---
title: Fold Products of Linear Forms
url: https://www.emergentmind.com/topics/fold-products-of-linear-forms
type: topic
---

# Fold Products of Linear Forms

Searching arXiv for recent papers on fold products of linear forms and related ideals.
to=arxiv_search.search  天天中彩票官方  彩神争霸平台json
{"query":"all:\"fold products of linear forms\" OR all:\"a-fold products of linear forms\" OR ti:\"Betti numbers of fold products of linear forms\"","max_results":10,"sort_by":"submittedDate","sort_order":"descending"}
Fold products of linear forms are a class of homogeneous ideals obtained by multiplying selected subsets of a finite collection of linear forms and taking the ideal generated by all such products. In the standard graded polynomial ring \(S=\mathbb{k}[x_0,x_1,\dots,x_n]\), or \(R=K[x_1,\dots,x_k]\), one fixes a collection \(\Sigma=(\ell_1,\dots,\ell_k)\) or \(E=(\ell_1,\dots,\ell_n)\) of linear forms and, for an integer \(r\) or \(a\), defines the corresponding fold-product ideal by the degree-\(r\) or degree-\(a\) products of distinct members of the collection [2507.07294], [2004.07430], [1807.08021]. These ideals interpolate between several familiar constructions in commutative algebra and algebraic geometry, notably complete intersections, star-configuration ideals, and Veronese-type ideals [2507.07294]. A central theme of the subject is that these ideals exhibit unexpectedly rigid homological behavior: in broad generality they have linear graded free resolutions, and in generic settings their graded Betti numbers admit explicit closed formulas [2004.07430], [2507.07294].

## 1. Definitions and basic construction

Let \(\Sigma=(\ell_1,\ell_2,\dots,\ell_k)\) be a finite collection of linear forms in \(S=\mathbb{k}[x_0,x_1,\dots,x_n]\). For any integer \(1\le r\le k\), an \(r\)-fold product is a monomial
\[
\ell_{i_1}\ell_{i_2}\cdots \ell_{i_r}, \qquad 1\le i_1<\cdots<i_r\le k,
\]
and the associated ideal is
\[
I_{r,k}=I_r(\Sigma)=\bigl\langle \ell_{i_1}\ell_{i_2}\cdots\ell_{i_r}\mid 1\le i_1<\cdots<i_r\le k\bigr\rangle\subset S
\]
[2507.07294]. In the notation of Burity–Tohăneanu–Xie, if \(R=K[x_1,\dots,x_k]\) and \(\Sigma=(\ell_1,\dots,\ell_n)\) is a finite multiset of linear forms, possibly with proportional repeats, then
\[
I_a(\Sigma):=(\ell_{i_1}\ell_{i_2}\cdots\ell_{i_a}:1\le i_1<\cdots<i_a\le n)\subset R,
\]
with the conventions \(I_0(\Sigma)=R\) and \(I_a(\Sigma)=0\) for \(a>n\) [2004.07430]. The products \(\ell_{i_1}\cdots\ell_{i_a}\) are called standard generators [2004.07430].

By construction, \(I_{r,k}\) is generated in degree \(r\), and in general position its minimal generators are exactly the \(\binom{k}{r}\) products \(\prod_{i\in J}\ell_i\) indexed by \(r\)-subsets \(J\subset\{1,\dots,k\}\) [2507.07294]. The number of generators is therefore
\[
\beta_{1,r}(I_{r,k})=\binom{k}{r}
\]
in the generic case [2507.07294].

The terminology “fold product” emphasizes that the ideal is defined by products of several distinct linear factors rather than by powers of a single ideal generator. This suggests an interpolation between combinatorial constructions indexed by subsets and classical homological invariants attached to powers and symbolic powers of linear-prime ideals.

## 2. Linear graded free resolutions

A homogeneous ideal \(I\subset R\) generated in degree \(d\) has a linear graded free resolution if its minimal free resolution has the form
\[
0\to R^{\beta_p}(-(d+p))\to\cdots\to R^{\beta_1}(-(d+1))\to R^{\beta_0}(-d)\to R/I\to 0
\]
[1807.08021]. For fold-product ideals, linearity is the principal structural theorem.

Burity–Tohăneanu–Xie proved that for any field \(K\) of characteristic \(0\), any collection \(\Sigma\) of \(n\) linear forms in \(R=K[x_1,\dots,x_k]\), and any \(1\le a\le n\), the ideal \(I_a(\Sigma)\) has a minimal graded free resolution which is linear of degree \(a\) [2004.07430]. Equivalently,
\[
\operatorname{reg} R/I_a(\Sigma)=a-1
\]
[2004.07430]. This result subsumes earlier cases and removes restrictions such as nonproportionality or low ambient dimension.

An earlier paper established two broad cases. First, when \(a=n-2\) and no two \(\ell_i\) are proportional, the ideal generated by all \((n-2)\)-fold products has linear graded free resolution [1807.08021]. Second, when \(k=2\), for any collection \(E\subset K[x,y]_1\), possibly with repetitions, and any \(1\le a\le n\), the ideal generated by \(a\)-fold products has linear free resolution [1807.08021]. The later theorem of [2004.07430] extends these patterns to arbitrary \(a\), arbitrary collections, and arbitrary polynomial rings in characteristic \(0\).

In the generic hyperplane-arrangement case, the resolution is not only linear but pure. If \(\Sigma=(\ell_1,\dots,\ell_k)\) spans the irrelevant maximal ideal \(\langle x_0,\dots,x_n\rangle\) and no two \(\ell_i\) are proportional, then for \(1\le r\le k\), \(I_{r,k}\) has a pure linear minimal free resolution of length
\[
p=\mathrm{pdim}_S(S/I_{r,k})=\min\{k,k-r+1\}=k-r+1
\]
[2507.07294]. This gives a particularly rigid form of the homological data.

## 3. Betti numbers and explicit minimal resolutions

The most explicit recent advance is the determination of graded Betti numbers in the generic case. For \(\Sigma=(\ell_1,\dots,\ell_k)\) spanning \(\langle x_0,\dots,x_n\rangle\), with no two proportional, the only nonzero graded Betti numbers of \(I_{r,k}\) are
\[
\beta_{i,i+r-1}(I_{r,k})=\binom{k}{i+r-1}\binom{i+r-2}{r-1}, \qquad i=1,2,\dots,k-r+1,
\]
and \(\beta_{i,j}(I_{r,k})=0\) otherwise [2507.07294]. The minimal free resolution is therefore pure and linear, with all shifts determined by the single binomial-product formula.

Earlier work had already given a detailed explicit resolution in the special case \(a=n-2\). If \(A=\{\ell_1,\dots,\ell_n\}\subset R_1\) is a hyperplane arrangement of rank \(k\), \(f=\ell_1\ell_2\cdots\ell_n\), and \(f_{i,j}=f/(\ell_i\ell_j)\), then \(R/I_{n-2}(A)\) has a linear free resolution of length \(3\):
\[
0\to R^{\beta_3}(-n)\to R^{\beta_2}(-(n-1))\to R^{\beta_1}(-(n-2))\to R\to R/I_{n-2}(A)\to 0,
\]
where
\[
\beta_1=m-p(A),\qquad \beta_2=2m-n-2p(A)=n(n-2)-2p(A),\qquad \beta_3=n-p(A)-1,
\]
with \(m=n(n-1)/2\) and \(p(A)\) the dimension of the space of \(3\)-dependencies among the \(\ell_i\) [1807.08021]. In this case, all syzygies among the generators \(f_{i,j}\) are linear [1807.08021].

The generic formula of [2507.07294] and the arrangement-theoretic resolution of [1807.08021] describe complementary regimes. The former gives closed Betti formulas for general \(r\) under genericity assumptions; the latter isolates the homological role of \(3\)-dependencies in the extremal \((n-2)\)-fold case. A plausible implication is that dependence data among the linear forms governs deviations from the generic binomial pattern.

## 4. Proof methods and structural formulas

The proof of linearity in full generality in [2004.07430] proceeds by induction on the pair \((n,\operatorname{rank}(\Sigma))\). The key exact-sequence input is the colon identity
\[
I_a(\Sigma):\ell = I_{a-1}(\Sigma\setminus\{\ell\}),
\]
which yields
\[
0\to R(-1)/I_{a-1}(\Sigma\setminus\{\ell\}) \to R/I_a(\Sigma)\to R/(\ell,I_a(\Sigma))\to 0
\]
[2004.07430]. Together with regularity bounds and a mapping-cone argument, this proves \(\operatorname{reg}R/I_a(\Sigma)=a-1\) and hence linearity [2004.07430].

The same work also proves the intersection formula
\[
I_a(\Sigma)=\bigcap_{p\in T(\Sigma)} p^{\,a-n+v_\Sigma(p)},
\]
where \(T(\Sigma)\) is the set of all linear primes generated by subsets of \(\Sigma\), counted with multiplicity, and
\[
v_\Sigma(p)=\#\{\ell\in \Sigma\mid \ell\in p\}
\]
[2004.07430]. This expresses fold-product ideals as intersections of powers of linear primes and supplies a bridge to symbolic-power phenomena and Hilbert-series calculations.

In the \((n-2)\)-fold arrangement case, the proof uses an exact complex built from circuit data. Writing \(A(A)=\bigoplus_{1\le i<j\le n}R\cdot e_{i,j}\) and \(A^3(A)=\bigoplus_{\text{circuits }\{a<b<c\}}R\cdot \gamma_{a,b,c}\), one constructs maps \(\phi\) and \(\psi\) such that \(\psi(e_{i,j})=f_{i,j}\), while \(\phi\) records the linear relations arising from unique \(3\)-dependencies \(c_a\ell_a+c_b\ell_b+c_c\ell_c=0\) [1807.08021]. The identity \(\ker\psi=\operatorname{im}\phi\) shows that the only linear relations among the \(f_{i,j}\) come from \(3\)-circuits [1807.08021]. Lifting the resulting complex by tensoring with Koszul resolutions and taking mapping cones yields the minimal linear resolution [1807.08021].

For the generic Betti-number theorem, [2507.07294] states a sketch based on linear powers, purity, and combinatorial enumeration. One first shows, by induction or by deletion–contraction, that \(I_{r,k}\) has linear powers, so each power \(I_{r,k}^u\) has a linear resolution. By Eisenbud–Goto or Herzog–Kühl one then deduces that the resolution of \(I_{r,k}\) is pure of degree \(r\), and a combinatorial count or direct mapping-cone argument yields the explicit binomial formula for \(\beta_{i,i+r-1}\) [2507.07294].

## 5. Examples and limiting cases

Two worked examples from [2507.07294] illustrate the generic formulas.

| Case | Ideal | Minimal free resolution |
|---|---|---|
| \(r=2,\ k=3\) | \(I_{2,3}=\langle \ell_1\ell_2,\ell_1\ell_3,\ell_2\ell_3\rangle\) | \(0\to S(-3)^2\to S(-2)^3\to I_{2,3}\to 0\) |
| \(r=2,\ k=4\) | \(I_{2,4}=\langle \ell_i\ell_j\mid i<j\rangle\) | \(0\to S(-4)^3\to S(-3)^8\to S(-2)^6\to I_{2,4}\to 0\) |

For \(r=2,\ k=3\), one has
\[
\beta_{1,2}=3,\qquad \beta_{2,3}=2,\qquad \beta_{3,4}=0,
\]
and the ideal is the classical star configuration of three lines in \(\mathbb{P}^2\) [2507.07294]. For \(r=2,\ k=4\), the six quadrics generate \(I_{2,4}\), and the Betti numbers are
\[
\beta_{1,2}=6,\qquad \beta_{2,3}=8,\qquad \beta_{3,4}=3,\qquad \beta_{4,5}=0
\]
[2507.07294].

A basic non-generic example appears in \(R=K[x,y]\) with \(\Sigma=\{x,y,x+y\}\) and \(a=2\). The standard generators are \(xy\), \(x(x+y)=x^2+xy\), and \(y(x+y)=xy+y^2\), and one checks directly that
\[
I_2(\Sigma)=(x^2,xy,y^2)=(x,y)^2
\]
[2004.07430]. Since \((x,y)^2\) has the linear resolution
\[
0\to R(-3)\to R^2(-2)\to (x,y)^2\to 0,
\]
this exhibits linearity concretely [2004.07430].

The limiting cases organize the construction. If \(r=1\), then \(I_{1,k}=(\ell_1,\dots,\ell_k)\), while if \(r=k\), then \(I_{k,k}=(\ell_1\ell_2\cdots\ell_k)\) is principal [2507.07294]. The first is described in [2507.07294] as a complete intersection of \(k\) linear forms, and the second is the one-generator extreme. In this sense fold-product ideals interpolate between linear ideals and principal products.

## 6. Relations to star configurations, Veronese-type ideals, and Orlik–Terao theory

Fold-product ideals are explicitly linked to several established constructions. If \(\Sigma\) is the defining linear forms of an essential arrangement of \(n\) hyperplanes in \(\mathbb{P}^{k-1}\), then the ideal
\[
I_{n-c+1}(\Sigma)
\]
cuts out the classical codimension-\(c\) star configuration of linear subspaces [2507.07294]. Thus star-configuration ideals arise as a distinguished family of fold-product ideals.

If \(\Sigma\) consists of \(m_i\) copies of the coordinate form \(x_i\), then
\[
I_{r,k}(\Sigma)=\bigl\langle x_0^{a_0}x_1^{a_1}\cdots x_n^{a_n}\mid 0\le a_i\le m_i,\ \sum a_i=r\bigr\rangle,
\]
which is exactly a Veronese-type ideal in the sense of Abdollmaleki–Zaare-Nahandi [2507.07294]. The Betti-number formula in [2507.07294] specializes to their known formulas in that case.

The \((n-2)\)-fold case is also tied to Orlik–Terao theory. For a hyperplane arrangement \(A\), the second-order Orlik–Terao algebra is
\[
OT(2,A)=K[1/(\ell_i\ell_j)\mid 1\le i<j\le n],
\]
and it is isomorphic to the special fiber \(F(I_{n-2}(A))=\bigoplus_{d\ge 0}(I_{n-2}^d/\mathfrak{m}I_{n-2}^d)\) [1807.08021]. The presentation ideal \(I(2,A)\) contains linear generators coming from \(3\)-circuits and quadratic Plücker-type generators indexed by four distinct indices [1807.08021]. Via Sylvester forms, one can recover generators of \(I(2,A)\) from the linear relations in the symmetric ideal \(\mathrm{Sym}(I_{n-2}(A))\) [1807.08021].

The 2020 paper further states that when \(\Sigma\) defines a line arrangement \(A\subset \mathbb{P}^2\) and \(a=|A|-2\), explicit generators of the defining ideal of the special fiber \(R[I_a(A)t]/\mathfrak{m}R[I_a(A)t]\) are determined, recovering and extending conjectures in the literature [2004.07430]. In the case \(k=3\) and \(a=s-2\), \(I_{s-2}(\Sigma)\) is shown to be of fiber type, meaning that the Rees ideal is generated by the linear syzygies plus the special-fiber equations [2004.07430].

## 7. Scope, applications, and adjacent notions

The applications described in [2004.07430] extend beyond free resolutions. For a collection of \(s\) hyperplanes in \(\mathbb{P}^N\) meeting properly, with defining ideal \(I\) of the codimension-\(c\) star configuration \(V_c\), the paper proves containments of symbolic and ordinary powers predicted by Harbourne–Huneke and verifies Chudnovsky’s and Demailly’s conjectures on \(a\)-invariants [2004.07430]. This places fold-product ideals within the study of symbolic powers and asymptotic invariants of subspace arrangements.

A potential source of confusion is the phrase “products of linear forms” in arithmetic geometry. Browning–Matthiesen study affine varieties defined by
\[
N_{K/\mathbb{Q}}(x_1,\dots,x_n)=P(t),
\]
where the field norm factors as
\[
N_{K/\mathbb{Q}}(x)=\prod_{i=1}^n \ell_i(x)
\]
after choosing an integral basis and embeddings \(K\hookrightarrow \mathbb{C}\) [1307.7641]. That setting concerns norm forms, descent on torsors, and the Brauer–Manin obstruction, rather than ideals generated by fold products in a graded polynomial ring [1307.7641]. The common phrase “product of linear forms” therefore refers to different mathematical structures in the commutative-algebraic and arithmetic contexts.

The recent paper “Betti numbers of fold products of linear forms” [2507.07294] indicates that the subject has moved from proving linearity toward finer homological invariants. Combined with the structural theorem of [2004.07430] and the arrangement-theoretic analysis of [1807.08021], this suggests a mature framework in which fold-product ideals are understood simultaneously through combinatorial generation, intersection decompositions by linear primes, and explicit syzygetic formulas.

Source: https://www.emergentmind.com/topics/fold-products-of-linear-forms