if and only if, on every open W' ⊆ W, every section of M is a unique
regular multiple of the restriction of s.
The nonvanishing locus is invariant under isomorphisms of invertible sheaves.
Isomorphic invertible sheaves have the same nonvanishing loci of sections over U: the set of
opens X_s, for s a section over U, is invariant under isomorphism.
On the structure sheaf 𝒪_X, regarded as an invertible sheaf, the nonvanishing locus of a
regular function r is its basic open set X_r.