Dimension of a variety (Tag D-5LJUX)
The dimension of a topological space is the supremum of the lengths \(n\) of chains \[ Z_0 \subsetneq Z_1 \subsetneq \cdots \subsetneq Z_n \] of irreducible closed subsets.
If \(Y \subseteq \AA^n\) is an affine algebraic set, then \(\dim Y = \krulldim A(Y)\). [@Har10a, Proposition I.1.7]
Let \(k\) be any field and \(B\) a finitely generated \(k\)-algebra that is a domain. Then \(\krulldim B = \trdeg_k \Frac B\). [@Har10a, Theorem I.1.8A]
If \(Y\) is a quasi-affine variety, then \(\dim Y = \dim \closure{Y}\), where \(\closure{Y}\) is the closure of \(Y\) in \(\AA^n\). [@Har10a, Proposition I.1.10]
Hence for an affine variety \(X\) over \(k\), \[ \dim X = \krulldim A(X) = \trdeg_k k(X) , \] and in particular \(\dim \AA^n = n\) [@Har10a, Proposition I.1.9]. For nonconstant \(f\in k[x_1,\ldots,x_n]\), every irreducible component of \(V(f)\subseteq\AA^n\) has dimension \(n-1\) [@Har10a, Proposition I.1.13]. In terms of transcendence degree: if \(g\) is an irreducible factor of \(f\) of positive degree in \(x_n\), then the images of \(x_1,\ldots,x_{n-1}\) in \(k[x_1,\ldots,x_n]/(g)\) are algebraically independent, because \(g\) divides no nonzero polynomial in \(x_1,\ldots,x_{n-1}\), and the image of \(x_n\) is algebraic over them.
For an affine variety \(X\), \(\dim X\) is the length of the longest chain of irreducible closed subsets, it equals the Krull dimension of \(A(X)\), and for \(X\) irreducible it equals \(\operatorname{trdeg}_kk(X)\).
Krull's principal ideal theorem, and codimension one (Tag PR-VARHT)
Let \(A\) be Noetherian and \(f \in A\) neither a unit nor a zero divisor. Every minimal prime over \((f)\) has height one. [@Har10a, Theorem I.1.11A]
(Consequences used on varieties)
For \(B\) a finitely generated domain over a field and \(\mfp \in \Spec B\), \[ \height \mfp + \krulldim (B/\mfp) = \krulldim B . \] [@Har10a, Theorem I.1.8A] A closed irreducible \(Y \subseteq \AA^n\) has \(\codim Y = 1\) exactly when \(Y = V(f)\) for a single irreducible nonconstant \(f \in k[x_1,\ldots,x_n]\). [@Har10a, Proposition I.1.13]
(Unique factorization via height-one primes)
A Noetherian domain \(A\) is a UFD if and only if every prime ideal of \(A\) of height one is principal. [@Har10a, Proposition I.1.12A]
The Hauptidealsatz says that one nonzerodivisor cuts codimension one; in affine space, height-one primes are principal because \(k[x_1,\ldots,x_n]\) is a UFD, so irreducible codimension-one subvarieties are hypersurfaces.
The additivity formula fails without finite generation over a field. Codimension does not bound the number of equations: the ideal of the image of \(t \mapsto (t^3,t^4,t^5)\) in \(\AA^3\) is a prime of height two that cannot be generated by two elements (P-AGHNOTLCI).
Degree, before the Hilbert polynomial
Degree as an intersection count (Tag D-VARDEG)
Let \(X \subseteq \PP^N\) be projective of dimension \(n\). The degree of \(X\) is the number of points of \[ X \intersect H_1 \intersect \cdots \intersect H_n \] for \(H_1, \ldots, H_n\) general hyperplanes.
This number is finite, independent of the general hyperplanes chosen, and equals \(n!\) times the leading coefficient of the Hilbert polynomial \(P_X\).
If a hyperplane \(H\) contains no irreducible component of \(X\), the sequence \(0\to\OO_X(-1)\to\OO_X\to\OO_{X\cap H}\to0\) is exact, so \(P_{X\cap H}(r)=P_X(r)-P_X(r-1)\). If \(P_X\) has leading term \(\frac{d}{n!}r^n\), then \(P_{X\cap H}\) has leading term \(\frac{d}{(n-1)!}r^{n-1}\), so \(X\cap H\) has dimension \(n-1\) and the same degree \(d\). After \(n\) such cuts, \(X\cap H_1\cap\cdots\cap H_n\) is a finite scheme of length \(d\); for general hyperplanes it is reduced, so it consists of \(d\) points.
For hyperplanes that are not general, the intersection can have fewer points: a line tangent to a smooth conic meets it in one point, of intersection multiplicity \(2\), as Bézout’s theorem requires. Degree depends on the embedding: the twisted cubic and a line are both isomorphic to \(\PP^1\) but have degrees \(3\) and \(1\) in their respective embeddings.
Rational normal scrolls (Tag D-VARSCROLL)
Let \(a_1, \ldots, a_k \geq 1\) and \(N = \sum_i a_i + k - 1\). Choose complementary linear subspaces \(\PP^{a_1}, \ldots, \PP^{a_k} \subseteq \PP^N\), a rational normal curve \(C_i \subseteq \PP^{a_i}\) of degree \(a_i\), and isomorphisms \(\phi_i \colon \PP^1 \to C_i\). The rational normal scroll \(S(a_1, \ldots, a_k) \subseteq \PP^N\) is the union over \(t \in \PP^1\) of the \((k-1)\)-planes spanned by \(\phi_1(t), \ldots, \phi_k(t)\).
\(S(a_1, \ldots, a_k)\) is the image of the projective bundle \(\PP\big(\OO_{\PP^1}(a_1) \oplus \cdots \oplus \OO_{\PP^1}(a_k)\big)\) under the complete linear system of its tautological bundle \(\OO(1)\). It is a smooth variety of dimension \(k\) and degree \(\sum_i a_i\), and its projective equivalence class depends only on the multiset \(\{a_1, \ldots, a_k\}\).
\(S(1,1) \subseteq \PP^3\) is the smooth quadric surface, ruled by the lines joining corresponding points of two skew lines. \(S(1,2) \subseteq \PP^4\) is the cubic scroll, the blowup of \(\PP^2\) at a point embedded by the conics through that point. It is cut out by the \(2 \times 2\) minors of \[ \begin{bmatrix} x_0 & x_1 & y_0 \\ x_1 & x_2 & y_1 \end{bmatrix} , \] and is the closure of the image of \((s, t) \mapsto [1 : s : s^2 : t : st]\). Allowing \(a_i = 0\) gives cones: \(S(0,2)\) is the quadric cone over a conic.
(Scroll)
A scroll is a projective variety \(X \subseteq \PP^N\) with a morphism \(X \to B\) onto a variety \(B\) whose fibres are linear subspaces of \(\PP^N\) of a fixed dimension.
The Hilbert polynomial
The Hilbert polynomial and its coefficients (Tag D-L6ERW)
For \(X \subseteq \PP^n\) closed with homogeneous coordinate ring \(S(X) = k[x_0,\ldots,x_n]/I(X)\), the Hilbert function is \[ h_X(r) \da \dim_k S(X)_r . \] For \(r \gg 0\) it agrees with a polynomial \(P_X(r)\), the Hilbert polynomial of \(X\).
For a coherent sheaf \(\mcf\) on a closed subscheme \(X \subseteq \PP^n_k\), the Hilbert function of \(\mcf\) is \(h_\mcf(m) \da h^0(X, \mcf(m))\).
For \(m \gg 0\), \(h_\mcf(m) = \chi(X, \mcf(m)) = P_\mcf(m)\), the value of the Hilbert polynomial of \(\mcf\) (D-COHEULER), and \(P_X = P_{\OO_X}\). [@Har10a, Exercise III.5.2]
(Reading off the coefficients)
Write \(\dim X = d\). Then \(\deg P_X = d\), and
the leading coefficient is \(\deg(X)/d!\); this defines the degree of \(X\) [@Har10a, §I.7];
the constant term is \(P_X(0) = \chi(\OO_X)\), and the arithmetic genus is \(p_a(X) = (-1)^d \qty(P_X(0) - 1)\), which is \(1 - \chi(\OO_X)\) for a curve.
The leading coefficient, and hence the degree, depends on the embedding: \(\PP^1\) embedded by \(\OO(3)\) has degree \(3\). The constant term \(\chi(X,\OO_X)\) and the arithmetic genus do not depend on the embedding [@Har10a, Exercise III.5.2].
(Hilbert polynomials of $\PP^n$ and the twisted cubic)
\(P_{\PP^n}(r) = \binom{r+n}{n}\), of degree \(n\) and leading coefficient \(1/n!\), so \(\deg \PP^n = 1\) and \(p_a(\PP^n) = 0\).
On \(\PP^1\), one has \(h^0(\PP^1,\OO(m))=m+1\) for \(m\ge0\) and zero for \(m<0\) [@Har10a, Proposition II.5.13]. Taking \(m=3r\) for \(r\ge0\) gives the Hilbert polynomial of the twisted cubic, \(P(r)=3r+1\), with degree \(3\) and arithmetic genus \(0\).
The leading coefficient determines the degree, while for a projective curve the constant term records the arithmetic genus through \(\chi(\OO_X)\).
A hypersurface of equation-degree $d$ has degree $d$ (Tag PR-JR7TS)
Let \(f \in k[x_0,\ldots,x_n]\) be homogeneous, irreducible, of degree \(d\), and let \(X = V(f) \subseteq \PP^n\). Then \(\deg X = d\).
(Degree of a hypersurface section)
Let \(Y \subseteq \PP^n\) be a projective variety of dimension \(\geq 1\) and \(H\) a hypersurface not containing \(Y\), with \(Y \cap H = Z_1 \cup \cdots \cup Z_s\) the irreducible components. Then \[ \sum_{j=1}^s i(Y, H; Z_j) \deg Z_j = (\deg Y)(\deg H) , \] where \(i(Y, H; Z_j)\) is the intersection multiplicity along \(Z_j\). For distinct plane curves \(Y, Z \subseteq \PP^2\) of degrees \(d\) and \(e\) with no common component, \(\sum_{P \in Y \cap Z} i(Y, Z; P) = de\) (Bézout). [@Har10a, Theorem I.7.7]
The proof is the exact sequence \[ 0 \to S(-d) \xrightarrow{\; \cdot f \;} S \to S/(f) \to 0 , \] which is exact because \(S\) is a domain and \(f \neq 0\), so multiplication by \(f\) is injective. Taking dimensions in degree \(r\), \[ P_X(r) = \binom{r+n}{n} - \binom{r-d+n}{n} , \] a polynomial of degree \(n-1\) whose leading coefficient is \(d/(n-1)!\).
The statement is the compatibility of two definitions of degree that look unrelated: the degree of the defining equation, and the normalised leading term of the Hilbert polynomial. The geometric count — the number of points in which \(X\) meets a general line — agrees with both, and that is Bezout in the first nontrivial case.
What depends on the embedding
Degree does, and so does the whole Hilbert polynomial, because the homogeneous coordinate ring does: \(\PP^1\) embedded as a line has degree \(1\), and embedded as the twisted cubic it has degree \(3\), and it is the same curve. Dimension does not. Neither does the arithmetic genus of a smooth projective curve, so the constant term remains unchanged when the embedding changes even though the leading coefficient need not.
Families of varieties
Hypersurfaces of degree \(d\) are the points of a projective space, and effective cycles of fixed dimension and degree are the points of the Chow variety.
General and very general hypersurfaces (Tag D-VARVERYGEN)
A hypersurface of degree \(d\) in \(\PP^n\) is the closed subscheme \(V(F)\) of a nonzero homogeneous polynomial \(F\) of degree \(d\). Two polynomials define the same hypersurface exactly when they differ by a nonzero scalar, so the hypersurfaces of degree \(d\) are the points of \[ \PP^{N} = \PP H^0(\PP^n, \OO(d)), \qquad N = \binom{n+d}{d} - 1 . \]
(General and very general)
A property holds for a general hypersurface of degree \(d\) if it holds for every hypersurface whose point lies outside a proper closed subset of \(\PP^N\). It holds for a very general hypersurface of degree \(d\) if it holds for every hypersurface whose point lies outside a countable union of proper closed subsets of \(\PP^N\).
Over an uncountable algebraically closed field, a countable union of proper closed subsets does not cover \(\PP^N\), so a very general hypersurface exists. Over \(\overline{\mathbb{Q}}\) or \(\overline{\mathbb{F}}_p\) the countably many hypersurfaces defined over the field can exhaust the parameter space, and “very general” can be vacuous.
Smoothness is a general property: the singular hypersurfaces form the discriminant hypersurface in \(\PP^N\). A property such as \(\Pic(X) = \ZZ \cdot \OO_X(1)\) for a surface \(X \subseteq \PP^3\) of degree \(d \geq 4\) holds only for a very general surface: the Noether–Lefschetz locus where the Picard number jumps is a countable union of proper closed subsets.
Chow forms and the Chow variety (Tag D-VARCHOW)
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 \(\mathcal{C}_{0,d}(\PP^n) = \Sym^d \PP^n\).