The Proj ChowScheme¶

AUTHORS:

• Manfred Lehn (2013)
• Christoph Sorger (2013)

sage.schemes.chow.library.proj.Proj(n, hyperplane_class='h', names=None, name=None, latex_name=None)¶

Return the projective space \(\mathbb{P}^n\) of dimension n.

INPUT:

• n – An integer, the dimension of the projective space.
• hyperplane_class - An (optional) name for the hyperplane class
• name – An optional string, the name of the ChowScheme
• latex_name– An optional string, the latex representation of the ChowScheme

OUTPUT:

• The ChowScheme corresponding to the projective space in the sense of Grothendieck, i.e. the rank 1 quotients of \(mathbb{C}^{n+1}\).

EXAMPLES:

sage: X = Proj(3)  # P3 of rank 1 quotients of a 4 dim. vector space
sage: X.sheaves["universal_sub"]
Bundle(Proj(3, 'h'), 3, [1, -h, h^2, -h^3])

sage.schemes.chow.library.proj.ProjBundle(E, hyperplane_class='h', names=None, name=None, latex_name=None)¶

Return the Proj of a bundle E.

INPUT:

• E – A sheaf on a ChowScheme
• hyperplane_class - An (optional) name for the hyperplane class
• name – An optional string, the name of the ProjBundle
• latex_name– An optional string, the latex representation of the ProjBundle

OUTPUT:

• The ChowScheme corresponding to \(\mathbb{P}(E)\) in the sense of Grothendieck, i.e. the rank 1 quotient modules of E.

EXAMPLES:

sage: P3 = Proj(3, name='P3')
sage: S = P3.sheaves['universal_sub']
sage: PS = ProjBundle(S); str(PS)
'Proj(Bundle(P3, 3, [1, -h, h^2, -h^3]))'