Gauss Algebra in Squarefree Veronese Algebras

• The paper details how the Gauss algebra serves as the homogeneous coordinate ring of the Gauss map image for squarefree Veronese algebras.
• Methodological insights include a toric presentation via Jacobian minors and explicit monomial generation criteria for dimensions d ≤ 7.
• Key implications involve establishing normality and Cohen–Macaulayness through the structure of discrete polymatroids and their combinatorial properties.

The Gauss algebra associated to squarefree Veronese algebras serves as a bridge between toric algebra, combinatorial commutative algebra, and the geometry of the Gauss map. For squarefree Veronese algebras generated in degree $r$, the structure and properties of the Gauss algebra are governed by the underlying combinatorics of squarefree monomials, and the connection to polymatroidal ideals allows precise determination of normality and Cohen-Macaulayness in many cases. Recent work provides a complete analysis of the degree %%%%1%%%% case for small dimension, illuminating the interplay between exponents, support, and algebraic properties.

1. Definition of Squarefree Veronese Algebras and the Gauss Algebra

Let $K$ be a field of characteristic zero, and $S = K[x_1, \ldots, x_d]$ the polynomial ring in $d$ variables. The squarefree Veronese algebra of degree $r$, denoted $A = K[V_{r,d}] \subset S$, is generated by all squarefree monomials of degree $r$:

$$V_{r,d} = \{x_{i_1} x_{i_2} \cdots x_{i_r} : 1 \leq i_1 < \cdots < i_r \leq d\},\qquad A = K[V_{r,d}].$$

The Gauss algebra $G(A)$ is defined as the $K$-subalgebra of $S$ generated by the $d \times d$ minors of the Jacobian matrix of the set of generators $g_1,\ldots,g_n$ of $A$, with $n = \binom{d}{r}$. Explicitly, considering the rational map

$$\varphi : \mathbb{P}^{d-1} \dashrightarrow \mathbb{P}^{n-1},\quad (x_1: \ldots : x_d) \mapsto (g_1(x) : \ldots : g_n(x)),$$

the Gauss algebra $G(A)$ is the homogeneous coordinate ring of the image of the Gauss map defined by $\varphi$ (Bandari et al., 25 Dec 2025, Herzog et al., 2018).

2. Toric Presentation and Monomial Generators

For $r=3$, every generator $g_i$ of $A$ is a squarefree monomial of degree $3$ in $S$, so $g_i = x_1^{a_{1i}}\cdots x_d^{a_{di}}$ with each $a_{ji}\in\{0,1\}$ and $\sum_{j=1}^d a_{ji}=3$. The differential matrix $\Theta(g_1,\ldots,g_n)$ has rank $d$ in characteristic zero, and its maximal minors generate $G(A)$. Due to the monomial nature of the generators, $G(A)$ is toric. The image of the Gauss map can be described combinatorially: the set of $d \times d$ minors of $\Theta$ corresponds—modulo monomial factors—to the minors of the $d \times n$ exponent matrix $\mathrm{Log}(g_1,\ldots,g_n)$, multiplied by $x_{j_1}\cdots x_{j_d}$ for each set of columns $j_1<\cdots<j_d$.

As a consequence, the generators of $G(A)$ are all monomials of the form

$m = \frac{g_{i_1} \cdots g_{i_d}}{x_1\cdots x_d}$

such that $\det \mathrm{Log}(g_{i_1},\ldots,g_{i_d}) \neq 0$ (Bandari et al., 25 Dec 2025). Because each $g_i$ has degree $3$, every such $m$ has degree $2d$ in $S$. Additional combinatorial constraints, detailed below, further specify which such monomials truly generate $G(A)$.

3. Explicit Generators for Small Dimension ($d \leq 7$)

For $d\leq 7$, a complete combinatorial description of $G(A)$ is established. Define

$$\mathrm{Mon}_S^*(4,2d) = \{x_1^{a_1} \cdots x_d^{a_d} : \deg(u)=2d,\, a_i \leq d-2\, \forall i,\, |\{i: a_i>0\}| \geq 4\}$$

and, for $d\geq 4$,

$$E_d = \{u \in \mathrm{Mon}_S^*(4,2d) : \mathrm{supp}(u) = 4\}.$$

A structural result shows that, for $d=5,6,7$,

$G(A) = K[\mathrm{Mon}_S^*(4,2d) \setminus E_d],$

where the explicit description of $E_d$ is as follows:

$d$	$\mathrm{Mon}_S^*(4,2d)$	$E_d$ (monomials excluded from $G(A)$)
$5$	$\deg=10$, $a_i\leq 3$	$x_{i_1}^3x_{i_2}^3x_{i_3}^3x_{i_4}$: $i_1<i_2<i_3<i_4$
$6$	$\deg=12$, $a_i\leq 4$	$x_{i_1}^4x_{i_2}^4x_{i_3}^3x_{i_4}$: $i_1<i_2<i_3<i_4$
$7$	$\deg=14$, $a_i\leq 5$	$x_{i_1}^5x_{i_2}^5x_{i_3}^3x_{i_4}$ and $x_{i_1}^5x_{i_2}^4x_{i_3}^4x_{i_4}$

No monomial in $E_d$ belongs to $G(A)$ by a rank argument, and all monomials outside $E_d$ do arise as suitable Jacobian minors. Thus, for $d \in \{5,6,7\}$, $G(A)$ is minimally generated (over $K$) by all degree-$2d$ monomials in $S$ with support at least $4$, exponents bounded above by $d-2$, except those in $E_d$ (Bandari et al., 25 Dec 2025).

4. Structural Properties: Normality and Cohen–Macaulayness

$G(A)$ is shown to be the base ring of a discrete polymatroid. Specifically, the ideal $I_d = (\mathrm{Mon}_S^*(4,2d) \setminus E_d)$ is polymatroidal: every time $u, v$ are minimal generators with $\deg_{x_i}(u) > \deg_{x_i}(v)$, there exists $j$ with $\deg_{x_j}(u) < \deg_{x_j}(v)$ such that $x_j(u/x_i) \in I_d$. The base ring of a polymatroid is normal and Cohen–Macaulay (Herzog–Hibi, Ch.~12), yielding:

For $d=5,6,7$, $G( K[V_{3,d}] )$ is a normal, Cohen–Macaulay toric $K$-algebra (Bandari et al., 25 Dec 2025).

5. Comparison: Squarefree $2$-Veronese and General Patterns

For $r=2$, the Gauss algebra has the following structure (Herzog et al., 2018):

• $G(A) = K[ \mathrm{Mons}(3,d) ]$ for $d\geq 5$, where $$\mathrm{Mons}(3,d) = \{u\in S: \deg(u)=d,\, |\mathrm{supp}(u)|\geq 3\}$$.
• For $d=4$, $G(A)$ omits exactly the monomial $x_1x_2x_3x_4$.

The embedding dimension in this case is $2^d - 1 - d - \binom{d}{2}$. $G(A)$ is the base ring of the polymatroid with ground set $[d]$ and rank $d$ subject to support at least $3$, and is normal Cohen–Macaulay for $d\geq 5$.

For general $r$, a phenomenon of similar type is conjectured: $G(A)$ is expected to be generated by all monomials of degree $(r-1)d$ in $S$ whose support is at least $r+1$ and each exponent is at most $d-1$. Complete verification of this formula for $r>2$ is an open problem (Herzog et al., 2018).

6. Combinatorial Aspects and the Polymatroid Connection

The combinatorial underpinnings of $G(A)$ rely crucially on the exchange property characteristic of discrete polymatroids. The polymatroidality of the ideal of monomial generators ensures both normality and Cohen–Macaulayness. The explicit exclusion of certain monomials with minimal support from $G(A)$ is dictated by the invertibility of the relevant exponent submatrices. The method delineated in size $d \leq 7$ reflects an induction on $d$ and exact construction of suitable $d$-tuples of degree $3$ squarefree monomials (Bandari et al., 25 Dec 2025).

7. Hilbert Series and Open Problems

The Hilbert series and further invariants (e.g., Betti numbers, binomial relations) for $G(A)$ in the squarefree $3$-Veronese and higher cases are not computed in the existing literature. The primary results to date concern the enumeration of generators and the proof of combinatorial and algebraic properties in low dimension (Bandari et al., 25 Dec 2025, Herzog et al., 2018). For $r\geq 3$ and large $d$, the exact form of $G(A)$'s generators remains the subject of continuing investigation. The broader connection to the geometry of the Gauss map and the combinatorics of polymatroids constitutes a significant strand of current research.