The Hidden Geometry of Full Perazzo Algebras and Jordan Types

In the quiet world where algebra and geometry meet, a family of polynomials called Perazzo forms hides a geometry that refuses to play by the usual rules. These forms generate shapes with a curious property: their Hessian determinant vanishes identically. The Hessian, a matrix of second derivatives, is a staple diagnostic in geometry—when it vanishes, the geometry often refuses to bend under the usual intuition. That odd fact triggers a cascade of questions about how spaces of polynomials encode shapes, and what hidden symmetries survive when the standard calculus logic stalls.

A trio of researchers—Pedro Macias Marques, Rosa Miró-Roig, and Josep Pérez—set out to study a particularly generous version of these objects, the full Perazzo algebras. These are Artinian Gorenstein algebras built from a Perazzo form via Macaulay–Macaulay duality. The central question is deceptively simple: if you pick any linear form and multiply through the algebra, how does the system break into Jordan blocks—the canonical blocks that encode linear maps? That block structure, the Jordan type, acts like a fingerprint that captures how the algebra organizes its internal states under a basic operation.

The authors, working across the Universidade de Évora and the Universitat de Barcelona, report a crisp classification: for full Perazzo algebras, the Jordan type of any linear form falls into one of three archetypes. That’s a striking simplification in a landscape where one might expect a zoo of possibilities. It’s the algebraic equivalent of discovering that a seemingly complex machine can only produce three distinct rhythms when you poke it with a switch. And because the Jordan type is tied to Lefschetz properties—the algebra’s version of a geometric “rigidity”—the result also helps map when and how these algebras exhibit the weak Lefschetz property, a topic of intense interest in the field.

What Perazzo algebras are and why Hessians matter

Perazzo hypersurfaces X inside projective space are defined by a form F that couples a sum of monomials in X-variables with polynomials in Y-variables. The Pi’s are degree d−1 polynomials in the Y’s, they are independent as a set, but bound together so that the entire form is not a cone. This constraint forces the Hessian det of F to vanish identically—a signature that the usual curvature-based reasoning about shapes must be abandoned. The upshot is a geometry that is rigid in one sense but flexible in another, offering a rare laboratory for investigating how algebraic structure encodes shape when the familiar tools fail.

From the algebra side, each Perazzo form F yields a standard graded Artinian Gorenstein algebra AF via Macaulay duality: AF is R/AnnR(F) with R a polynomial ring. The h-vector of AF captures how many independent states live in each degree, and for Perazzo algebras it is symmetric because AF is Gorenstein. The special case called “full” means the Pi’s form a basis of the space of degree d−1 Y-forms; this fullness gives a precise combinatorial handle on the algebra’s size and structure. In such a full setting, the Hilbert function—the sequence of dimensions in each degree—takes a compact, almost geometric shape, which helps guide what linear operators can do inside the algebra.

Why go through all this energy for Hessians and duals? Because the Lefschetz properties—particularly the weak form that many people care about—tie directly to how multiplication by a linear form behaves. For Perazzo algebras, the Hessian vanishes, so the strong Lefschetz property cannot hold. Yet many Perazzo algebras can still satisfy the weaker Lefschetz property, which is enough to unlock interesting algebraic and geometric consequences. The paper positions itself at this crossroads: by pinning down Jordan types, it clarifies when WLP can appear and how often.

Jordan type and its finer cousin the Jordan degree type

If you take a graded Artinian algebra A and pick a linear form l, the operation of multiplying by l becomes a linear transformation on A. The Jordan type is simply the partition that lists the sizes of the Jordan blocks of this transformation, arranged from largest to smallest. Think of it as a spectral fingerprint: not the eigenvalues themselves but how the action unfolds across the chain of subspaces. In this language, the story of a Perazzo algebra becomes a tale of beads and strings—the blocks (strings) where multiplying by l moves you along beads (basis elements) of increasing degree until you hit a dead end.

There’s a finer invariant also defined in this realm called the Jordan degree type. It remembers not just how long each block is but where it starts—at which degree the corresponding string begins. That extra datum matters because two different Jordan types might share the same sizes but differ in where those blocks kick off in the graded structure. In short: Jordan type tells you the lengths of the blocks, Jordan degree type tells you where each block lives in the degree ladder.

For most algebras, listing all possible Jordan types that can occur is a tough, sometimes intractable problem. But one can say a lot by looking at the generic situation: for a general linear form, you expect the largest possible Jordan type with respect to a natural dominance order. When the algebra is a Perazzo one, the question becomes whether this generic type is as big as possible, and how many distinct types can appear as you move through the space of linear forms. The authors sharpen this picture by focusing on full Perazzo algebras and computing what the possibilities are.

The main result in plain language

The authors’ central theorem is a clean trinity. In a full Perazzo algebra AF with dual generator F in the canonical form, the Jordan type of any linear form l within AF’s degree-one piece is one of three shapes. The first two are long, highly structured sequences of block sizes driven by the two integers d (the degree) and m (the number of Y-variables). These two families are designed to reflect the repeats and symmetry baked into the Perazzo construction. The third shape is a simpler two-block form whose exact split is governed by how big the contraction of l with F produces in the first derivative space.

Concretely, the first two possibilities yield a chain of blocks that signal a robust, almost maximal way the algebra can respond to a linear input. They come with corresponding Jordan degree types that follow a precise pattern: the starting degrees and the subsequent starts of each string line up in a regular cadence. The third possibility is the minimalist option: multiplication by l produces exactly two types of strings—some of length two and the rest of length one—so the action has a much simpler, almost binary rhythm. This triad is remarkable because it cuts through what could be a wild zoo of Jordan behavior and imposes a disciplined structure across all full Perazzo algebras.

Beyond the theorem, the paper also gives corollaries that illuminate what lies behind the scenes. For example, they show that the generic linear Jordan type in this family has a predictable number of blocks—roughly twice the count of degrees of freedom in the Y-variables—reflecting a universal rhythm offered by the full Perazzo construction. They also explore what happens in a two-variable subcase, where the contraction space becomes two-dimensional and the algebra becomes a complete intersection in a precise sense. All of these pieces reinforce the sense that Perazzo algebras, despite their reputation for vanishing Hessians, harbor a surprisingly tidy internal logic when you watch them through the Jordan-type lens.

Why this matters beyond the page

Classification is not just a bookkeeping exercise. It has practical implications for how we understand Lefschetz properties in Gorenstein algebras, a class of objects that frequently arises in algebraic geometry, singularity theory, and even questions about polynomial decompositions. By pinning down the exact possibilities for Jordan types in full Perazzo algebras, the paper provides a map for where WLP can hold and where it cannot. That, in turn, helps mathematicians test conjectures about how geometry and algebra co-evolve when Hessians misbehave in systematic ways.

The story also has a constructive flavor. Knowing the precise Jordan types guides computational experiments: one can design linear forms to probe the boundaries of the three cases, or to search for instances where the Jordan degree type reveals nuances the coarser Jordan type misses. The work makes a strong case that even in tricky families—where the Hessian vanishes and the underlying geometric object resists being a cone—the algebraic fingerprint remains trackable and meaningful. It’s a reminder that structure can survive constraint, and that the math community’s toolkit—duality, Hilbert functions, and the language of blocks—can still reveal order beneath complexity.

Looking ahead, several avenues buzz with potential. Extending the classification to broader families of Perazzo algebras or to different degrees could test how general the three-case pattern is. Researchers might also explore how these Jordan fingerprints interplay with growth conditions of the Hilbert function, or with deeper aspects of the Lefschetz lore, like the precise loci of weak Lefschetz property in parameter spaces. And there’s the tantalizing possibility that these algebraic fingerprints could inspire analogous questions in combinatorics, representation theory, or even physics where Hessian-like constraints whisper about stability and symmetry.

The work of Marques, Miró-Roig, and Pérez—anchored in the Universidade de Évora and the Universitat de Barcelona—paints a vivid portrait of a mathematical landscape where constraints sharpen insight rather than diminish it. The three-family Jordan-type classification for full Perazzo algebras is not merely a technical victory; it’s a narrative about how structure persists when surfaces refuse to bend, and how a simple act—multiplying by a linear form—unlocks a surprisingly clear map of hidden order. For curious readers, that’s the thrill at the heart of modern algebra: a quiet algebraic machine, when poked gently, reveals a universe of elegant, accessible rhythm.