Complete intersection

In commutative algebra and algebraic geometry, a complete intersection refers to an ideal or a subvariety defined by the minimal number of equations necessary to achieve its codimension, ensuring a particularly simple and well-behaved structure. Specifically, in a commutative Noetherian ring R, an ideal I \neq R is a complete intersection if its height h(I) equals the minimal number of generators \mu(I), meaning I can be generated by a regular sequence of length equal to its height. Geometrically, for a subvariety Y \subset X in an algebraic variety X, Y is a complete intersection if the ideal sheaf \mathcal{I}(Y) in the structure sheaf \mathcal{O}(X) is an ideal-theoretic complete intersection, corresponding to Y being the common zero set of codimension-many hypersurfaces. The concept distinguishes between local and global complete intersections: an ideal I is a local complete intersection if, for every maximal ideal m of R, the localization I_m is a complete intersection in R_m. In the geometric setting, a variety is a local complete intersection if it is locally defined this way at every point, which includes all smooth subvarieties but excludes more singular or complicated objects. A related but weaker notion is the set-theoretic complete intersection, where the radical of I is generated by codimension-many elements, allowing for non-ideal-theoretic definitions but still capturing the "minimal intersection" idea geometrically. Complete intersections are fundamental due to their strong homological properties; for instance, quotient rings by complete intersection ideals are Cohen-Macaulay, with depth equal to dimension, and they exhibit finite projective dimension over regular rings. In algebraic geometry, they arise naturally in intersection theory, deformation theory, and enumerative problems, where their simplicity facilitates computations of invariants like Hilbert series or genus; for example, the normal bundle of a complete intersection subvariety in projective space is a direct sum of line bundles, providing a simple structure that aids in such calculations. These objects contrast with Gorenstein rings or other classes, highlighting their role in classifying rings and varieties with "nice" singularities.

Fundamentals

Definition

In algebraic geometry, a closed subvariety X of an algebraic variety Y is a local complete intersection if the ideal sheaf \mathcal{I}_X of X in Y is locally generated by a regular sequence whose length equals the codimension of X in Y. A sequence of elements f_1, \dots, f_k in a Noetherian ring A (such as the stalk of the structure sheaf at a point) forms a regular sequence if f_1 is a non-zerodivisor in A, f_2 is a non-zerodivisor in A/(f_1), and inductively, each f_i is a non-zerodivisor in A/(f_1, \dots, f_{i-1}) for i = 2, \dots, k. This condition implies that the ideal (f_1, \dots, f_k) has height exactly k, ensuring the quotient ring has the expected dimension. The local complete intersection property is defined pointwise via local rings, but a global complete intersection arises when the ideal is generated globally by such a regular sequence, often as the intersection of hypersurfaces in the case of projective varieties. In projective space \mathbb{P}^n, for instance, a subvariety is a global complete intersection if its homogeneous ideal is generated by c homogeneous polynomials forming a regular sequence, where c is the codimension. The equality between the length of the regular sequence and the codimension guarantees that X is equidimensional, as the minimal number of generators matches the height of the ideal, preventing embedded components or unexpected dimension drops. Local complete intersections are locally Cohen-Macaulay.

Basic properties

A complete intersection ring, defined as the quotient of a commutative ring by an ideal generated by a regular sequence, exhibits the Cohen-Macaulay property, wherein the depth equals the dimension.^[1] This equality ensures that the ring behaves well homologically, with no discrepancies between local cohomology dimensions and the ring's geometric dimension.^[2] The ideal of a complete intersection admits a minimal free resolution given by the Koszul complex on its generators.^[1] This resolution is exact because the generators form a regular sequence, providing a concrete and efficient way to compute projective dimensions and other homological invariants.^[3] For schemes that are local complete intersections, Serre duality applies directly, pairing cohomology groups via the dualizing sheaf, which simplifies due to the regular sequence defining the embedding.^[4] This duality manifests straightforwardly without additional complications from non-regular elements, facilitating computations in sheaf cohomology. Complete intersection rings are equidimensional, meaning all prime ideals have the same height relative to the dimension, and they contain no embedded points, as the Cohen-Macaulay condition precludes associated primes of lower dimension.^[2] This purity of support ensures that the scheme's components are uniformly dimensioned without extraneous lower-dimensional loci.^[5]

Illustrative cases

Examples

A hypersurface in affine or projective space, defined as the zero locus of a single polynomial equation, is a complete intersection of codimension 1, as it satisfies the regular sequence condition with a single generator.^[6] For instance, the circle defined by x^2 + y^2 = [1](/page/1) in \mathbb{A}^2 over the complex numbers is such a hypersurface.^[7] The intersection of two hypersurfaces provides another fundamental example, particularly when the defining polynomials form a regular sequence of length equal to the codimension. In \mathbb{P}^3, the zero locus of two homogeneous quadratic polynomials, assuming they intersect transversely, yields a smooth curve of degree 4 and genus 1, known as an elliptic curve, which is a complete intersection of codimension 2.^[8] This embedding realizes the elliptic curve as the common zeros of the two quadrics, illustrating how complete intersections can capture low-genus curves in projective space.^[9] Certain determinantal varieties, which are loci where a generic matrix has rank at most a fixed value, arise as complete intersections in low-dimensional cases. For example, the Segre embedding of \mathbb{P}^1 \times \mathbb{P}^1 into \mathbb{P}^3 defines a quadric surface as the zero set of a single quadratic form, corresponding to the 2-by-2 minors of a $2 \times 2 matrix of coordinates, making it a determinantal complete intersection of codimension 1.^[10] This variety exemplifies how bilinear conditions can produce complete intersections via minor determinants. Over finite fields, complete intersections retain their algebraic structure, with hypersurfaces serving as straightforward instances. The Fermat curve x^d + y^d + z^d = 0 in \mathbb{P}^2 over \mathbb{F}_q is a hypersurface complete intersection whose rational points can be enumerated using character sums, highlighting arithmetic applications. Such examples demonstrate the uniformity of the complete intersection property across base fields.^[11]

Non-examples

The twisted cubic curve in \mathbb{P}^3, parametrized by (1 : t : t^2 : t^3), provides a classical example of a curve that is not a complete intersection. This curve has codimension 2 in \mathbb{P}^3, so it would be a complete intersection if its homogeneous ideal were generated by exactly two elements forming a regular sequence. However, the ideal is minimally generated by three independent quadrics, specifically the $2 \times 2 minors of the matrix

\begin{pmatrix} x_0 & x_1 & x_2 \\ x_1 & x_2 & x_3 \end{pmatrix},

which are x_1^2 - x_0 x_2, x_2^2 - x_1 x_3, and x_0 x_3 - x_1 x_2. This excess of generators over the codimension prevents the ideal from being generated by a regular sequence of length 2. A related non-example is the rational normal curve of degree 4 in \mathbb{P}^4, parametrized by (1 : t : t^2 : t^3 : t^4). This curve also has codimension 3, but its homogeneous ideal requires six minimal generators, consisting of the $2 \times 2 minors of the catalecticant matrix

\begin{pmatrix} x_0 & x_1 & x_2 & x_3 \\ x_1 & x_2 & x_3 & x_4 \end{pmatrix}.

The number of these minors is \binom{4}{2} = 6, exceeding the codimension and thus violating the condition for being a complete intersection, as the generators do not form a regular sequence of length 3.^[12] Another illustration of failure arises from schemes that are not equidimensional, such as the union of a curve and an isolated point in \mathbb{P}^3. Suppose C is an irreducible curve of dimension 1 and p is a point not on C; their union V = C \cup \{p\} has components of dimensions 1 and 0. A complete intersection in projective space must be equidimensional, with all components having the expected dimension equal to the ambient dimension minus the length of the defining regular sequence. Here, the ideal of V cannot be generated by a regular sequence of length 2 (to achieve codimension 2 for the curve part), as the embedded point disrupts the purity of dimension, making the quotient ring non-Cohen--Macaulay in a way incompatible with a complete intersection structure.^[13] Rational scroll surfaces offer further examples where the complete intersection condition fails, particularly due to the structure of their ideals and, in singular cases, non-Cohen--Macaulay local rings at certain points. Consider the rational normal scroll S(1,2) \subset \mathbb{P}^4, a smooth ruled surface of dimension 2 and codimension 2, obtained as the union of lines joining corresponding points on rational normal curves of degrees 1 and 2 in disjoint projective spaces. Its homogeneous ideal is generated by the three $2 \times 2 minors of a $2 \times 3 matrix, exceeding the codimension and preventing generation by a regular sequence of length 2. For singular scrolls like S(0,3) \subset \mathbb{P}^3, a cone over the twisted cubic with vertex at the origin, the local ring at the singular vertex fails to be Cohen--Macaulay, as the depth does not equal the dimension there, further ruling out a complete intersection structure despite the global codimension matching the number of potential generators.^[14]

Algebraic invariants

Multidegree

In algebraic geometry, a complete intersection X in a smooth variety Y is often defined as the common zero locus of global sections s_1, \dots, s_k of line bundles L_1, \dots, L_k on Y, where the sections form a regular sequence and k = \codim_Y X. The cycle class [X] \in A_k(Y) is given by the cap product of the ambient fundamental class [Y] with the top Chern class of the virtual normal bundle, which for this setup equals the product \prod_{i=1}^k c_1(L_i) \cap [Y]. The multidegree of X refers to the vector of coefficients of this class when expressed in a basis for the Chow group A_k(Y).^[15] In the specific case of hypersurfaces of degrees d_1, \dots, d_k in projective space \mathbb{P}^n, the line bundles are powers of the hyperplane bundle \mathcal{O}(1), so c_1(L_i) = d_i h where h is the class of a hyperplane. The Chow ring A^*(\mathbb{P}^n) is \mathbb{Z}/(h^{n+1}), and the class simplifies to [X] = \left( \prod_{i=1}^k d_i \right) h^k \cap [\mathbb{P}^n]. Thus, the multidegree is the scalar coefficient \prod_{i=1}^k d_i, which equals the degree of X as a subvariety of \mathbb{P}^n.^[15] For example, consider a curve C in \mathbb{P}^3 arising as the complete intersection of two surfaces of degrees a and b. The class is [C] = ab \, h^2 \cap [\mathbb{P}^3], so the multidegree (degree of C) is ab.^[15] This notion generalizes Bézout's theorem from classical algebraic geometry, where the intersection number of k = n hypersurfaces in general position in \mathbb{P}^n is \prod_{i=1}^n d_i; for a positive-dimensional complete intersection, the degree \prod_{i=1}^k d_i plays an analogous role as the intersection multiplicity with a general linear subspace complementary to the dimension of X.^[15]

General position

In algebraic geometry, a collection of hypersurfaces is said to be in general position if their scheme-theoretic intersection has the expected dimension and is transverse, meaning that at every point of the intersection, the differentials of the defining equations are linearly independent, resulting in a reduced scheme with no multiple components.^[16] This condition ensures that the intersection is a smooth complete intersection variety when the ambient space is smooth.^[17] Bertini's theorem plays a central role in establishing this genericity: for a smooth projective variety over an algebraically closed field of characteristic zero, a generic hyperplane section is smooth and thus transverse to the variety.^[16] Iterating this result, the intersection of multiple generic hypersurfaces of prescribed degrees in projective space yields a smooth complete intersection of the expected dimension, as the defining sections form a regular sequence and the intersection remains reduced away from any base locus.^[17] Specifically, in \mathbb{P}^n, the generic intersection of r hypersurfaces of degrees d_1, \dots, d_r (with r \leq n) is an irreducible smooth complete intersection of codimension r.^[16] When hypersurfaces are not in general position, their intersection may still be a complete intersection if the defining equations generate an ideal of the correct height and form a regular sequence, but the scheme can have positive-dimensional components with higher multiplicity due to tangencies or singularities.^[18] In such cases, the multiplicity at intersection points exceeds one, reflecting non-transversality, though the overall codimension is preserved.^[18] This contrasts with the reduced structure guaranteed in the general position scenario.

Topological features

Homology

For a smooth complete intersection X \subset \mathbb{P}^N of dimension n, the sheaf cohomology groups H^i(X, \mathcal{O}_X) vanish for $0 < i < n. This intermediate vanishing follows from the Koszul resolution of \mathcal{O}_X as an \mathcal{O}_{\mathbb{P}^N}-module, which is a finite complex of locally free sheaves whose cohomology is known from Bott's theorem on projective space. An adapted form of Kodaira's vanishing theorem applies to smooth complete intersections in projective space. If the anticanonical bundle -\omega_X is ample (which holds when the multidegree satisfies \sum d_j \le N)^[19], then Kodaira's theorem implies H^i(X, \mathcal{O}_X \otimes L) = 0 for i > 0 and ample line bundles L on X, providing vanishing for twists of the structure sheaf.^[20] The topological Betti numbers of smooth complete intersections can be computed using the Koszul complex associated to the defining equations. For a hypersurface (codimension 1 complete intersection), the Lefschetz hyperplane theorem states that the restriction map in cohomology H^i(\mathbb{P}^n, \mathbb{Z}) \to H^i(X, \mathbb{Z}) is an isomorphism for i < n-1 and surjective for i = n-1, where \dim X = n-1.^[21] Thus, the Betti numbers agree with those of \mathbb{P}^n outside the middle dimension, and the dimension of the primitive cohomology H^{n-1}(X)_{\mathrm{prim}} is given by (-1)^{n-1} (\chi(X) - n), with the Euler characteristic \chi(X) computed via the adjunction formula. For general codimension, iterative applications of the weak Lefschetz theorem yield isomorphisms in cohomology up to degrees i < n, where n is the dimension of X, and the Koszul complex provides a spectral sequence converging to the cohomology, allowing explicit computation of primitive Betti numbers in terms of the degrees d_1, \dots, d_c. Regarding singular homology, the complex points of a smooth complete intersection X of dimension n have H_{2n}(X(\mathbb{C}), \mathbb{Z}) \cong \mathbb{Z}, generated by the fundamental class. For the real points X(\mathbb{R}), even-dimensional complete intersections often exhibit non-trivial top-dimensional homology. For example, a smooth complete intersection of two quadrics in \mathbb{CP}^{2m+2} (dimension $2m) has H_{2m}(X(\mathbb{R}), \mathbb{Z}) \cong \mathbb{Z}^k for some k \geq 1, depending on the real topology, such as the number of connected components of the real locus. In singular cases with isolated singularities, the top Betti number adjusts by the Milnor numbers at singular points, yielding b_n(V) = b_n(V_{\text{smooth}}) - \sum \mu(V, a_i), where \mu is the Milnor number.^[22]

Euler characteristic

The topological Euler characteristic \chi(V) of a complete intersection variety V is defined as the alternating sum \sum_{i} (-1)^i b_i(V), where b_i(V) are the Betti numbers of V. For smooth projective complete intersections over \mathbb{C}, this invariant is determined solely by the dimension of the ambient projective space and the degrees of the defining hypersurfaces, reflecting the rigidity of their topology via the Lefschetz hyperplane theorem. This theorem implies that the homology of V agrees with that of the ambient space \mathbb{P}^n outside the middle dimension, with the difference arising from the primitive cohomology in the middle degree, allowing explicit computation of \chi(V). For a smooth hypersurface V \subset \mathbb{P}^{n+1} of degree d \geq 1 (so \dim V = n), the Euler characteristic admits the closed-form expression

\chi(V) = n + 2 - \frac{1}{d} \left[ 1 + (-1)^{n+1} (d-1)^{n+2} \right].

This formula arises from integrating the top Chern class c_n(TV) over V, using the adjunction formula for the tangent bundle TV = TP^{n+1}|_V \otimes N_{V/\mathbb{P}^{n+1}}^\vee and the known Chern classes of \mathbb{P}^{n+1} and the normal bundle N_{V/\mathbb{P}^{n+1}} = \mathcal{O}_V(d). For instance, when n=1 (plane curves), it recovers \chi(V) = d(3-d), consistent with the arithmetic genus formula. When n=2 (surfaces in \mathbb{P}^3), a degree-3 hypersurface yields \chi(V) = 9, reflecting its Picard rank and 27 lines. For general smooth complete intersections defined by hypersurfaces of degrees d_1, \dots, d_r in \mathbb{P}^{n+r} (with \dim V = n), no simple closed-form expression exists for arbitrary r, but \chi(V) can be computed recursively by viewing V as a hypersurface in the complete intersection of the first r-1 equations, applying the hypersurface formula iteratively along with the Künneth formula for the Betti numbers. Explicit algorithms for this recursion, based on Segre classes and projective degrees, are available and implementable for computational verification, even extending to mildly singular cases via the Chern-Schwartz-MacPherson class. These computations highlight how \chi(V) grows polynomially with the degrees, providing key data for enumerative geometry and mirror symmetry applications.