homogeneous polynomial and homogeneous ideal

Depends on

• The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution
• Nonnegatively graded rings and modules, homogeneous elements, and twists

Used by

• A plane intersection with no common component is nonempty and zero-dimensional Corollary
• General hypersurfaces give smooth complete intersections Corollary
• affine cone projective set Definition
• degree projective hypersurface Definition
• homogeneous ideal saturation Definition
• Linear systems of plane curves and their base loci Definition
• Multiplicity of a hypersurface equation at a rational point Definition
• Plane projective curves and their components Definition
• projective algebraic set Definition
• Projective scheme of a homogeneous quotient and its standard affine charts Definition
• Resultant of two plane forms, viewed in one variable Definition
• Tangent cone and tangent lines at a point Definition
• A node has two distinct tangent directions Example
• A nondegenerate projective quadric Example
• A projective cone with a smooth conic base Example
• A quadratic-cubic plane complete intersection has eventual Hilbert value six Example
• Different singularities can share a tangent cone Example
• The cusp retains a doubled tangent line Example
• The rank-one 2 by 2 determinantal cone Example
• Three axes are not three coplanar lines Example
• All initial forms define the tangent cone Lemma
• Coprime positive-degree plane forms form a regular sequence Lemma
• Coprime tangent cones force a power of the maximal ideal into the local ideal Lemma
• Field extension preserves the graded pieces and the total length of a zero-dimensional projective quotient Lemma
• homogeneous polynomial zero locus well defined Lemma
• homogenization dehomogenization correspondence Lemma
• Prime and local-ring correspondence on standard projective charts Lemma
• projective irreducibility homogeneous prime Lemma
• Scaling, specialization, and the affine and infinite charts of a binary resultant Lemma
• The resultant detects finitely many common projective points Lemma
• The standard open D_+(f) of a projective quotient is the affine chart Spec((S_f)₀) Lemma
• The truncated multiplication map is injective exactly when the tangent cones are coprime Lemma
• Bertini smoothness away from the base locus Theorem
• Two coprime projective plane forms meet in total length equal to their degree product Theorem

Dependency tree · two levels

4 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• J. S. Milne, Algebraic Geometry, Chapter 6 (standard reference, not scraped)
• Michael Artin, Algebraic Geometry, Chapter 3 (standard reference, not scraped)