Veronese Embedding in Projective Geometry

• Veronese embedding is a process that maps points in projective space to collections of homogeneous monomials of fixed degree, effectively linearizing nonlinear conditions.
• Recent research has detailed its algebraic, geometric, and combinatorial properties, revealing precise determinantal relations and syzygies governed by representation theory.
• Its applications span tensor decomposition, coding theory, and the study of Ulrich bundles, offering key insights into secant and tangential varieties in higher dimensions.

The Veronese embedding is a fundamental construction in algebraic geometry that canonically embeds a projective space into a higher-dimensional projective space by mapping a point to the collection of all homogeneous monomials of a fixed degree in its coordinates. This construction underpins the paper of projective embeddings, syzygies, secant and tangential varieties, and properties such as determinantal generation and Ulrich bundles. Recent research builds upon classical results to clarify the algebraic, geometric, combinatorial, and representation-theoretic features of Veronese embeddings and their generalizations.

1. Algebraic and Geometric Structure

Given a projective space $\mathbb{P}^n$ over an algebraically closed field $\mathbb{K}$, the $d$-uple Veronese embedding is the morphism

$$\nu_d : \mathbb{P}^n \to \mathbb{P}^N, \quad [x_0 : \cdots : x_n] \mapsto [\cdots : x_0^{a_0}x_1^{a_1}\cdots x_n^{a_n} : \cdots],$$

where $a_0 + \cdots + a_n = d$ and $N = \binom{n+d}{d} - 1$.

This map is a very ample line bundle embedding associated to $\mathcal{O}_{\mathbb{P}^n}(d)$. The image, called the Veronese variety $V_{n,d}$, parameterizes degree $d$ forms on $\mathbb{P}^n$. The ideal of $V_{n,d}$ is cut out by explicit determinantal relations: for $d=2$, these are given by the $2\times 2$ minors of a symmetric matrix (the "quadratic Veronese"), and higher $d$ correspond to similar determinantal descriptions via catalecticant matrices and their generalizations.

The image enjoys the property that the intersection of all degree-$d$ hypersurfaces containing a subset $M$ in $\mathbb{P}^n$ corresponds to the linear span of its images under $\nu_d$, thus transforming degree-$d$ conditions into linear ones (Havlicek et al., 2012).

2. Syzygies and Free Resolutions

Extensive research has elucidated the minimal free resolutions for Veronese varieties and their subvarieties, showing how these are governed by both the combinatorics of monomials and the representation theory of $\mathrm{GL}(V)$ (Netay, 2016, Fulger et al., 2012, Bruce et al., 2017). In particular:

• For the quadratic Veronese embedding $\nu_2: \mathbb{P}^n\to\mathbb{P}^{N}$ ($N=\binom{n+2}{2}-1$), the ideal is generated by the $2\times2$ minors of the symmetric matrix formed from the quadratic monomials.
• The syzygies (modules of relations) among these generators are organized into a Koszul-type complex, and they admit rich decompositions into Schur functors (irreducible $\mathrm{GL}(V)$-representations), whose asymptotic structure as $d\to\infty$ is governed by the asymptotics of classical plethysms (Fulger et al., 2012).
• For the case of quadrics in characteristic $\mathrm{char}\neq2$, the ideal is generated by quadrics of rank at most $3$ for all $d\ge 3$, $n\ge 2$, and has rank index $4$ for $d=2$, $n\ge 3$ in characteristic $3$ (Han et al., 2020, Lee et al., 11 Sep 2025).

Free resolutions for Veronese embedded plane curves have been obtained explicitly, with the shape depending on the parity of the degree and extra equations arising from the defining polynomial of the curve (Kanhere, 2010).

3. Secant, Tangential, and Higher-Order Varieties

The geometry of secant and tangential varieties to the Veronese embedding is deeply connected to questions of tensor decomposition, identifiability, and interpolation.

• For the $k$-th secant variety $\sigma_k(V_{n,d})$, powerful classification results stipulate when these are non-defective (i.e., have the expected dimension). The celebrated Alexander–Hirschowitz theorem gives a complete answer in the classical Veronese case (Laface et al., 2011). For Segre–Veronese and mixed settings, similar asymptotic bounds have been established, showing that such varieties are non-defective up to explicit combinatorial bounds on $k$ (Araujo et al., 2016).
• The prime ideals of small-degree secant varieties have been determined explicitly in low-dimensional cases, with tools including prolongation and weight space decomposition (Furukawa et al., 1 Oct 2024).
• For tangential varieties (union of tangent spaces at all points of a Veronese variety), the dimensions of their secant varieties were conjectured and proven (except for a few exceptional defective cases). Resolution of the conjecture required highly optimized computer algebra, with the most computationally intensive cases involving dimensions as high as $\mathbb{P}^{79}$ embedded in $\mathbb{P}^{88,559}$ (Abo et al., 2015).

The singular locus of $\sigma_3(V_{n,d})$ has been shown to coincide scheme-theoretically with $\sigma_2(V_{n,d})$ except in isolated exceptional cases, notably for $d=4$, $n\geq 3$ where extra components corresponding to degenerate forms exist (Han, 2014).

4. Rank and Generation Properties

Recent advances have focused on the "rank index" and QR(3) property (quadratic rank three), investigating when all quadratic equations defining a Veronese embedding can be generated by forms of rank at most $3$.

• For characteristic $\neq 2,3$, the rank index for $(\mathbb{P}^n,\mathcal{O}_{\mathbb{P}^n}(d))$ is $3$ for $d\geq2$, but for $\operatorname{char}=3$, this fails for the second Veronese ($d=2$, $n\geq3$), where the minimal index jumps to $4$ and the span of rank-3 quadrics is of codimension $\binom{n+1}{4}$ in the space of all quadratics (Lee et al., 11 Sep 2025). This codimension count provides a precise measure of the exceptional behavior in low degrees and special characteristics.

5. Generalizations and Applications

A number of generalizations and connections have been developed:

• The $(d,\boldsymbol{\sigma})$-Veronese variety introduces automorphism twists, yielding varieties in which, for fixed parameters, any $d+1$ points are independent and $d+2$ are dependent if and only if they arise from a line in the base space. This has applications in coding theory, where associated linear codes exhibit MDS and almost-MDS behavior (Durante et al., 2021).
• Reduced arc schemes for Veronese embeddings have been studied in the context of infinite jet spaces, yielding coordinate rings whose graded components are cocyclic modules for current algebras, with connections to Demazure modules (Dumanski et al., 2019).
• The explicit structure of Veronese varieties over finite fields has been leveraged for higher weight spectrum calculations of projective Reed–Muller codes, with the combinatorial geometry of configurations on the Veronese image governing the ranks of codeword supports (Kaipa et al., 20 May 2024).

6. Vector Bundles, Ulrich Bundles, and Derived Categories

Ulrich bundles play a vital role in understanding the linear syzygies and the homological complexity of a projectively embedded variety:

• For the Veronese embeddings $v_a: \mathbb{P}^n\to\mathbb{P}^N$, it has been established that no Ulrich bundle of rank $r\le 3$ exists for $n\ge 4$, regardless of divisibility conditions on $a$. This indicates high Ulrich complexity and limits the possible linearity of free resolutions for these varieties in higher dimensions (Lopez et al., 12 Jun 2024).
• When Ulrich bundles do exist (in lower dimension or special situations), they yield maximal linearity in resolutions and optimal cohomological vanishing, highlighting a nuanced distinction between low- and high-dimensional cases.

7. Classification, Characterizations, and Uniqueness

The Veronese embedding is uniquely characterized by its linearization of nonlinear conditions and by its extremal incidence properties:

• Severi's theorem (and its generalizations) classifies the Veronese embeddings among all non-degenerate varieties of the same degree and dimension by local incidence axioms involving rational normal curves and their tangents (Schillewaert et al., 2014).
• The geometric characterization of quadratic Veronese embeddings via image properties (e.g., image of lines as non-degenerate conics) and dimension-preserving mappings provides a combinatorial and geometric recognition theorem for this class of embeddings (Havlicek et al., 2012).
• The uniqueness of the Veronese surface among surfaces in $\mathbb{P}^5$ admitting isomorphic projection to $\mathbb{P}^4$ exemplifies the rigidity of the classical $d$-uple embedding (Schillewaert et al., 2014).

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This synthesis outlines the main structural, homological, combinatorial, and geometric aspects of Veronese embeddings as currently understood, highlighting the explicit computations, inductive proofs, and constructive methods that support ongoing research in algebraic geometry, combinatorics, coding theory, and representation theory.