关于代数几何射影簇部分‘形式集合（set of forms）’的疑问

Hey there! Let's break down what a set of forms means in the context of projective varieties—this is a foundational term that ties directly to how we define projective spaces and their subvarieties.

First, let's start with the building block: a form (or homogeneous polynomial) is a polynomial where every single term has the same total degree. For example, in the polynomial ring k[x₀, x₁, ..., xₙ] over a field k:

• x₀² + x₁x₂ is a degree-2 form (both terms add up to degree 2)
• x₀ + x₁² is NOT a form (its terms have degrees 1 and 2, which don't match)

A set of forms is simply a collection of these homogeneous polynomials. But why does this specific collection matter so much for projective geometry?

The Critical Role in Defining Projective Varieties #

Projective varieties are exactly the common zero sets of a set of forms in projective space ℙⁿ(k). Here's why homogeneity is non-negotiable:

• In projective space, points are equivalence classes written as [x₀ : x₁ : ... : xₙ], where (x₀,...,xₙ) ≠ (0,...,0) and scaling all coordinates by a non-zero λ ∈ k gives the same point: [λx₀ : ... : λxₙ] = [x₀ : ... : xₙ].
• A homogeneous polynomial f has the key property: f(λx₀, ..., λxₙ) = λᵈf(x₀,...,xₙ) (where d is the degree of f). So if f vanishes at a representative point (x₀,...,xₙ), it vanishes at every scaled version of that point too. This means the zero set of f is well-defined on projective space—it doesn't depend on which representative we pick for the point.

Quick Examples to Make It Stick #

• Take the set of forms {x₀ - x₁, x₂} in k[x₀,x₁,x₂]. Their common zeros in ℙ²(k) are all points [a : a : 0] (with a ≠ 0)—this is a projective line, a 1-dimensional projective variety.
• A single degree-2 form like {x₀² + x₁² - x₂²} defines a conic section in ℙ²(k)—think of an ellipse, parabola, or hyperbola, but embedded in projective space.

A Couple of Related Notes #

• We often care about the ideal generated by a set of forms: since sums and products of homogeneous polynomials are also homogeneous, this ideal is a homogeneous ideal. Thanks to Hilbert's Projective Nullstellensatz, there's a bijection between projective varieties and radical homogeneous ideals—so sets of forms are the starting point for defining all projective varieties.
• Don't mix forms up with non-homogeneous polynomials! Non-homogeneous polynomials don't work for projective space because their zero sets aren't invariant under coordinate scaling—vanishing at (x₀,...,xₙ) doesn't mean they vanish at (λx₀,...,λxₙ).

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