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Generic point

In algebraic geometry, particularly within the framework of scheme theory, a generic point of an irreducible topological space or scheme is defined as a point whose closure under the Zariski topology is the entire space.^[1] This point belongs to every nonempty open subset and represents the "most general" or densest element, capturing properties that hold throughout the space except possibly on lower-dimensional subsets.^[2] In the affine case, for a scheme X = \Spec(A) where A is an integral domain, the generic point corresponds to the zero ideal (0), and its residue field is the function field of X.^[1] The concept formalizes the idea of studying varieties or schemes "generically," allowing mathematicians to focus on behaviors that are true for a dense open set rather than specific points. For instance, in \Spec \mathbb{Z}, the prime ideal (0) serves as the generic point, with its closure being the entire spectrum, contrasting with closed points corresponding to prime ideals (p) for primes p.^[3] Similarly, for the affine plane \Spec k[x, y] over a field k, the generic point is the zero ideal, while points like the prime ideal (y - x^2) represent the generic points of irreducible curves within it.^[2] This structure ensures that every irreducible scheme has a unique generic point, enabling precise descriptions of specializations and general positions in morphisms between schemes.^[1] Generic points play a foundational role in modern algebraic geometry, underpinning concepts like the function field, fibers of morphisms, and the "yoga of generic points" for handling limits and valuations. The residue field at the generic point of an integral scheme is the function field of the scheme, facilitating the study of rational functions and étale cohomology.^[2] In non-affine settings, such as projective varieties, the generic point extends this framework, ensuring uniformity in treating classical and arithmetic geometries.^[3]

Introduction and Basics

Definition

In algebraic geometry, the generic point of an irreducible algebraic variety X over a field k is the unique point \eta \in X whose closure is the entire variety X. For an affine variety X = V(I) \subseteq \mathbb{A}^n_k, this corresponds to the prime ideal (0) in the coordinate ring k[X] = k[x_1, \dots, x_n]/I, as the spectrum of k[X] identifies points of X with prime ideals, and the closure of \{(0)\} is \operatorname{Spec} k[X].^[4] The generic point \eta represents X in a "general" sense, embodying properties that hold on a dense open subset of X. It is distinct from closed points, which are maximal ideals corresponding to specific points in the variety, and from special points where exceptional behaviors occur; instead, \eta satisfies generic properties—those true outside a proper closed subset—such as being a point of general position where no unnecessary algebraic relations hold among the coordinates.^[4]^[5] For a reducible variety X, which decomposes into irreducible components X_i, each component X_i has its own generic point \eta_i, whose closure is precisely X_i. These generic points are the minimal elements in the partially ordered set of points under specialization, ensuring that the variety's structure is captured by its irreducible parts.^[5] In the scheme-theoretic framework, the generic point extends this notion to the spectrum of a ring, where for an integral scheme, it corresponds to the zero ideal in the structure sheaf.^[4]

Motivation

In classical algebraic geometry, numerous properties of algebraic varieties, such as smoothness or the behavior of morphisms, hold universally except on proper closed subvarieties of strictly lower dimension, which form a negligible subset in the Zariski topology. The introduction of generic points addresses this by encapsulating the "typical" or predominant case within a single representative point, thereby streamlining proofs and conceptualizations without the need to enumerate or exclude exceptional loci explicitly. This approach shifts emphasis from specific points to the overarching structure, facilitating a more unified treatment of geometric phenomena.^[6] The utility of generic points draws an analogy to dense subsets in topological spaces, where the generic point of an irreducible variety is dense, meaning every nonempty open set intersects its closure, which coincides with the entire variety. Consequently, a property verified at the generic point extends to hold on a dense open subset, mirroring how "almost everywhere" assertions in analysis or measure theory capture essential behaviors without addressing every individual case. This formalizes the intuitive notion of generality prevalent in early twentieth-century Italian algebraic geometry, where statements about "almost all" points could be rigorously localized to a canonical representative.^[6]^[7] By abstracting away from explicit coordinate systems, generic points enable intrinsic, coordinate-free analyses of key attributes like dimension or intersection theory, which often manifest generically across the variety. For instance, the dimension of a variety can be determined at its generic point via the transcendence degree of its function field, avoiding ad hoc choices of embeddings or bases that might obscure universal patterns. This abstraction not only enhances elegance in theoretical developments but also aligns with the broader goals of modern algebraic geometry in prioritizing structural invariants over representational details.^[6]^[7]

Formal Framework

In algebraic varieties

In the classical framework of algebraic varieties over an algebraically closed field k, an algebraic variety X is defined as an irreducible closed algebraic subset of affine or projective space, endowed with the Zariski topology where the closed sets are the algebraic subsets of X. The generic point \eta of X is the unique point whose closure \overline{\{\eta\}} equals the entire variety X. For an affine variety with coordinate ring A, this corresponds to the point for the zero ideal in \Spec(A). This identifies \eta as the minimal element in the partially ordered set of prime ideals under inclusion, ensuring that every nonempty open subset of X contains \eta, thereby capturing the "generic" behavior across the variety.^[8] The residue field \kappa(\eta) at the generic point \eta is canonically isomorphic to the function field k(X) of the variety X, which consists of the field of fractions of the coordinate ring of X and represents all rational functions defined on a dense open subset of X. This association underscores the role of \eta in encoding the global rational structure of the variety, as any rational function on X is regular at \eta.^[8] In terms of dimension, the generic point \eta has codimension zero in X, since the Krull dimension of X equals the transcendence degree of k(X) over k, and \eta corresponds to the full-dimensional irreducible component without further specialization. This positioning highlights \eta as the top-dimensional representative, with specializations to closed points corresponding to embeddings of residue fields \kappa(\eta) \hookrightarrow \kappa(\xi) where \xi are maximal ideals with \kappa(\xi) finite extensions of k. This framework for varieties over fields extends naturally to the more general setting of schemes.^[8]

In schemes

In the scheme-theoretic framework of algebraic geometry, points of a scheme are identified with prime ideals in the spectra of the rings sheaf associated to its affine open covers, generalizing the classical notion from varieties to allow for more flexible structures including non-reduced schemes and non-separated morphisms. The Zariski topology on such a scheme endows it with a structure where irreducible closed subsets correspond to prime ideals, and the generic point of an irreducible closed subset Z is the unique point \eta whose closure \overline{\{\eta\}} = Z. This point \eta lies in every nonempty open subset of Z, capturing the "most general" behavior within that component.^[1]^[9] For an affine scheme X = \operatorname{Spec}(A) where A is an integral domain, the scheme is integral, meaning it is both irreducible and reduced, and possesses a unique generic point corresponding to the zero ideal (0). The closure of this generic point is the entire space X, as the zero ideal is contained in every prime ideal of A, ensuring that \{\eta\} is dense in the Zariski topology. The stalk of the structure sheaf at this generic point \mathcal{O}_{X,\eta} is the field of fractions K(A) of A, often called the function field of X, which encodes the rational functions on the scheme. This setup highlights how the generic point serves as the "generic stalk" representing the global function field in the integral case.^[1]^[2] In the more general case of a non-irreducible scheme, which decomposes into a finite union of irreducible components, each such component admits its own unique generic point, with the generic points of the scheme being precisely those of its irreducible components. For instance, if X has multiple irreducible components Z_1, \dots, Z_n, then the generic points \eta_1, \dots, \eta_n satisfy \overline{\{\eta_i\}} = Z_i for each i, and no single point is dense in the whole X unless it is irreducible. This multiplicity reflects the scheme's structure as a disjoint union in the topological sense, allowing the generic points to parametrize the distinct "generic fibers" or behaviors across components.^[5]^[1]

Properties and Characteristics

Key properties

In algebraic geometry, the generic point of an irreducible closed subset Z of a scheme X is defined as the point \xi \in Z such that the closure of \{\xi\} equals Z.^[10] This property implies that the generic point is dense in Z, meaning every nonempty open subset of Z contains \xi, in the sense of the Zariski topology.^[10] For an irreducible scheme X, the closure of its generic point \eta is the entire space X = \overline{\{\eta\}}, emphasizing the density of \eta within X.^[11] Moreover, schemes are sober topological spaces, so every irreducible closed subset has a unique generic point.^[12] The residue field of the generic point \eta of an integral scheme X is the function field of X, which is maximal among the residue fields of all points in X.^[13] This maximality reflects the fact that the residue field extension degrees increase along specializations from the generic point to closed points.^[13]

Closure and specialization

In the Zariski topology on a scheme X, the specialization order is defined such that a point \eta specializes to a point \xi (denoted \eta \leadsto \xi) if \xi lies in the closure of the singleton \{\eta\}.^[14] For the generic point \eta of an irreducible closed subset Z \subseteq X, the closure of \{\eta\} is Z itself, so \eta specializes to every other point \xi \in Z.^[15] Equivalently, every point \xi \in Z has \eta as a generalization, since the specialization relation is the reverse of the generalization order.^[14] This structure positions the generic point as the most general element in the partial order restricted to Z, with all other points being its specializations.^[16] Specialization chains in an irreducible scheme illustrate the hierarchical structure from the generic point to closed points, where each step corresponds to a proper inclusion of closures and a drop in dimension. Specifically, a chain \eta = \eta_0 \leadsto \eta_1 \leadsto \cdots \leadsto \eta_n = \xi, with \xi a closed point, has length n equal to the dimension of the local ring \mathcal{O}_{X, \xi} minus the dimension of the local ring \mathcal{O}_{X, \eta}, reflecting the codimension of \xi relative to the generic point. Such chains are maximal in length for the dimension of the scheme, as the Krull dimension at the generic point equals the dimension of X.^[17] In Noetherian schemes, these chains are finite, ensuring well-defined dimensions.^[18] A topological space (or scheme) is irreducible if and only if it is nonempty and admits a unique generic point, whose closure is the entire space.^[15] Conversely, the existence of a unique generic point characterizes irreducibility, as multiple generic points would imply a decomposition into distinct irreducible components.^[15] Schemes are sober topological spaces, meaning every irreducible closed subset has precisely one generic point, which underpins the bijection between irreducible closed subsets and their generic points.^[15]

Examples and Illustrations

Classical examples

In classical algebraic geometry over an algebraically closed field k, the affine line \mathbb{A}^1_k = \operatorname{Spec} k provides a fundamental example of a generic point. The points of \mathbb{A}^1_k are the prime ideals of k, which factor as (x - a) for a \in k or the zero ideal (0). The closed points correspond to the maximal ideals (x - a), each representing a specific point a on the line. The generic point is the zero ideal (0), whose closure is the entire affine line \mathbb{A}^1_k = V((0)), as it is dense in the Zariski topology.^[12] Another illustrative example arises in the projective plane \mathbb{P}^2_k = \operatorname{Proj} k[x, y, z], where a smooth conic curve C is defined by the zero locus of an irreducible homogeneous quadratic polynomial Q(x, y, z) = 0. The generic point of C corresponds to the homogeneous prime ideal (Q) in k[x, y, z], and its closure is the entire curve V(Q). This generic point represents the function field k(C) of the conic, which is the field of fractions of the degree-zero part of the graded coordinate ring k[x, y, z]_{(Q)}, isomorphic to the rational function field k(t) via a parametrization such as the stereographic projection. For an irreducible hypersurface V(f) in affine or projective space over k, where f is an irreducible polynomial, the generic point is the principal prime ideal (f). Its closure is the whole hypersurface V(f), which is irreducible by assumption. This generic point lies outside the singular locus \operatorname{Sing}(V(f)) = V(f, \partial f / \partial x_1, \dots, \partial f / \partial x_n), a proper closed subvariety of lower dimension, ensuring that the local ring at the generic point is a field (the function field k(V(f))) and thus regular.

Scheme-theoretic examples

In scheme theory, the spectrum of an integral domain exemplifies the generic point, where the zero ideal serves as the unique generic point whose Zariski closure is the entire affine scheme. For X = \Spec R with R an integral domain, the point \eta corresponding to the prime ideal (0) is dense, and the residue field at \eta is the fraction field of R.^[12] This structure generalizes the classical notion of a generic point in varieties, but extends to non-geometric base rings.^[11] A prominent example is X = \Spec \mathbb{Z}, the affine plane over the integers, which models arithmetic surfaces. The generic point \eta = (0) captures the "global" behavior over \mathbb{Q}, with its closure comprising the whole scheme, including specializations to closed points like (p, f(x)) for primes p \in \mathbb{Z} and irreducible polynomials f. The stalk \mathcal{O}_{X, \eta} is the fraction field \mathbb{Q}(x), emphasizing the scheme's integral nature over \Spec \mathbb{Z}.^[12]^[3] For non-reduced schemes, the generic point of an irreducible component corresponds to the minimal prime ideal containing the nilradical, and nilpotents can persist in the stalk at this point, distinguishing it from reduced cases. Consider X = \Spec k/(x^2), a non-reduced thickening of the origin in the affine line over a field k. This scheme is irreducible with a single point \eta (the image of (x)), serving as its generic point, whose closure is X itself; however, the stalk \mathcal{O}_{X, \eta} \cong k_{(x)}/(x^2) contains nilpotents, reflecting infinitesimal structure at the "generic" level. Product schemes illustrate generic points across multiple components, highlighting schemes' capacity for disconnected spaces. The scheme X = \Spec(k \times k) decomposes as the disjoint union of two affine lines \Spec k \sqcup \Spec k, each an irreducible component. The generic point of the first component is \mathfrak{p}_1 = (0 \times k), dense in \Spec k, with residue field k(x); similarly, \mathfrak{p}_2 = (k \times 0) is generic for the second, with residue field k(y). These points' closures are their respective components, underscoring the scheme's reducibility.^[12]

Applications

In intersection theory

In intersection theory, generic points play a crucial role in ensuring transversality and properness of intersections between subvarieties or cycles. When two subvarieties V and W of a scheme X intersect properly, meaning the codimension of each irreducible component of V \cap W equals the sum of the codimensions of V and W, the intersection multiplicity at the generic point \xi of such a component Z is computed using the Tor formula: e_Z(V, W) = \sum_i (-1)^i \length_{\mathcal{O}_{X,\xi}} \Tor_i^{\mathcal{O}_{X,\xi}}(\mathcal{O}_{V,\xi}, \mathcal{O}_{W,\xi}).^[19] This local computation at the generic point simplifies the determination of intersection numbers, which in turn yield degrees via extensions of function fields; for example, over algebraically closed fields, the degree of the intersection class pushed forward to a point is the dimension of the function field of the generic intersection point over the base field.^[20] The moving lemma further leverages generic points by allowing the perturbation of cycles to achieve transverse intersections. Specifically, for cycles \alpha of dimension r and \beta of dimension s on a projective scheme, there exists a rationally equivalent cycle \alpha' \sim_{\rat} \alpha such that \alpha' and \beta intersect properly, with the support of the intersection consisting of points generic to their components, avoiding embedded or excess intersections.^[21] This perturbation ensures that the intersection product \alpha' \cdot \beta is well-defined and computes the same class as the original, facilitating calculations in the Chow groups without special cases. For self-intersections, generic points enable the definition of multiplicities directly on the scheme without needing to resolve singularities. The self-intersection class of a subvariety C embedded in a smooth ambient space is an element of the Chow group A_*(C), supported at the generic point of C with multiplicity given by the degree of the normal bundle or the refined intersection product along the embedding; for a curve C on a surface, this yields C \cdot C copies of the generic point \eta_C of C. This approach, via the deformation to the normal cone, assigns multiplicities intrinsically, preserving properties like closure under specialization.^[20]

In deformation theory

In deformation theory, families of algebraic varieties are often parametrized by a base scheme B, where the generic point \eta of B defines the generic fiber X_\eta, which is the pullback of the family over the function field k(B). This generic fiber captures the "general" member of the family, frequently exhibiting desirable properties such as smoothness, even when special fibers over closed points develop singularities. For a flat proper morphism f: X \to B with geometrically integral fibers, the generic fiber X_\eta is smooth if the total space X is smooth over B, as smoothness is an open condition in flat families. This contrasts with special fibers, which may acquire mild singularities, allowing deformation theory to analyze how general smooth objects specialize.^[22]^[23] The role of generic points extends to understanding obstructions in lifting deformations. In a versal deformation of a variety X_0 over a local Artin ring, the obstruction to extending a deformation from a smaller ring lies in a cohomology group such as H^2(X_0, T_{X_0}), where T_{X_0} is the tangent sheaf. If these obstruction classes vanish when evaluated at the generic point of the versal base, it often implies that the deformation functor is unobstructed locally, enabling the construction of a smooth formal versal space. For reduced schemes that are generically smooth, the Kodaira-Spencer map identifies the tangent space of the deformation functor with \operatorname{Ext}^1_{O_{X_0}}(\Omega_{X_0}, O_{X_0}), and vanishing obstructions at generic points ensure liftability along small extensions, as seen in examples like smooth projective varieties.^[23]^[22] In moduli spaces, generic points represent typical objects in the parameter space, free from the special symmetries or degenerations that characterize closed points. For instance, in the moduli space of curves M_g for genus g \geq 2, the generic point corresponds to a smooth curve without automorphisms beyond the identity, and the tangent space at this point has dimension $3g-3, matching the expected dimension from deformation theory. This perspective allows computations of global invariants, such as the dimension of the moduli space, by focusing on the generic case, where properties like unobstructedness hold universally. Miniversal deformations further illustrate this, as the generic fiber over the versal ring provides a model for general members, aiding in the prorepresentability of the deformation functor under conditions like finite-dimensional tangent spaces.^[22]^[23]

Historical Development

Origins

The concept of the generic point originated in the mid-19th century with Bernhard Riemann's development of Riemann surfaces as a tool to study multivalued functions, particularly meromorphic functions on algebraic curves. In his 1851 habilitation thesis and subsequent 1857 paper on abelian functions, Riemann considered the typical or non-special behavior of these functions at points away from branch points, poles, or ramification loci, anticipating later ideas of generic points. This allowed him to quantify the genus of the surface and establish foundational results like the Riemann-Roch theorem, which relates the dimension of the space of meromorphic functions with prescribed poles to topological invariants evaluated at such general positions.^[24] David Hilbert's Nullstellensatz, introduced in 1893 as part of his work on invariant theory, further advanced the distinction between generic and special points by linking polynomial ideals to the zero sets they define over algebraically closed fields. The theorem asserts that the radical of an ideal equals the intersection of the maximal ideals corresponding to points in the variety, implicitly separating the generic solutions—captured by the function field of the variety, analogous to evaluation at a general point—from special solutions confined to proper subvarieties of lower dimension. This correspondence provided an algebraic framework for understanding the structure of solution sets, highlighting how properties holding at the generic level propagate to most points while failing at special ones. Precursors to the Zariski topology and the explicit role of generic points emerged in Oscar Zariski's early 20th-century investigations of ideals and algebraic varieties during the 1930s, particularly in his studies of algebraic surfaces and birational geometry over fields of arbitrary characteristic. In works such as his 1935 monograph Algebraic Surfaces and related papers on resolution of singularities, Zariski adopted an ideal-theoretic approach to varieties, treating irreducible components via prime ideals and emphasizing points whose closures fill the entire component—the generic points—as central to describing general membership and dimension. These ideas culminated in his 1944 paper on the compactness of Riemann manifolds for abstract algebraic function fields, where he formalized a topology on the set of prime ideals, making the generic point the unique dense point of each irreducible variety.^[25]

Key developments

In the 1960s, Alexander Grothendieck formalized the notion of generic points within the emerging theory of schemes, as detailed in his Éléments de géométrie algébrique (EGA). There, the spectrum Spec(R) of a ring R is equipped with a Zariski topology where the generic point of an irreducible closed subscheme corresponds to the prime ideal (0) in the case of an integral domain, whose closure is the entire space; this structure unifies classical geometric intuition with algebraic foundations, enabling the treatment of points as prime ideals with associated residue fields.^[26] This formalization in EGA IV further extends to local properties and morphisms, where generic points capture the "general" behavior along fibers, as seen in theorems on flatness and dimension.^[27] During the late 1960s and 1970s, the Séminaires de Géométrie Algébrique (SGA) advanced the role of generic points in étale cohomology, particularly in descent and purity contexts. In SGA 1, effective descent criteria for étale morphisms rely on verifying conditions at generic points of base schemes, ensuring that objects over the total space descend from the generic fiber when the morphism is representable and separated. Purity theorems, developed by Pierre Deligne in SGA 4½, utilize generic points to establish exact sequences relating the étale cohomology of a scheme to that of its open and closed subschemes, such as in the localization sequence where the generic point of a normal crossing divisor facilitates isomorphisms in higher cohomology groups.^[28] These developments provided tools for computing cohomology via specialization at generic points, bridging algebraic and topological aspects. In the 1980s, extensions to Arakelov geometry introduced analogues of generic points at infinite places within arithmetic schemes. Building on Suren Arakelov's foundational intersection theory for arithmetic surfaces, which incorporates metrics on fibers over infinite primes to define degrees, later works by Alexander Parshin and Serge Lang generalized this to higher-dimensional varieties by treating the "generic fiber at infinity" as a uniformizing element analogous to the classical generic point, enabling height functions and intersection multiplicities that account for archimedean completions. This arithmetic analogue, refined in Gillet's arithmetic intersection theory, allows generic points over the rationals to pair with infinite metrics, yielding Arakelov-Green functions that measure "points at infinity" in diophantine settings.^[29]

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