(Chow form)
Let \(X \subseteq \PP^n\) be an irreducible projective variety of dimension \(k\) and degree \(d\) over an algebraically closed field. The set of \((n-k-1)\)-planes in \(\PP^n\) that meet \(X\) is an irreducible hypersurface \[ Z(X) \subseteq \Gr(n-k-1, n), \] and it is the zero locus of a section of \(\OO(d)\) in the Plücker embedding, unique up to a scalar. This section is the Chow form of \(X\). For an effective \(k\)-cycle \(\sum_i m_i X_i\), the Chow form is the product of the Chow forms of the \(X_i\) raised to the powers \(m_i\).
(Chow variety)
The Chow form defines an injective map from the set of effective \(k\)-cycles of degree \(d\) in \(\PP^n\) to the projective space \(\PP H^0(\Gr(n-k-1,n), \OO(d))\). Its image is a closed algebraic subset, the Chow variety \(\mathcal{C}_{k,d}(\PP^n)\), which parametrizes effective \(k\)-cycles of degree \(d\) in \(\PP^n\).
For \(k = n-1\), a \((n-1)\)-cycle of degree \(d\) is a hypersurface counted with multiplicity, a \(0\)-plane meets it exactly when the point lies on it, and \(\mathcal{C}_{n-1,d}(\PP^n) = \PP H^0(\PP^n, \OO(d))\) is the space of degree-\(d\) hypersurfaces. For \(k = 0\), the \((n-1)\)-planes meeting a point \(p\) form a hyperplane in the dual projective space, and \(\mathcal{C}_{0,d}(\PP^n) = \Sym^d \PP^n\).
Dependencies
D-VARDEGDegree as an intersection countuses