Projective variety

A projective variety is an irreducible closed subset of projective space \mathbb{P}^n over an algebraically closed field, defined as the zero set of a collection of homogeneous polynomials, equipped with the induced Zariski topology.^[1]^[2]^[3] Projective space \mathbb{P}^n itself consists of lines through the origin in affine (n+1)-space, with points represented by homogeneous coordinates [x_0 : \cdots : x_n], where not all coordinates vanish.^[2] This structure ensures that projective varieties are "compact" in the classical sense when defined over the complex numbers, analogous to compact Riemann surfaces, and they incorporate points at infinity, making them suitable for studying global geometric properties without boundary issues inherent in affine varieties.^[1] The coordinate ring of a projective variety X \subseteq \mathbb{P}^n is the homogeneous quotient ring k[X] = k[x_0, \dots, x_n]/I(X), where I(X) is the homogeneous prime ideal of polynomials vanishing on X, and by the projective Nullstellensatz, this ideal is the radical of the vanishing ideal for non-empty sets.^[2]^[3] Projective varieties can be covered by finitely many affine open sets via the standard affine charts U_i = X \cap D(x_i), where D(x_i) is the principal open set excluding the hyperplane V(x_i), allowing the use of affine techniques while preserving projective properties.^[1] Regular functions on the entire variety are constant, reflecting their rigidity, with the sheaf of regular functions \mathcal{O}_X providing a local description.^[1] In algebraic geometry, projective varieties form the core objects for investigating phenomena like intersection theory, dimension (defined as the transcendence degree of the function field), and smoothness (where the Zariski tangent space dimension equals the variety's dimension at smooth points).^[2] Notable examples include projective curves such as elliptic curves in \mathbb{P}^2, defined by homogeneous cubics, and higher-dimensional objects like hypersurfaces.^[3] Their study traces back to foundational work by figures like Dedekind on ideals (1876), Hilbert's Nullstellensatz, and Noether's developments, evolving into modern tools for enumerative geometry and moduli problems.^[3] Theorems like Bézout's, stating that two plane curves of degrees m and n without common components intersect in mn points (counting multiplicity), underscore their role in counting geometric invariants.^[2]

Definitions and Basic Constructions

Projective Space and Homogenization

Projective space \mathbb{P}^n_k over an algebraically closed field k is defined as the set of all one-dimensional subspaces (lines through the origin) of the vector space k^{n+1}, or equivalently, the quotient space (k^{n+1} \setminus \{0\}) / k^\times, where k^\times acts by scalar multiplication.^[2]^[4] Points in \mathbb{P}^n_k are represented using homogeneous coordinates [x_0 : x_1 : \dots : x_n], where not all x_i = 0, and two tuples (x_0, \dots, x_n) and (y_0, \dots, y_n) represent the same point if there exists \lambda \in k^\times such that x_i = \lambda y_i for all i.^[2] This construction identifies \mathbb{P}^n_k with the space of lines in k^{n+1}, providing a geometric interpretation that generalizes the classical projective plane.^[2] To define projective varieties from affine ones, the process of homogenization extends polynomials from affine space \mathbb{A}^n_k to projective space \mathbb{P}^n_k. For a polynomial f \in k[x_1, \dots, x_n] of degree d, its homogenization F with respect to a new variable X_0 is obtained by writing f = f_d + f_{d-1} + \dots + f_0, where each f_i is the homogeneous component of degree i, and setting F(X_0, x_1, \dots, x_n) = X_0^d f(x_1/X_0, \dots, x_n/X_0) = f_d(x_1, \dots, x_n) + X_0 f_{d-1}(x_1, \dots, x_n) + \dots + X_0^d f_0.^[5] For an affine variety V \subset \mathbb{A}^n_k defined by an ideal I(V) = (f_1, \dots, f_m), the projective closure \overline{V} \subset \mathbb{P}^n_k is the zero set of the ideal generated by the homogenizations F_1, \dots, F_m, ensuring \overline{V} \cap \mathbb{A}^n_k = V.^[5] For example, the affine curve V(y^2 - x^3) \subset \mathbb{A}^2_k homogenizes to V(y^2 z - x^3) \subset \mathbb{P}^2_k, where z is the homogenizing variable.^[2] Homogenization incorporates points at infinity into the affine variety, which are points in \overline{V} where the homogenizing coordinate vanishes (e.g., z = 0 in \mathbb{P}^2_k). These points resolve behaviors at "infinity" in the affine setting, such as asymptotic directions of curves.^[5] For instance, the homogenization of the hyperbola V(xy - 1) \subset \mathbb{A}^2_k adds two points at infinity, [1:0:0] and [0:1:0], closing the curve in \mathbb{P}^2_k.^[6] Topologically, \mathbb{P}^n_k serves as a compactification of \mathbb{A}^n_k, embedding \mathbb{A}^n_k as the dense open set where x_0 \neq 0 (via the chart [1 : x_1 : \dots : x_n]), with the hyperplane at infinity \mathbb{P}^{n-1}_k = V(x_0) compactifying the space in the Zariski topology.^[5] This structure ensures that projective varieties, as closed subsets of \mathbb{P}^n_k, are compact in the classical topology when k = \mathbb{C}, providing a foundation for studying global properties.^[4]

Homogeneous Ideals and Proj Construction

A graded ring S is a commutative ring equipped with a direct sum decomposition S = \bigoplus_{d \geq 0} S_d, where each S_d is an abelian group and the multiplication map satisfies S_m \cdot S_n \subseteq S_{m+n} for all m, n \geq 0.^[7] Elements of S_d are called homogeneous of degree d, and the decomposition allows for a natural \mathbb{Z}_{\geq 0}-grading on the ring. A prototypical example is the polynomial ring S = k[x_0, \dots, x_n] over a field k, graded by total degree, where each S_d consists of homogeneous polynomials of degree d.^[8] A homogeneous ideal I in a graded ring S is an ideal generated by homogeneous elements, equivalently, I = \bigoplus_{d \geq 0} (I \cap S_d).^[9] The irrelevant ideal, denoted S_+, is the homogeneous ideal \bigoplus_{d > 0} S_d, which consists of all elements of positive degree.^[7] For the polynomial ring k[x_0, \dots, x_n], S_+ is the ideal generated by x_0, \dots, x_n. Homogeneous ideals play a central role in defining subschemes, as the quotient S/I inherits a grading whenever I is homogeneous.^[8] The Proj construction associates to a graded ring S (with S_+ \neq S) the space \operatorname{Proj} S, defined as the set of all homogeneous prime ideals of S that do not contain S_+.^[7] This set is equipped with the Zariski topology, whose basic open sets are the standard opens D_+(f) = \{ p \in \operatorname{Proj} S \mid f \notin p \} for homogeneous elements f \in S_d with d \geq 1. Each D_+(f) is homeomorphic to \operatorname{Spec} S_{(f)}, where S_{(f)} denotes the degree-zero part of the localization of S at the multiplicative set generated by f.^[9] To endow \operatorname{Proj} S with a scheme structure, the affine schemes \operatorname{Spec} S_{(f)} on the D_+(f) are glued along their intersections: D_+(f) \cap D_+(g) = D_+(fg) for homogeneous f, g, with the natural localization maps S_{(f)} \to S_{(fg)} and S_{(g)} \to S_{(fg)} ensuring compatibility.^[7] The structure sheaf \mathcal{O}_{\operatorname{Proj} S} is defined by \Gamma(D_+(f), \mathcal{O}_{\operatorname{Proj} S}) = S_{(f)} on basic opens, extended uniquely to all opens, yielding a ringed space that is a scheme.^[9] For twisting, the sheaf \mathcal{O}_{\operatorname{Proj} S}(d) arises from the graded S-module S(d) with components S(d)_i = S_{d+i}, and it satisfies \mathcal{O}_{\operatorname{Proj} S}(m) \otimes \mathcal{O}_{\operatorname{Proj} S}(n) \cong \mathcal{O}_{\operatorname{Proj} S}(m+n); in the case of projective space \mathbb{P}^n = \operatorname{Proj} k[x_0, \dots, x_n], these are the invertible sheaves \mathcal{O}_{\mathbb{P}^n}(d).^[9] Projective schemes are precisely those schemes isomorphic to \operatorname{Proj} S for some graded ring S, and there is an equivalence between closed subschemes of \mathbb{P}^n and quotients of k[x_0, \dots, x_n] by saturated homogeneous ideals, where saturation ensures the ideal is properly defined modulo units in degree zero.^[7] Specifically, a homogeneous ideal I in k[x_0, \dots, x_n] defines a closed immersion \operatorname{Proj} k[x_0, \dots, x_n]/I \hookrightarrow \mathbb{P}^n.^[9]

Relation to Affine Varieties

Projective space \mathbb{P}^n over an algebraically closed field k admits a standard affine open cover consisting of n+1 sets U_i = \{ [x_0 : \cdots : x_n] \mid x_i \neq 0 \} for i = 0, \dots, n, where each U_i is isomorphic to affine n-space \mathbb{A}^n via the map sending [x_0 : \cdots : x_n] to (x_0/x_i, \dots, \hat{x_i}/x_i, \dots, x_n/x_i).^[10] This cover demonstrates that projective space is locally affine, allowing the study of projective varieties through their intersections with these affine opens.^[11] Dehomogenization provides a explicit isomorphism between a projective variety X \subset \mathbb{P}^n intersected with U_i and an affine variety in \mathbb{A}^n. For a point in X \cap U_i, setting the i-th homogeneous coordinate to 1 yields affine coordinates, and the defining homogeneous equations of X restrict to polynomial equations on this affine chart after dehomogenization.^[11] Conversely, any affine variety embeds into projective space via homogenization of its ideal, yielding its projective closure.^[12] Projective varieties serve as compactifications of affine varieties by adjoining a "hyperplane at infinity." Specifically, for the affine open U_0 \cong \mathbb{A}^n in \mathbb{P}^n, the complement is the hyperplane H_\infty = \{x_0 = 0\} \cong \mathbb{P}^{n-1}, which adds points at infinity to "complete" the affine space in the Zariski topology.^[12] The projective closure \tilde{Y} of an affine variety Y \subset \mathbb{A}^n is thus obtained by homogenizing the equations of Y and taking the zero set in \mathbb{P}^n, with Y = \tilde{Y} \cap U_0 and the points at infinity forming \tilde{Y} \cap H_\infty.^[11] This construction ensures that \tilde{Y} is proper (compact in the classical sense over \mathbb{C}), unlike the non-compact affine Y.^[12] Quasi-projective varieties are defined as open subsets of projective varieties, bridging the gap between affine and projective geometry.^[13] For instance, every affine variety is quasi-projective, as it arises as the intersection of its projective closure with an affine open in projective space.^[13] This class includes all varieties that can be embedded locally into projective space while retaining affine-like behavior on opens.^[10]

Fundamental Properties

Completeness and Projective Morphisms

In algebraic geometry, a variety X over a field k is defined to be complete if, for every variety Y over k, the projection morphism p_Y: X \times_k Y \to Y is a closed map.^[14] This property ensures that the image of any morphism from a complete variety X to another variety Y is closed in Y.^[14] Equivalently, over an algebraically closed field, completeness implies that regular functions on a connected complete variety are constant.^[14] Properness generalizes completeness to morphisms. A morphism f: X \to Y of varieties over a field k is proper if it is of finite type, separated, and universally closed, meaning that for any base change Y' \to Y, the induced morphism X' \to Y' (where X' = X \times_Y Y') has closed image.^[15] An equivalent characterization, known as the valuative criterion of properness, states that for Noetherian schemes (or varieties), f is proper if and only if, for every discrete valuation ring R with fraction field K and any commutative diagram involving a morphism \operatorname{Spec} K \to X and \operatorname{Spec} R \to Y, there exists a unique lift \operatorname{Spec} R \to X making the diagram commute.^[15] This criterion captures the "rigidity" of proper morphisms under extensions of valuation rings. Over an algebraically closed field k, every projective variety is complete.^[14] Specifically, if X \subset \mathbb{P}^n_k is a closed subvariety, then the projection \mathbb{P}^n_k \times Y \to Y is closed for any Y, implying that X inherits this property as a closed subscheme.^[14] A morphism f: X \to Y is projective if X is a closed subscheme of a projective space bundle over Y, or equivalently, if there exists an ample line bundle on X such that f factors through the associated projective bundle.^[14] Projective morphisms are proper: any such f is of finite type, separated, and universally closed, as the projection from projective space satisfies these conditions.^[15] This properness yields a universal property for morphisms into projective varieties. If Z is a proper variety over an algebraically closed field k and f: Z \to X is a morphism to a projective variety X \subset \mathbb{P}^n_k, then the image f(Z) is closed in X, and f factors uniquely through the closed embedding of f(Z) into X.^[14] Properness can be verified locally using affine covers of the target variety.^[16]

Homogeneous Coordinate Ring

For a projective variety X embedded in the projective space \mathbb{P}^n_k over an algebraically closed field k, the homogeneous coordinate ring is the graded k-algebra S(X) = \bigoplus_{d \geq 0} H^0(X, \mathcal{O}_X(d)), where \mathcal{O}_X(d) denotes the d-th power of the Serre twisting sheaf associated to the embedding, and H^0(X, \mathcal{O}_X(d)) is the k-vector space of global sections of this sheaf.^[17] This ring encodes the algebraic structure of X compatibly with its projective embedding, as the Proj construction applied to S(X) recovers X by quotienting out the irrelevant ideal generated by the degree-1 elements.^[17] The ideal defining X in the polynomial ring k[x_0, \dots, x_n] is a saturated homogeneous ideal, meaning it equals its saturation with respect to the maximal irrelevant ideal (x_0, \dots, x_n), which ensures that the associated sheaf \tilde{I}(X) on \mathbb{P}^n is the ideal sheaf of X.^[17] Consequently, S(X) is isomorphic to k[x_0, \dots, x_n]/I(X), where I(X) is this saturated ideal. The ring S(X) is particularly relevant when it is generated as a k-algebra by its degree-1 component, a condition equivalent to X being projectively normal in the embedding; in this case, higher-degree sections are generated by linear forms, facilitating computations of invariants.^[17] The Hilbert function of X is defined as h_X(d) = \dim_k S(X)_d, measuring the dimension of the degree-d graded piece of the homogeneous coordinate ring. For small d, h_X(d) grows combinatorially, reflecting the initial constraints imposed by the embedding, but for sufficiently large d, it stabilizes to a polynomial behavior that captures asymptotic growth related to the geometry of X.^[17] This ring relates to the ideal sheaf \tilde{I}(X) via the graded module structure, where the minimal number of generators of I(X) as a homogeneous ideal determines the embedding codimension of X in \mathbb{P}^n, providing a measure of how the variety sits in the ambient space.^[17]

Hilbert Polynomial and Degree

The Hilbert function h_X(d) of a projective variety X \subset \mathbb{P}^n over an algebraically closed field is defined as the dimension of the degree-d component of its homogeneous coordinate ring S(X) = k[x_0, \dots, x_n]/I(X), where I(X) is the homogeneous ideal of X.^[18] This function measures the growth of sections of the line bundle \mathcal{O}_X(d). For large d, h_X(d) stabilizes and equals a polynomial P_X(t) \in \mathbb{Q} of degree equal to \dim X, known as the Hilbert polynomial of X.^[18] The leading coefficient of P_X(t) encodes key geometric invariants, specifically with leading term \frac{\deg X}{(\dim X)!} t^{\dim X}.^[18] The degree \deg X of an r-dimensional projective variety X is the integer such that the leading coefficient of P_X(t) is \frac{\deg X}{r!}. Geometrically, this is the intersection multiplicity of X with a general linear subspace of complementary dimension n - r, or the number of points in a general such intersection counted with multiplicity.^[18] This definition aligns the algebraic Hilbert polynomial with classical intersection theory on projective space.^[19] For curves (\dim X = 1), the Hilbert polynomial takes the form P_X(t) = (\deg X) t + 1 - p_a(X), where p_a(X) is the arithmetic genus, an invariant reflecting the topology of X via p_a(X) = 1 - \chi(\mathcal{O}_X).^[18] The arithmetic genus is non-negative and vanishes if and only if X \cong \mathbb{P}^1.^[18] Representative examples illustrate these concepts. The Veronese surface, the image of the Veronese embedding \nu_2: \mathbb{P}^2 \to \mathbb{P}^5, is a surface of degree 4, so its Hilbert polynomial has leading term \frac{4}{2!} t^2 = 2 t^2.^[18] Likewise, the Grassmannian \mathrm{Gr}(2,5), parametrizing 2-planes in \mathbb{C}^5 and embedded via the Plücker map into \mathbb{P}^9, is 6-dimensional with degree 5, yielding leading term \frac{5}{6!} t^6.^[20]

Examples and Invariants

Projective Curves

Projective curves are one-dimensional projective varieties, typically realized as closed subschemes of projective space \mathbb{P}^n defined by homogeneous equations. These objects provide foundational examples in algebraic geometry, bridging affine curves to compact settings through homogenization and exhibiting rich invariants like genus and degree. Unlike higher-dimensional varieties, curves admit complete classification by genus, with smoothness ensuring they are Riemann surfaces over the complex numbers. The simplest projective curve is the rational curve, exemplified by the projective line \mathbb{P}^1, which has genus 0 and serves as the model for all smooth projective curves of genus 0. \mathbb{P}^1 embeds into \mathbb{P}^2 as a line, a degree 1 curve, and any birational map from \mathbb{P}^1 to a plane curve of degree d parametrizes rational curves of that degree.^[21] Rational curves are characterized by their parametrization by rational functions, reflecting the function field isomorphic to k(t) for a field k.^[22] Elliptic curves represent the next level of complexity, defined as smooth projective curves of genus 1 equipped with a base point, though the embedding often omits explicit reference to the point. A standard embedding into \mathbb{P}^2 arises from homogenizing the affine Weierstrass form y^2 = x^3 + ax + b (with discriminant nonzero to ensure smoothness), yielding the projective equation Y^2 Z = X^3 + a X Z^2 + b Z^3. This cubic model has degree 3 and genus 1, capturing the curve's complete structure in projective space.^[23] For smooth plane projective curves embedded in \mathbb{P}^2, the genus g relates directly to the degree d via the formula

g = \frac{(d-1)(d-2)}{2}.

This relation, classical in algebraic geometry, quantifies how higher-degree embeddings increase topological complexity, with d=1 or d=2 yielding genus 0 (lines and conics, isomorphic to \mathbb{P}^1) and d=3 giving genus 1 (elliptic curves). The degree here aligns with the Hilbert polynomial's leading coefficient, aiding genus computations through intersection theory.^[24] Singular projective curves, defined by homogeneous ideals where the scheme has non-reduced structure or self-intersections, require normalization to recover a smooth model. The normalization \tilde{C} \to C is the unique birational morphism from a smooth projective curve \tilde{C} (the integral closure of the coordinate ring) that resolves singularities, such as nodes or cusps, by separating branches at singular points. For instance, both a nodal cubic and a cuspidal cubic in \mathbb{P}^2 normalize to \mathbb{P}^1 (genus 0), preserving the arithmetic genus but adjusting the geometric genus to reflect the smooth topology. This process is always possible for curves, as singularities lie in codimension 1.^[25]

Projective Hypersurfaces

A projective hypersurface in \mathbb{P}^n over an algebraically closed field k is defined as the zero locus V(f) = \{ [x_0 : \cdots : x_n] \in \mathbb{P}^n \mid f(x_0, \dots, x_n) = 0 \}, where f \in k[x_0, \dots, x_n] is a homogeneous polynomial of positive degree.$$] If f is irreducible, then V(f) is an irreducible projective variety of dimension n-1 and degree \deg(f). These arise naturally from affine hypersurfaces via homogenization, where an affine equation g(y_1, \dots, y_n) = 0 in \mathbb{A}^n is extended to the projective closure by setting f(x_1, \dots, x_n, x_0) = x_0^{\deg(g)} g(x_1/x_0, \dots, x_n/x_0). Singularities on a projective hypersurface V(f) occur at points where f and all its partial derivatives \partial f / \partial x_i vanish simultaneously. For an isolated hypersurface singularity at the origin in the local affine model, the Milnor number \mu(f) quantifies the topological complexity of the singularity and is given by \mu(f) = \dim_k \mathcal{O}_{n,0} / (f, \partial f / \partial x_1, \dots, \partial f / \partial x_n), where \mathcal{O}_{n,0} is the local ring of the germ at the origin in \mathbb{C}^n; this invariant is constant under small deformations and equals the rank of the middle homology of the Milnor fiber.[

The Tyurina algebra, associated to the versal deformation of the singularity, is the quotient $\mathcal{O}_{n,0} / T(f)$, where the Tyurina ideal $T(f)$ is generated by $f$ and the partial derivatives together with relations from the [embedding](/page/Embedding); its dimension, the Tjurina number $\tau(f)$, satisfies $\tau(f) \leq \mu(f)$, with equality holding for quasi-homogeneous singularities.

] These numbers classify the local equisingularity type and bound the dimension of the moduli space of deformations. Resolution of singularities for projective hypersurfaces over fields of characteristic zero can be achieved through a finite sequence of blow-ups along nonsingular subvarieties, yielding a smooth proper birational model \tilde{V} \to V(f) with exceptional divisors that are themselves projective hypersurfaces.[ Hironaka's theorem guarantees such a resolution exists for any algebraic variety, but for hypersurfaces, the process often simplifies due to codimension one, with blow-ups centered at the singular locus iteratively reducing multiplicity until smoothness is attained; for example, blowing up the maximal ideal at a point singularity replaces it with a projective space bundle over the exceptional divisor.] The Fano scheme F_1(V(f)) of a projective hypersurface parametrizes the 1-dimensional linear subspaces (lines) contained in V(f), constructed as the zero locus of a universal section of a vector bundle on the Grassmannian \mathrm{Gr}(2, n+1).[

For a [smooth](/page/Smooth) cubic [hypersurface](/page/Hypersurface) in $\mathbb{P}^4$, this Fano surface is a [smooth](/page/Smooth) projective surface of geometric [genus](/page/Genus) 5 whose [geometry](/page/Geometry) encodes [rationality](/page/Rationality) properties of the threefold, as studied via its relation to the intermediate Jacobian.

] ^[26] In higher dimensions, such as cubics in \mathbb{P}^5, the Fano scheme of lines is a fourfold that aids in understanding birational invariants and period domains.

Abelian Varieties

An abelian variety over a field k is defined as a nonsingular projective algebraic variety A that is also a commutative algebraic group, meaning the group operations of addition and inversion are given by morphisms of varieties.^[27] This structure ensures that A is complete and geometrically connected, inheriting projectivity from its proper morphism properties as a group scheme.^[27] The commutativity follows from the rigidity of group laws on projective varieties, making the addition map symmetric.^[28] A key feature of abelian varieties is the notion of polarization, which provides a positivity structure compatible with the group law. A polarization on an abelian variety A of dimension g is an ample line bundle L on A, up to translation by points of A, that induces a homomorphism \lambda_L: A \to \hat{A} to the dual abelian variety \hat{A} = \Pic^0(A), where the kernel is finite and the induced map on the N-torsion is an isogeny of degree N^{2g}.^[27] A principal polarization occurs when this homomorphism is an isomorphism, corresponding to a degree-1 ample line bundle; in such cases, the zero section of L defines an effective ample divisor known as the theta divisor \Theta, which embeds A into projective space via the complete linear system |n\Theta| for sufficiently large n.^[27] For example, on a principally polarized abelian variety, the theta divisor \Theta is ample and its associated line bundle L(\Theta) satisfies \chi(L(\Theta)) = 1, highlighting the principal nature.^[27] A fundamental example of an abelian variety is the Jacobian of a smooth projective curve C of genus g \geq 1 over k. The Jacobian J(C) is the moduli space parametrizing degree-zero line bundles on C, realized as a projective variety of dimension g with a natural principal polarization induced by the theta divisor, which corresponds to the embedding of effective divisors of degree g-1 on C.^[27] This construction shows that every abelian variety over an infinite field admits a surjective homomorphism from some Jacobian, underscoring their role in the geometry of curves.^[27] Abelian varieties have dimension g \geq 1, where the tangent space at the identity is a vector space of that dimension, and the group law is analytic in local coordinates. The endomorphism ring \End(A) consists of all morphisms from A to itself as a group scheme, forming a ring that acts faithfully on the tangent space at the identity; over algebraically closed fields of characteristic zero, \End(A) \otimes \mathbb{Q} is a semisimple \mathbb{Q}-algebra of finite rank, often commutative for simple abelian varieties.^[27] For instance, when A is an elliptic curve (dimension 1), \End(A) is either \mathbb{Z} or an order in a quadratic imaginary field.^[27]

Morphisms and Embeddings

Projections and Veronese Embeddings

Projections from a point or a linear subspace provide essential rational maps in the study of projective varieties, allowing reduction of embedding dimensions while preserving key geometric properties. Consider a projective variety X \subset \mathbb{P}^n over an algebraically closed field and a linear subspace C \subset \mathbb{P}^n disjoint from X. The projection \pi_C: \mathbb{P}^n \dashrightarrow \mathbb{P}^{n - \dim C - 1} with center C is a rational map defined by sending a point p \in \mathbb{P}^n \setminus C to the intersection of the line joining p to a generic point in C with a complementary linear subspace. Restricting to X, this induces a rational map \pi_C|_X: X \dashrightarrow Y \subset \mathbb{P}^{n - \dim C - 1}, where Y is the closure of the image, provided the center avoids the tangent spaces to X.^[29] Such projections are undefined along the cone over X with vertex C, but resolve to morphisms after blowing up the center.^[29] The Veronese embedding offers a canonical method to embed \mathbb{P}^n into a higher-dimensional projective space using homogeneous polynomials of fixed degree. For d \geq 1, the d-th Veronese map v_d: \mathbb{P}^n \to \mathbb{P}^N, where N = \binom{n+d}{d} - 1, sends [x_0 : \cdots : x_n] to the point whose coordinates are all monomials of degree d in the x_i, up to scalar. This morphism is defined by the complete linear system | \mathcal{O}_{\mathbb{P}^n}(d) |, and the image v_d(\mathbb{P}^n) is a projective variety of dimension n and degree d^n. The map v_d is an embedding, hence very ample, meaning \mathcal{O}_{\mathbb{P}^n}(d) generates the embedding and separates points and tangent vectors. More generally, for a projective variety X \subset \mathbb{P}^n, the restriction of v_d to X yields an embedding if the restriction of \mathcal{O}_{\mathbb{P}^n}(d) to X is very ample.^[30] A line bundle L on a projective variety X is very ample if the associated morphism \phi_{|L|}: X \to \mathbb{P}^H to projective space, given by a basis of global sections H^0(X, L), is an embedding. While the Nakai-Moishezon criterion characterizes ampleness—a prerequisite for very ampleness—via intersection numbers, stating that L is ample if and only if L^{\dim V} \cdot V > 0 for every irreducible subvariety V \subset X, very ampleness requires additional separation conditions. Briefly, L is very ample if it is ample and the map separates points and tangents, with the criterion providing a numerical test for the underlying positivity.^[31] Birational projections, particularly generic linear projections from a point or subspace outside the variety, play a key role in studying invariants under birational equivalence. For a non-degenerate projective variety X \subset \mathbb{P}^r of dimension n and codimension c \geq 2, a generic projection from a point not on X or its tangent spaces induces a birational map onto its image in \mathbb{P}^{r-1}, provided c \geq 2. Such projections preserve the birational type of X, and consequently, the degree of X—defined as the intersection multiplicity with a general linear subspace of complementary dimension—remains unchanged, as birational morphisms between smooth projective varieties of the same dimension are degree-preserving isomorphisms in codimension 1.^[32] This invariance facilitates computations of degrees in lower embeddings without altering intrinsic properties.^[32]

Linear Systems and Dual Varieties

In algebraic geometry, the complete linear system associated to a Cartier divisor D on a projective variety X is the projective space \mathbb{P} H^0(X, \mathcal{O}_X(D)), parametrizing the effective divisors linearly equivalent to D.^[33] This space consists of all global sections of the line bundle \mathcal{O}_X(D), up to scalar multiple, and its dimension is h^0(X, \mathcal{O}_X(D)) - 1.^[33] If the linear system is basepoint-free, it defines a morphism \phi_{|D|}: X \to \mathbb{P}^H, where H = h^0(X, \mathcal{O}_X(D)) - 1, embedding X into projective space via the evaluation map that sends a point x \in X to the hyperplane of sections vanishing at x. By the theorem on very ample line bundles, if \mathcal{O}_X(D) is very ample, this morphism is an embedding. The dual variety of a projective variety X \subset \mathbb{P}^N is defined as the closure in the dual projective space (\mathbb{P}^N)^\vee of the set of all hyperplanes tangent to X at some smooth point.^[34] A hyperplane H \in (\mathbb{P}^N)^\vee is tangent to X at x \in X_{\text{smooth}} if it contains the embedded tangent space T_x X.^[34] For a smooth irreducible X, the dual variety X^\vee is typically a hypersurface, but its codimension in (\mathbb{P}^N)^\vee may exceed 1, leading to the notion of defect. The defect of X, denoted \delta(X), is given by \delta(X) = \dim(\mathbb{P}^N)^\vee - \dim X^\vee - 1.^[34] Varieties with positive defect are ruled by linear spaces of dimension \delta(X), and \delta(X) > 0 implies that the canonical bundle K_X is not nef.^[34] For smooth projective varieties, the biduality theorem asserts that the dual of the dual variety recovers the original: X^{\vee\vee} = X.^[34] This reflexivity holds because the Gauss map \gamma: X \to \mathrm{Gr}(\dim X + 1, N + 1), sending each smooth point to its tangent space, is finite and birational onto its image when X is smooth, ensuring the incidence correspondence between points and tangent hyperplanes is proper.^[34] If both X and X^\vee are smooth, then \dim X = \dim X^\vee.^[34] Examples of varieties achieving biduality with positive defect include quadrics and certain Segre embeddings, such as \mathbb{P}^n \times \mathbb{P}^1.^[34] Bertini's theorem provides a genericity result for linear systems on projective varieties. For a basepoint-free linear system |D| on a smooth projective variety X over an algebraically closed field of characteristic zero, there exists a Zariski-open dense subset U \subset |D| such that every effective divisor E \in U is smooth of codimension 1 in X.^[35] More generally, the theorem applies to ample line bundles, ensuring that general hyperplane sections are smooth and connected, away from the base locus.^[35] This result relies on the properness of the morphism induced by |D| and the semicontinuity of the dimension of singular loci in families. In positive characteristic, additional hypotheses like reducedness may be needed to avoid pathologies.^[35]

Coherent Sheaves and Cohomology

Structure of Coherent Sheaves

A coherent sheaf on a projective variety X is a sheaf \mathcal{F} of \mathcal{O}_X-modules that is locally of finite presentation, meaning that on every affine open subset U = \operatorname{Spec}(A) of X, the restriction \mathcal{F}|_U is a finitely presented A-module.^[36] Since projective varieties are Noetherian schemes, coherent sheaves arise as the sheafification of finitely generated graded modules over the homogeneous coordinate ring of X, and they can be either locally free (vector bundles) or torsion (supported on proper subvarieties).^[36] This structure ensures that coherent sheaves capture the essential algebraic data of subschemes and bundles on X, forming the category \operatorname{Coh}(X) which is abelian.^[36] The Hilbert syzygy theorem provides a key resolution property for coherent sheaves on projective space \mathbb{P}^n. It states that every coherent sheaf \mathcal{F} on \mathbb{P}^n admits a finite resolution by locally free sheaves of length at most n+1, where n is the dimension of \mathbb{P}^n.^[37] More precisely, associating to \mathcal{F} the graded module E = \bigoplus_m H^0(\mathbb{P}^n, \mathcal{F}(m)) over the polynomial ring S = k[z_0, \dots, z_n], the projective dimension of E is bounded by n+1, reflecting the global generation of sheaves on projective varieties embedded in \mathbb{P}^n.^[37] This bound extends to coherent sheaves on arbitrary projective varieties via embeddings, limiting the complexity of syzygies in their minimal free resolutions.^[38] For vector bundles on \mathbb{P}^n, the Beilinson monad offers an explicit resolution using exterior powers of the tautological bundle. Specifically, any coherent sheaf \mathcal{F} on \mathbb{P}^n can be resolved by a monad of the form [ 0 \to \bigoplus_i \Omega^j(n_j) \to \bigoplus_i \mathcal{O}{\mathbb{P}^n}(m_i) \to \bigoplus_i \mathcal{O}{\mathbb{P}^n}(l_i) \to \mathcal{F} \to 0,

where $\Omega^j$ denotes the $j$-th exterior power of the [cotangent bundle](/page/Cotangent_bundle), and the shifts are determined by the [cohomology](/page/Cohomology) of $\mathcal{F}$.[](https://link.springer.com/article/10.1007/BF01681436) This construction, derived from the bounded [derived category](/page/Derived_category) of coherent sheaves, provides a uniform way to describe indecomposable vector bundles and facilitates computations of Ext groups.[](https://link.springer.com/article/10.1007/BF01681436) Stability notions refine the structure of coherent sheaves, enabling the study of moduli spaces. Slope stability, introduced for vector bundles, defines a torsion-free sheaf $\mathcal{E}$ on a polarized projective variety $(X, \mathcal{O}_X(1))$ as $\mu$-stable if for every proper subsheaf $\mathcal{F} \subset \mathcal{E}$, the slope $\mu(\mathcal{F}) < \mu(\mathcal{E})$, where $\mu(\mathcal{F}) = \frac{c_1(\mathcal{F}) \cdot H}{\operatorname{rk}(\mathcal{F})}$ with $H$ the hyperplane class.[](https://www.dam.brown.edu/people/mumford/alg_geom/papers/1977a--StabilityLecturesIHES-Swiss.pdf) Gieseker stability generalizes this to a Hilbert polynomial comparison: a sheaf $\mathcal{E}$ is Gieseker-stable if for every proper subsheaf $\mathcal{F} \subset \mathcal{E}$, the normalized Hilbert polynomial $P(\mathcal{F}, m)/\operatorname{rk}(\mathcal{F}) < P(\mathcal{E}, m)/\operatorname{rk}(\mathcal{E})$ for large $m$, prioritizing higher-degree terms over slope alone.[](https://projecteuclid.org/journals/journal-of-differential-geometry/volume-27/issue-1/A-construction-of-stable-bundles-on-an-algebraic-surface/10.4310/jdg/1214441654.pdf) These conditions ensure boundedness and the existence of projective moduli spaces for semistable sheaves on smooth projective varieties.[](https://projecteuclid.org/journals/journal-of-differential-geometry/volume-27/issue-1/A-construction-of-stable-bundles-on-an-algebraic-surface/10.4310/jdg/1214441654.pdf) ### Cohomology Groups on Projective Varieties Sheaf cohomology provides a fundamental tool for studying global properties of coherent sheaves on projective varieties, extending the notion of cohomology from topology to algebraic geometry. On a projective variety $X$, such as the projective space $\mathbb{P}^n$, sheaf cohomology groups $H^i(X, \mathcal{F})$ for a coherent sheaf $\mathcal{F}$ can be computed using Čech cohomology with respect to a suitable open cover. Specifically, Čech cohomology is defined via an open cover $\mathcal{U} = \{U_\alpha\}$ of $X$, where the Čech complex is formed by taking alternating sums of sections over intersections: the 0-th term is $\prod_\alpha \mathcal{F}(U_\alpha)$, the 1-st term is $\prod_{\alpha < \beta} \mathcal{F}(U_{\alpha\beta})$, and higher terms similarly, with the cohomology of this complex yielding the Čech groups $\check{H}^i(\mathcal{U}, \mathcal{F})$. For projective space $\mathbb{P}^n$ over an algebraically closed field $k$, the standard affine cover $\mathcal{U} = \{D(x_i)\}_{i=0}^n$, where $D(x_i)$ are the principal open sets defined by the homogeneous coordinates $x_i$, has the property that all intersections $U_{i_0 \dots i_j} = D(x_{i_0}) \cap \cdots \cap D(x_{i_j})$ are affine schemes. Since higher cohomology vanishes on affine schemes (i.e., $H^j(U, \mathcal{G}) = 0$ for $j > 0$ and quasi-coherent $\mathcal{G}$), the Leray theorem ensures that $\check{H}^i(\mathcal{U}, \mathcal{F}) \cong H^i(\mathbb{P}^n, \mathcal{F})$ for coherent $\mathcal{F}$, allowing explicit computations via the finite-dimensional Čech complex.[](https://stacks.math.columbia.edu/tag/01XS) A key result is the finite-dimensionality of these cohomology groups on projective varieties. For $\mathbb{P}^n$ over a [field](/page/Field) $k$ and [coherent sheaf](/page/Coherent_sheaf) $\mathcal{F}$, each $H^i(\mathbb{P}^n, \mathcal{F})$ is a finite-dimensional $k$-[vector space](/page/Vector_space), and moreover, $H^i(\mathbb{P}^n, \mathcal{F}) = 0$ for all $i > n$. This follows from the structure of the Čech complex for the standard affine cover, which has length $n+1$ (yielding potential non-zero cohomology up to degree $n$), combined with the vanishing of higher [cohomology](/page/Cohomology) on the affine intersections; the argument proceeds by [induction](/page/Induction) on the [dimension](/page/Dimension) $n$, reducing to lower-dimensional projective spaces via exact sequences or spectral sequences associated to the cover. More generally, on any projective variety $X$ of [dimension](/page/Dimension) $d$ over $k$, Serre's theorem asserts that $H^i(X, \mathcal{F})$ is finite-dimensional for coherent $\mathcal{F}$ and vanishes for $i > d$, establishing the cohomological [dimension](/page/Dimension) bounded by the geometry of $X$. These properties distinguish projective varieties from affine ones, where higher [cohomology](/page/Cohomology) always vanishes, and enable inductive computations across [algebraic geometry](/page/Algebraic_geometry). Explicit computations of cohomology groups are available for important classes of sheaves on $\mathbb{P}^n$, such as the twisted differentials $\Omega^p(k) = \Omega^p_{\mathbb{P}^n} \otimes \mathcal{O}_{\mathbb{P}^n}(k)$, where $\Omega^p_{\mathbb{P}^n}$ is the sheaf of holomorphic (or algebraic) $p$-forms. The Bott formula provides the precise dimensions: for $k = 0$, $\dim H^q(\mathbb{P}^n, \Omega^p) = 1$ if $p = q$ and 0 otherwise; for $k > 0$, $H^q(\mathbb{P}^n, \Omega^p(k)) = 0$ unless $q = 0$ and $k > p$, in which case $\dim H^0(\mathbb{P}^n, \Omega^p(k)) = \binom{n + k - p}{k} \binom{k - 1}{p}$; for $k < 0$, $H^q(\mathbb{P}^n, \Omega^p(k)) = 0$ unless $q = n$ and $k < p - n$, in which case $\dim H^n(\mathbb{P}^n, \Omega^p(k)) = \binom{-k + p - 1}{-k} \binom{-k - 1}{n - p}$. This formula, derived using the Euler sequence and induction on $p$, highlights strong vanishing phenomena, such as $H^q(\mathbb{P}^n, \Omega^p(k)) = 0$ for $q > 0$ and $k \gg 0$, and is crucial for studying deformations and obstructions in moduli problems.[](https://arxiv.org/pdf/math/9904110) The long exact sequence in cohomology arises from short exact sequences of sheaves and is instrumental for relating cohomology groups across extensions. Given a short exact sequence $0 \to \mathcal{A} \to \mathcal{B} \to \mathcal{C} \to 0$ of coherent sheaves on $\mathbb{P}^n$, the associated long exact sequence is

\cdots \to H^{i-1}(\mathbb{P}^n, \mathcal{C}) \to H^i(\mathbb{P}^n, \mathcal{A}) \to H^i(\mathbb{P}^n, \mathcal{B}) \to H^i(\mathbb{P}^n, \mathcal{C}) \to H^{i+1}(\mathbb{P}^n, \mathcal{A}) \to \cdots,

which preserves exactness and finiteness. This sequence applies to study extensions of sheaves: the extensions of $\mathcal{C}$ by $\mathcal{A}$ (i.e., short exact sequences with middle term varying) are classified up to isomorphism by the cohomology group $H^1(\mathbb{P}^n, \mathcal{E}xt^0(\mathcal{C}, \mathcal{A}))$, where $\mathcal{E}xt^0(\mathcal{C}, \mathcal{A}) = \underline{\Hom}(\mathcal{C}, \mathcal{A})$ is the sheaf Hom; for locally free sheaves (vector bundles), this simplifies to $H^1(\mathbb{P}^n, \mathcal{C}^\vee \otimes \mathcal{A})$. Such applications allow inductive determination of cohomology for more complex sheaves from simpler ones, as in resolving a coherent sheaf via a finite free resolution and using the sequence repeatedly. ### Ring of Global Sections The ring of global sections $ H^0(X, \mathcal{O}_X(d)) $ for a projective variety $ X \subset \mathbb{P}^N $ over [an algebraically closed field](/page/Algebraically_closed_field) forms the degree-$ d $ component of the homogeneous coordinate [ring](/page/Ring) $ S(X) = \bigoplus_{d=0}^\infty H^0(X, \mathcal{O}_X(d)) $, which encodes the algebraic structure of $ X $ as a subvariety.[](https://www.college-de-france.fr/media/jean-pierre-serre/UPL5435398796951750634_Serre_FAC.pdf) This [graded ring](/page/Graded_ring) is finitely generated as [an algebra](/page/Algebra) over the [polynomial ring](/page/Polynomial_ring) $ k[x_0, \dots, x_N] $, with the irrelevant [ideal](/page/Ideal) defining $ X $ as its Proj.[](https://www.college-de-france.fr/media/jean-pierre-serre/UPL5435398796951750634_Serre_FAC.pdf) When $ X $ is embedded via a very ample line bundle $ \mathcal{L} $, identified with $ \mathcal{O}_X(1) $, the global sections $ H^0(X, \mathcal{L}^{\otimes d}) $ generate the [symmetric algebra](/page/Symmetric_algebra) $ \mathrm{Sym}(H^0(X, \mathcal{L})) $ in degree $ d $, providing the relations that cut out $ X $ in the [projective space](/page/Projective_space) $ \mathbb{P}(H^0(X, \mathcal{L})^*) $.[](https://www.college-de-france.fr/media/jean-pierre-serre/UPL5435398796951750634_Serre_FAC.pdf) For [normal](/page/Normal) projective varieties, Serre's [theorem](/page/Theorem) asserts that there exists an integer $ d_0 $ such that for all $ d \geq d_0 $, the sheaf $ \mathcal{O}_X(d) $ is generated by its global sections $ H^0(X, \mathcal{O}_X(d)) $ at every point, meaning the evaluation map $ \mathcal{O}_X \otimes H^0(X, \mathcal{O}_X(d)) \to \mathcal{O}_X(d) $ is surjective.[](https://www.college-de-france.fr/media/jean-pierre-serre/UPL5435398796951750634_Serre_FAC.pdf) Moreover, the higher [cohomology](/page/Cohomology) groups vanish: $ H^i(X, \mathcal{O}_X(d)) = 0 $ for all $ i > 0 $ and $ d \gg 0 $, ensuring that the global sections fully capture the sheaf without cohomological obstructions.[](https://www.college-de-france.fr/media/jean-pierre-serre/UPL5435398796951750634_Serre_FAC.pdf) This vanishing of higher cohomology for large twists parallels later analytic results like Kodaira's vanishing theorem in the [complex](/page/Complex) case. The generation property implies that for sufficiently large $ d $, the complete [linear system](/page/Linear_system) $ |\mathcal{O}_X(d)| $ is very ample, yielding an [embedding](/page/Embedding) of $ X $ into [projective space](/page/Projective_space) that realizes $ X $ as a closed subvariety defined by the relations among the sections.[](https://www.college-de-france.fr/media/jean-pierre-serre/UPL5435398796951750634_Serre_FAC.pdf) In the context of singularities, the [normalization](/page/Normalization) $ \tilde{X} $ of a projective variety $ X $ is itself projective, as the normalization morphism is finite, preserving projectivity; the global sections of $ \tilde{\mathcal{O}}_{\tilde{X}}(d) $, pulled back from ample bundles on $ X $, generate an [embedding](/page/Embedding) of the resolved variety, allowing singularities to be addressed while maintaining the projective structure.[](https://stacks.math.columbia.edu/tag/0GK4) ## Advanced Theorems for Smooth Varieties ### Serre Duality Serre duality establishes a profound connection between the cohomology groups of coherent sheaves on a [smooth](/page/Smooth) projective variety and those of their duals twisted by the canonical sheaf. This theorem, originally proved by [Jean-Pierre Serre](/page/Jean-Pierre_Serre) in 1955, applies to a [smooth](/page/Smooth) projective variety $X$ of dimension $n$ over an [algebraically closed field](/page/Algebraically_closed_field) $k$, and a [coherent sheaf](/page/Coherent_sheaf) $\mathcal{F}$ on $X$. The duality asserts that there is a natural [isomorphism](/page/Isomorphism) of [vector space](/page/Vector_space)s

H^i(X, \mathcal{F})^\vee \cong H^{n-i}(X, \mathcal{H}om(\mathcal{F}, \mathcal{O}_X) \otimes \omega_X),

where the superscript $\vee$ on the left denotes the $k$-dual vector space, $\mathcal{H}om(\mathcal{F}, \mathcal{O}_X)$ is the sheaf Hom to the structure sheaf, and $\omega_X$ is the canonical sheaf of $X$.[](https://www.college-de-france.fr/media/jean-pierre-serre/UPL5435398796951750634_Serre_FAC.pdf) This isomorphism is functorial in $\mathcal{F}$ and compatible with the cup-product structure on [cohomology](/page/Cohomology).[](https://www.college-de-france.fr/media/jean-pierre-serre/UPL5435398796951750634_Serre_FAC.pdf) The canonical sheaf $\omega_X$ is defined as the [line bundle](/page/Line_bundle) $\det \Omega_X^1 = \bigwedge^n \Omega_X^1$, where $\Omega_X^1$ is the cotangent sheaf of $X$; locally, it is generated by a nowhere-vanishing $n$-form serving as a [volume element](/page/Volume_element).[](https://www.college-de-france.fr/media/jean-pierre-serre/UPL5435398796951750634_Serre_FAC.pdf) A sketch of the proof relies on [Čech cohomology](/page/Čech_cohomology) computed with respect to an open affine cover of $X$. The key construction involves a non-degenerate bilinear [pairing](/page/Pairing) between $H^i(X, \mathcal{F})$ and $H^{n-i}(X, \mathcal{H}om(\mathcal{F}, \mathcal{O}_X) \otimes \omega_X)$, induced by residue maps. These residue maps are defined locally using a system of parameters and Thom-like residues along coordinate hyperplanes, extending to global [cohomology](/page/Cohomology) via the trace map from $H^n(X, \omega_X)$ to $k$. The pairing's non-degeneracy follows from the vanishing of cohomology on affines and the projective embedding properties, yielding the desired [isomorphism](/page/Isomorphism).[](https://www.college-de-france.fr/media/jean-pierre-serre/UPL5435398796951750634_Serre_FAC.pdf) One immediate application arises in the study of [canonical](/page/Canonical) divisors: Serre duality implies that $H^n(X, \omega_X) \cong k$, identifying the top [cohomology](/page/Cohomology) of the [canonical](/page/Canonical) sheaf with the [ground](/page/Ground) [field](/page/Field), which reflects the [existence](/page/Existence) of a global [volume form](/page/Volume_form) up to scalar.[](https://www.college-de-france.fr/media/jean-pierre-serre/UPL5435398796951750634_Serre_FAC.pdf) For smooth hypersurfaces, the [adjunction formula](/page/Adjunction_formula) computes the [canonical](/page/Canonical) sheaf explicitly: if $Y \subset X$ is a smooth effective [Cartier](/page/Cartier) divisor on a smooth projective variety $X$, then $\omega_Y \cong (\omega_X \otimes \mathcal{O}_X(Y))|_Y$. This relation derives from the short [exact sequence](/page/Exact_sequence) of cotangent sheaves $0 \to \mathcal{O}_X(-Y) \to \Omega_X^1|_Y \to \Omega_Y^1 \to 0$, taking determinants and using the [identification](/page/Identification) $\det \mathcal{O}_X(-Y) = \mathcal{O}_X(-Y)$.[](https://www.college-de-france.fr/media/jean-pierre-serre/UPL5435398796951750634_Serre_FAC.pdf) In the case of a smooth [hypersurface](/page/Hypersurface) of degree $d$ in $\mathbb{P}^{n+1}$, this yields $\omega_Y \cong \mathcal{O}_Y(d - n - 2)$, highlighting how duality constrains the geometry of embeddings.[](https://www.college-de-france.fr/media/jean-pierre-serre/UPL5435398796951750634_Serre_FAC.pdf) ### Riemann-Roch Theorem The Hirzebruch-Riemann-Roch theorem provides a formula for the Euler characteristic of a coherent sheaf on a smooth projective variety, generalizing the classical Riemann-Roch theorem from dimension one to arbitrary dimensions. For a smooth projective variety $X$ over the complex numbers and a coherent sheaf $\mathcal{F}$ on $X$, the theorem states that

\chi(X, \mathcal{F}) = \int_X \operatorname{ch}(\mathcal{F}) \operatorname{td}(T_X),

where $\chi(X, \mathcal{F})$ is the Euler characteristic, defined via Serre duality as the alternating sum of dimensions of cohomology groups $\sum_i (-1)^i h^i(X, \mathcal{F})$, $\operatorname{ch}(\mathcal{F})$ is the Chern character of $\mathcal{F}$, $T_X$ is the tangent sheaf of $X$, and $\operatorname{td}(T_X)$ is the Todd class of $T_X$. This integral is taken in the Chow ring or cohomology ring of $X$, yielding a rational number equal to the holomorphic Euler characteristic. The theorem was proved by Friedrich Hirzebruch using topological methods involving the $\hat{A}$-genus and cobordism theory. In the special case where $X$ is a smooth projective curve of genus $g$ and $\mathcal{F} = \mathcal{L}$ is a line bundle of degree $d$, the formula simplifies to

\chi(X, \mathcal{L}) = d + 1 - g.