Fact-checked by Grok 2 weeks ago

Veronese surface

The Veronese surface is a fundamental object in , defined as the image of the Veronese map \nu_2: \mathbb{P}^2 \to \mathbb{P}^5, which embeds the \mathbb{P}^2 into five-dimensional using all monomials of 2 in the [X_0 : X_1 : X_2], mapping to [X_0^2 : X_1^2 : X_2^2 : X_0 X_1 : X_0 X_2 : X_1 X_2]. This embedding produces a projective surface of degree 4 that is isomorphic to \mathbb{P}^2, serving as a prototypical example of a Veronese variety and illustrating how higher-degree maps resolve singularities or embed varieties non-trivially. Named after the Italian mathematician Giuseppe Veronese (1854–1917), who introduced the concept in 1884 as a "normal homaloidal surface of degree four in ," the surface arose from his pioneering work on higher-dimensional , where he demonstrated simplifications achievable by extending dimensions beyond three. Veronese's approach emphasized geometric intuition over purely algebraic methods, founding the field of n-dimensional projective spaces and highlighting projections of high-dimensional objects into lower spaces. Algebraically, the Veronese surface can be realized as the determinantal variety consisting of points in \mathbb{P}^5 corresponding to symmetric matrices of at most 1, defined by the vanishing of all 2×2 minors; this perspective underscores its role in parametrizing rank-one conics in \mathbb{P}^2. It is projectively normal, arithmetically Cohen–Macaulay, and homogeneous under the action of PGL(3), making it a key example for studying varieties, sections (which correspond to conics in \mathbb{P}^2), and enumerative problems in classical .

Historical context

Giuseppe Veronese and his contributions

Giuseppe Veronese (1854–1917) was an Italian mathematician whose work significantly influenced the development of modern , particularly in higher dimensions and foundational aspects. Born on May 7, 1854, in , a town near , he received his early education at the Technical Institute in before studying engineering at the Polytechnic Institute in in 1873. He later transferred to the University of Rome, where he graduated in mathematics in 1877 and obtained his doctorate the following year. After serving as an assistant in analytical geometry at Rome, Veronese spent time researching at the University of in 1880–1881 before being appointed professor of at the in 1881, a role he maintained until his death on July 17, 1917, in . Veronese's major contributions centered on the foundations of , where he emphasized geometric and to address complexities in multidimensional spaces. In works such as his 1880 paper on n-dimensional , he introduced methods for simplifying problems in higher dimensions by treating them as extensions of familiar two- and three-dimensional concepts, reducing reliance on algebraic computations. He also pioneered non-Archimedean geometry around 1890, developing a framework that incorporated and infinite quantities without contradicting , a concept later validated by and influencing debates on the continuum. Additionally, Veronese advanced studies in transfinite numbers and , contributing to the logical during a period of intense foundational scrutiny. Particularly relevant to , Veronese first described the Veronese surface in 1884 in his paper "La superficie omaloide normale a due dimensioni e del quarto ordine dello spazio a cinque dimensioni e le sue proiezioni nel piano e nello spazio ordinario," published in Memorie dell'Accademia dei Lincei (3) xix, 344–371, as a means to explore higher-degree embeddings of the into higher-dimensional spaces. This surface served as a key example for illustrating the projection of complex geometric forms and simplifying the analysis of varieties in projective settings, and he further developed these ideas in his late 19th-century publications, including the 1891 treatise Fondamenti di geometria a più dimensioni e a più specie di unità rettilinee. The idea has since been generalized to the Veronese embedding, a broader construction applicable to projective spaces of arbitrary dimension.

Development of the Veronese embedding

Giuseppe Veronese introduced the embedding in his 1884 paper in Memorie dell'Accademia dei Lincei and further developed it in his seminal 1891 treatise Fondamenti di geometria a più dimensioni e a più specie di unità rettilinee, where it served as a tool in to manage quadratic relations by projecting geometric objects into higher-dimensional spaces. This approach allowed for simplifications in the analysis of hyperspaces, building on his earlier 1882 paper in Mathematische Annalen that emphasized projections from higher dimensions. The concept gained traction and was generalized in the early by Corrado Segre, who in characterized the Veronese surface as the unique non-developable surface in \mathbb{P}^5, integrating it into studies of classifications within the Italian school of . Federigo Enriques further extended its applications, employing the embedding in his classifications of algebraic surfaces and explorations of rational varieties during the 1910s and 1920s, as seen in his collaborative works with Guido Castelnuovo on birational transformations. Key milestones in the embedding's evolution occurred in the 1930s through Oscar Zariski's rigorous algebraic treatments of surfaces, where it informed discussions of birational equivalence and in his 1935 monograph Algebraic Surfaces. By the 1960s, incorporated the Veronese embedding into scheme theory in his , highlighting its projective normality and role in embedding projective varieties via complete linear systems, thus bridging classical and modern .

Mathematical prerequisites

Projective spaces and varieties

In , the projective space \mathbb{P}^n over a k is defined as the set of all 1-dimensional linear subspaces (lines through the origin) of the k^{n+1}. This construction identifies points in \mathbb{P}^n with equivalence classes of nonzero vectors in k^{n+1}, where two vectors are equivalent if one is a nonzero scalar multiple of the other. Points in \mathbb{P}^n are represented using [x_0 : x_1 : \dots : x_n], with (x_0, \dots, x_n) \in k^{n+1} \setminus \{0\}, allowing for a compact description that naturally incorporates points at infinity. The topology on \mathbb{P}^n is the , where closed sets are zero loci of homogeneous polynomials. Projective varieties are fundamental objects within \mathbb{P}^n, defined as the common zero sets of a collection of homogeneous polynomials in the variables x_0, \dots, x_n. More precisely, a is an irreducible closed algebraic subset of \mathbb{P}^n, corresponding to the vanishing of an of homogeneous polynomials in the k[x_0, \dots, x_n]. The homogeneity ensures that the defining equations are well-defined on the , as scaling the coordinates does not alter the zero locus. Unlike , which are zero sets of polynomials in the \mathbb{A}^n = k^n, projective varieties provide a compactification by including hyperplanes at infinity, obtained via homogenization of affine equations. Key properties of projective varieties include , irreducibility, and birational equivalence, which underpin their and study. The of a projective variety X \subset \mathbb{P}^n is the of its homogeneous coordinate ring, equivalently the maximum among its affine open subsets. Irreducibility means X cannot be expressed as the union of two nonempty proper closed subvarieties, ensuring it behaves like an in its function theory. Two projective varieties are birationally equivalent if there exists a rational map between them with a rational inverse, inducing an of their fields of rational functions; this equivalence preserves and many geometric invariants. These prerequisites form the foundation for morphisms and embeddings in , such as those used to realize higher-degree varieties within projective spaces.

Embeddings and maps in algebraic geometry

In , an of a is a that is an onto its , where the image is a closed subvariety of the target space. This ensures the map is injective on points and induces an isomorphism of structure sheaves between the variety and its image. For projective varieties, embeddings are typically closed immersions, meaning the morphism factors through a closed subscheme that is isomorphic to the original variety. Veronese-type maps generalize embeddings by associating to a projective space \mathbb{P}^n a into a higher-dimensional projective space \mathbb{P}^N, where N = \binom{n+d}{d} - 1, using all monomials of fixed d in the homogeneous coordinates. These maps are defined by sending a point [x_0 : \cdots : x_n] to the coordinates given by the -d monomials, forming a morphism from \mathbb{P}^n to \mathbb{P}^N. Such maps provide a standard way to embed projective spaces via homogeneous s of uniform , often used to linearize higher- equations. Key properties of these embeddings include the preservation of : if the is a (birational to ), the image under a Veronese-type map remains rational, as the map induces an of fields up to birational equivalence. Veronese maps are birational onto their images, being closed immersions that are isomorphisms with the embedded subvariety. They differ from immersions, which are locally isomorphisms onto their images but may not be globally injective or closed; embeddings require global bijectivity onto a closed subvariety.

Definition and construction

The Veronese map for the surface

The Veronese map for the surface is a specific instance of the more general Veronese , which associates to a a higher-dimensional via homogeneous polynomials of fixed degree. This map, denoted \nu_2: \mathbb{P}^2 \to \mathbb{P}^5, sends a point [x : y : z] \in \mathbb{P}^2 to [x^2 : xy : xz : y^2 : yz : z^2] \in \mathbb{P}^5. In general form, \nu_2 is constructed using all monomials of degree 2 in the three x, y, z, namely x^2, xy, xz, y^2, yz, z^2, which yield 6 coordinates corresponding to the projective dimension \mathbb{P}^5 since the number of such monomials is given by the \binom{2+2}{2} = 6. The map is well-defined on \mathbb{P}^2 because the monomials are homogeneous of the same degree, ensuring that scaling the input coordinates by a nonzero scalar \lambda \in k results in the output coordinates scaling by \lambda^2, preserving the projective equivalence. Furthermore, \nu_2 is injective, establishing it as a that embeds \mathbb{P}^2 isomorphically onto its image in \mathbb{P}^5.

Parametric representation and coordinates

The Veronese surface V \subset \mathbb{P}^5 is the image of the Veronese map \nu_2: \mathbb{P}^2 \to \mathbb{P}^5, which embeds the into five-dimensional via quadratic monomials and provides a parametric representation of V. This map is defined by sending a point [x : y : z] \in \mathbb{P}^2 to the point in \mathbb{P}^5 with [x^2 : xy : xz : y^2 : yz : z^2], where the coordinates on \mathbb{P}^5 are labeled as [u : r : s : v : t : w] corresponding to the monomials x^2, xy, xz, y^2, yz, z^2, respectively. These coordinates satisfy relations derived from the products of the original variables, such as uv = r^2, uw = s^2, and vw = t^2. More completely, the homogeneous defining [V](/page/V.) in \mathbb{P}^5 is generated by the $2 \times 2 of the catalecticant , which is the $3 \times 3 \begin{pmatrix} u & r & s \\ r & v & t \\ s & t & w \end{pmatrix} formed by arranging the coordinates in this Hankel pattern. For example, one such minor is the of the top-left submatrix, yielding the equation uv - r^2 = 0; similarly, the bottom-right minor gives vw - t^2 = 0, and the relation uw - s^2 = 0 arises from the minor of the first and third rows and columns. These nine equations (with redundancies) collectively define [V](/page/V.) set-theoretically and ideal-theoretically as the rank-1 locus of the matrix.

Properties

Geometric characteristics

The Veronese surface is a 2-dimensional embedded in \mathbb{P}^5, specifically the image of the Veronese embedding of \mathbb{P}^2 of degree 2. As such, it has degree 4, meaning that the intersection of the surface with a general in \mathbb{P}^5 yields a curve of degree 4. This embedding realizes the surface as a compact, non-degenerate subvariety that captures the quadratic relations inherent to the source \mathbb{P}^2. The surface is smooth and non-singular at every point, owing to the injectivity and properness of the Veronese map, which ensures no self-intersections or singularities arise in the image. Furthermore, it is a surface, being birational to \mathbb{P}^2 via the itself, which preserves the rationality while linearizing the structure. This birational equivalence highlights its geometric simplicity despite the higher-dimensional ambient space. Regarding curves on the surface, points in \mathbb{P}^2 map to points on the Veronese surface, while lines in \mathbb{P}^2 map to conics—degree-2 curves—on the surface. These conics are the images under the degree-2 Veronese map and represent the fundamental 1-dimensional loci that foliate the surface geometrically.

Algebraic invariants and equations

The Veronese surface V \subset \mathbb{P}^5 is isomorphic to \mathbb{P}^2 via the Veronese embedding of degree 2 and is therefore a rational surface with arithmetic genus 0. Its general hyperplane section is a smooth conic curve of genus 0, yielding sectional genus 0. The canonical divisor class K_V on V intersects the hyperplane class h in degree -6, as determined by the applied to a hyperplane section C with C^2 = 4 and genus 0, giving $2 \cdot 0 - 2 = C^2 + C \cdot K_V. The Veronese surface is projectively normal in \mathbb{P}^5, so its saturated homogeneous ideal has no components in degrees other than 2 and higher, and the homogeneous coordinate ring \bigoplus_k H^0(V, \mathcal{O}_V(k)) is generated as a k-algebra by the degree-1 piece with relations appearing in degree 2. The ideal I(V) in the polynomial ring k[u,v,w,s,t,r], where the variables correspond to the monomials x_0^2, x_0 x_1, x_0 x_2, x_1^2, x_1 x_2, x_2^2, is generated by quadrics. Specifically, these are the $2 \times 2 minors of the catalecticant matrix \begin{pmatrix} u & v & w \\ v & s & t \\ w & t & r \end{pmatrix}. There are 9 such minors, which span the 6-dimensional space of all quadrics vanishing on V (since \dim \mathrm{Sym}^2(k^6) = 21 and \dim H^0(\mathcal{O}_{\mathbb{P}^2}(4)) = 15).

Applications and motivations

Linearization of quadratic forms and conics

The Veronese surface provides a fundamental tool for linearizing the study of quadratic forms and conics in the \mathbb{P}^2. The space of all conics in \mathbb{P}^2 is parametrized by \mathbb{P}^5, the projectivization of the 6-dimensional of homogeneous polynomials in three variables. A general point [a : b : c : f : g : h] in \mathbb{P}^5 corresponds to the conic defined by the a x^2 + b y^2 + c z^2 + 2 f y z + 2 g z x + 2 h x y = 0. This equation arises from the associated symmetric $3 \times 3 \begin{pmatrix} a & h & g \\ h & b & f \\ g & f & c \end{pmatrix}, where the conic is the set of points [x : y : z] such that the vanishes. Points on the Veronese surface V \subset \mathbb{P}^5 specifically parametrize the rank-one conics, which are the most degenerate case: double lines in \mathbb{P}^2. These correspond to quadratic forms that are perfect squares of s, i.e., (l_1 x + l_2 y + l_3 z)^2 = 0 for some linear form with coefficients [l_1 : l_2 : l_3] \in \mathbb{P}^2. The Veronese map \nu_2: \mathbb{P}^2 \to \mathbb{P}^5 realizes this parametrization by sending [x : y : z] to [x^2 : y^2 : z^2 : x y : x z : y z], the point whose coordinates are the quadratic monomials. For instance, the point [1 : 1 : 1 : 1 : 1 : 1] on V corresponds to the conic (x + y + z)^2 = 0, a double line through the points where x + y + z = 0. Degeneracy conditions, such as the conic reducing to a double line (), are thus captured geometrically by membership in V, while broader conditions like passing through a fixed point in \mathbb{P}^2 impose linear constraints directly on the coefficients in \mathbb{P}^5. This setup linearizes the multiplication of linear forms into quadratic forms. The coordinates of \mathbb{P}^5 form a basis for the space of quadratic forms, which is the symmetric square \mathrm{Sym}^2(\mathbb{C}^3)^*. The product of two linear forms, a bilinear yielding a , becomes expressible as a of these basis elements when viewed through the embedded points on V. Consequently, relations in the original variables of \mathbb{P}^2—such as those defining intersections or factorizations of conics—translate to linear equations or sections intersecting V in \mathbb{P}^5. For example, the condition that a factors as a product of two distinct linear forms (rank 2 degeneracy) lies in the secant variety of V, but the foundational -1 case on V itself demonstrates how the embedding turns inherently phenomena into linear algebraic problems on the surface.

Role in secant varieties and projections

The second variety \sigma_2(V) of the Veronese surface V \subset \mathbb{P}^5, which is the Zariski closure of the union of all lines joining pairs of points on V, is defective. While the expected dimension is 5, the actual dimension is 4, resulting in a defect of 1; this defectivity arises because points on \sigma_2(V) correspond to symmetric $3 \times 3 matrices of at most 2, and the is in \mathbb{P}^5 defined by the of such a matrix vanishing. This property is part of the Alexander-Hirschowitz of defective varieties of Veronese embeddings, where V is one of the exceptional cases for the Veronese map. In , \sigma_2(V) parametrizes decompositions of degree-2 homogeneous polynomials ( forms) into sums of at most two squares of linear forms, with the defect reflecting non-uniqueness in such representations for certain tensors. Projections of the Veronese surface provide insights into its geometric flexibility. A generic projection from a point in \mathbb{P}^5 to \mathbb{P}^4 yields an isomorphic embedding of V as a surface of degree 4, which is the unique non-degenerate smooth surface in \mathbb{P}^4 projectable isomorphically from \mathbb{P}^5. A specific projection from a general point on V itself to \mathbb{P}^3 produces the Steiner surface, a quartic surface with three cuspidal edges and ten nodal points, serving as a classical example of a ruled surface with singularities arising from the projection center lying on the variety. In applications, the Veronese surface facilitates advanced geometric computations. In computer vision, it underlies the multibody fundamental matrix for segmenting dynamic scenes into multiple rigid motions from two views; points on V correspond to fundamental matrices for individual motions, enabling estimation via linear algebra on the secant variety for mixture models. In enumerative geometry, V appears as a degeneracy locus in problems like counting plane conics tangent to five given conics, where the famous 3264 solutions include extraneous double lines parametrized by V, resolved by blowing up along the surface to compactify the moduli space.

Generalizations and related concepts

Veronese varieties in higher dimensions

The Veronese varieties in higher dimensions provide a natural generalization of the Veronese surface, extending the d-uple embedding to projective spaces of arbitrary dimension. The general Veronese variety V_{n,d} is defined as the image of the Veronese map \nu_d: \mathbb{P}^n \to \mathbb{P}^m, where m = \binom{n+d}{d} - 1 and the map sends a point with [x_0 : \cdots : x_n] to the point whose coordinates are all monomials of total degree d in the x_i. For n=2 and d=2, this construction yields the Veronese surface embedded in \mathbb{P}^5. These varieties inherit key geometric and algebraic from the embedding. The map \nu_d is an isomorphism onto its image, ensuring that V_{n,d} is smooth and rational, as it is biholomorphic (or birational in the algebraic sense) to \mathbb{P}^n. The degree of V_{n,d} is d^n, computed via as the number of intersection points with a general linear subspace of complementary dimension. Additionally, V_{n,d} is projectively normal, meaning that the homogeneous coordinate ring is integrally closed in its graded , a property that holds over any field and facilitates the study of its ideals and . A prominent example in higher dimensions is the Veronese threefold V_{3,2}, obtained by \mathbb{P}^3 via quadratic monomials into \mathbb{P}^9, where m = \binom{5}{2} - 1 = 9. This has $2^3 = 8 and exemplifies the role of Veronese varieties in embedding higher-dimensional spaces non-degenerately while preserving and .

Rational normal curves as special cases

The rational normal curve arises as a special case of the Veronese embedding when considering the projective line \mathbb{P}^1 instead of higher-dimensional projective spaces, providing a lower-dimensional analog to the Veronese surface. Specifically, the d-th Veronese embedding \nu_d: \mathbb{P}^1 \to \mathbb{P}^d maps the into \mathbb{P}^d, and its image V_{1,d} is known as the rational normal curve of degree d. This curve is , rational, and non-degenerate in \mathbb{P}^d, meaning it spans the full and has no linear dependencies among its points beyond the projective dimension. The parametrization of the rational normal curve is given explicitly by the Veronese map, which sends a point [s:t] \in \mathbb{P}^1 to the point in \mathbb{P}^d with homogeneous coordinates corresponding to the monomials of degree d: \nu_d([s:t]) = [s^d : s^{d-1}t : s^{d-2}t^2 : \cdots : st^{d-1} : t^d]. This embedding realizes the complete linear series |\mathcal{O}_{\mathbb{P}^1}(d)| on \mathbb{P}^1, ensuring the curve has degree d and is projectively normal. For d=2, the rational normal curve is a conic in \mathbb{P}^2, which in affine coordinates corresponds to a parabola, serving as the simplest non-linear example. For d=3, it becomes the curve in \mathbb{P}^3, a curve that twists through the without lying in any , illustrating the non-planar nature of higher-degree . These examples highlight how the Veronese map linearizes the degree-d forms on \mathbb{P}^1, analogous to the role of the Veronese surface in embedding quadratic forms on \mathbb{P}^2. In the context of the Veronese surface V = \nu_2(\mathbb{P}^2) \subset \mathbb{P}^5, the images of lines in \mathbb{P}^2 under this embedding provide a direct connection to rational normal curves. Each line \ell \cong \mathbb{P}^1 \subset \mathbb{P}^2 maps via the restriction of \nu_2 to \nu_2|_\ell: \mathbb{P}^1 \to \mathbb{P}^5, which is projectively equivalent to the degree-2 Veronese embedding \nu_2: \mathbb{P}^1 \to \mathbb{P}^2 \subset \mathbb{P}^5. Thus, these images are rational normal curves of degree 2, or conics, lying in 2-planes within \mathbb{P}^5 and covering the surface. This structure motivates the Veronese embedding of the surface by generalizing the curve case, where lines generate the geometry through their embedded images.