Fold Products of Linear Forms

• Fold products of linear forms are homogeneous ideals generated by multiplying distinct linear forms, bridging the gap between complete intersections and principal products.
• They exhibit rigid linear graded free resolutions with explicit binomial formulas for Betti numbers that underline their homological structure.
• These ideals connect to star configurations, Veronese-type constructions, and Orlik–Terao theory, offering insights into symbolic powers and arrangement theory.

Searching arXiv for 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 {"^^^^^^^^4query4 products of linear forms\" OR ^^^^^^^^4all:\4 products of linear forms\"^^^^^^^^4 OR ti:\4^^^^^^^^"Betti numbers of fold products of linear forms\"","max_results":^^^^4all:\4query4^^^^,"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 PRESERVED_PLACEHOLDER_^^^^4query4^^^^, or PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^, one fixes a collection PRESERVED_PLACEHOLDER_^^^^4 OR all:\4^^^^ or PRESERVED_PLACEHOLDER_^^^^4 OR ti:\4^^^^ 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 (&&&^^^^4query4^^^^&&&, &&&^^^^4all:\4^^^^&&&, &&&^^^^4 OR all:\4^^^^&&&). These ideals interpolate between several familiar constructions in commutative algebra and algebraic geometry, notably complete intersections, star-configuration ideals, and Veronese-type ideals (&&&^^^^4query4^^^^&&&). 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 (&&&^^^^4all:\4^^^^&&&, &&&^^^^4query4^^^^&&&).

^^^^4all:\4^^^^. 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 PRESERVED_PLACEHOLDER_^^^^4all:\4query4^^^^, an PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4^^^^-fold product is a monomial

PRESERVED_PLACEHOLDER_^^^^4all:\4 OR all:\4^^^^

and the associated ideal is

PRESERVED_PLACEHOLDER_^^^^4all:\4 OR ti:\4^^^^

(&&&^^^^4query4^^^^&&&). In the notation of Burity–Tohăneanu–Xie, if PRESERVED_PLACEHOLDER_^^^^4all:\44^^^^ and PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^5 is a finite multiset of linear forms, possibly with proportional repeats, then

PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^6

with the conventions PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^7 and PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^8 for PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^9 (&&&^^^^4all:\4^^^^&&&). The products PRESERVED_PLACEHOLDER_^^^^4 OR all:\4query4^^^^ are called standard generators (&&&^^^^4all:\4^^^^&&&).

By construction, PRESERVED_PLACEHOLDER_^^^^4 OR all:\4all:\4^^^^ is generated in degree PRESERVED_PLACEHOLDER_^^^^4 OR all:\4 OR all:\4^^^^, and in general position its minimal generators are exactly the PRESERVED_PLACEHOLDER_^^^^4 OR all:\4 OR ti:\4^^^^ products PRESERVED_PLACEHOLDER_^^^^4 OR all:\44^^^^ indexed by PRESERVED_PLACEHOLDER_^^^^4 OR all:\4^^^^5-subsets PRESERVED_PLACEHOLDER_^^^^4 OR all:\4^^^^6 (&&&^^^^4query4^^^^&&&). The number of generators is therefore

PRESERVED_PLACEHOLDER_^^^^4 OR all:\4^^^^7

in the generic case (&&&^^^^4query4^^^^&&&).

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.

^^^^4 OR all:\4^^^^. Linear graded free resolutions

A homogeneous ideal PRESERVED_PLACEHOLDER_^^^^4 OR all:\4^^^^8 generated in degree PRESERVED_PLACEHOLDER_^^^^4 OR all:\4^^^^9 has a linear graded free resolution if its minimal free resolution has the form

PRESERVED_PLACEHOLDER_^^^^4 OR ti:\4query4^^^^

(&&&^^^^4 OR all:\4^^^^&&&). For fold-product ideals, linearity is the principal structural theorem.

Burity–Tohăneanu–Xie proved that for any field PRESERVED_PLACEHOLDER_^^^^4 OR ti:\4all:\4^^^^ of characteristic PRESERVED_PLACEHOLDER_^^^^4 OR ti:\4 OR all:\4^^^^, any collection PRESERVED_PLACEHOLDER_^^^^4 OR ti:\4 OR ti:\4^^^^ of PRESERVED_PLACEHOLDER_^^^^4 OR ti:\44^^^^ linear forms in PRESERVED_PLACEHOLDER_^^^^4 OR ti:\4^^^^5, and any PRESERVED_PLACEHOLDER_^^^^4 OR ti:\4^^^^6, the ideal PRESERVED_PLACEHOLDER_^^^^4 OR ti:\4^^^^7 has a minimal graded free resolution which is linear of degree PRESERVED_PLACEHOLDER_^^^^4 OR ti:\4^^^^8 (&&&^^^^4all:\4^^^^&&&). Equivalently,

PRESERVED_PLACEHOLDER_^^^^4 OR ti:\4^^^^9

(&&&^^^^4all:\4^^^^&&&). This result subsumes earlier cases and removes restrictions such as nonproportionality or low ambient dimension.

An earlier paper established two broad cases. First, when $r$^^^^4query4^^^^ and no two $r$^^^^4all:\4^^^^ are proportional, the ideal generated by all $r$^^^^4 OR all:\4^^^^-fold products has linear graded free resolution (&&&^^^^4 OR all:\4^^^^&&&). Second, when $r$^^^^4 OR ti:\4^^^^, for any collection $r$4, possibly with repetitions, and any $r$5, the ideal generated by $r$6-fold products has linear free resolution (&&&^^^^4 OR all:\4^^^^&&&). The later theorem of (&&&^^^^4all:\4^^^^&&&) extends these patterns to arbitrary $r$7, arbitrary collections, and arbitrary polynomial rings in characteristic $r$8.

In the generic hyperplane-arrangement case, the resolution is not only linear but pure. If $r$9 spans the irrelevant maximal ideal $a$^^^^4query4^^^^ and no two $a$^^^^4all:\4^^^^ are proportional, then for $a$^^^^4 OR all:\4^^^^, $a$^^^^4 OR ti:\4^^^^ has a pure linear minimal free resolution of length

$a$4

(&&&^^^^4query4^^^^&&&). This gives a particularly rigid form of the homological data.

^^^^4 OR ti:\4^^^^. Betti numbers and explicit minimal resolutions

The most explicit recent advance is the determination of graded Betti numbers in the generic case. For $a$5 spanning $a$6, with no two proportional, the only nonzero graded Betti numbers of $a$7 are

$a$8

and $a$9 otherwise (&&&^^^^4query4^^^^&&&). 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 $r$^^^^4query4^^^^. If $r$^^^^4all:\4^^^^ is a hyperplane arrangement of rank $r$^^^^4 OR all:\4^^^^, $r$^^^^4 OR ti:\4^^^^, and $r$4, then $r$5 has a linear free resolution of length $r$6: $r$7 where

$r$8

with $r$9 and $a$^^^^4query4^^^^ the dimension of the space of $a$^^^^4all:\4^^^^-dependencies among the $a$^^^^4 OR all:\4^^^^ (&&&^^^^4 OR all:\4^^^^&&&). In this case, all syzygies among the generators $a$^^^^4 OR ti:\4^^^^ are linear (&&&^^^^4 OR all:\4^^^^&&&).

The generic formula of (&&&^^^^4query4^^^^&&&) and the arrangement-theoretic resolution of (&&&^^^^4 OR all:\4^^^^&&&) describe complementary regimes. The former gives closed Betti formulas for general $a$4 under genericity assumptions; the latter isolates the homological role of $a$5-dependencies in the extremal $a$6-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 (&&&^^^^4all:\4^^^^&&&) proceeds by induction on the pair $a$7. The key exact-sequence input is the colon identity

$a$8

which yields

$a$9

(&&&^^^^4all:\4^^^^&&&). Together with regularity bounds and a mapping-cone argument, this proves $\Sigma=(\ell_1,\ell_2,\dots,\ell_k)$^^^^4query4^^^^ and hence linearity (&&&^^^^4all:\4^^^^&&&).

The same work also proves the intersection formula

$\Sigma=(\ell_1,\ell_2,\dots,\ell_k)$^^^^4all:\4^^^^

where $\Sigma=(\ell_1,\ell_2,\dots,\ell_k)$^^^^4 OR all:\4^^^^ is the set of all linear primes generated by subsets of $\Sigma=(\ell_1,\ell_2,\dots,\ell_k)$^^^^4 OR ti:\4^^^^, counted with multiplicity, and

$\Sigma=(\ell_1,\ell_2,\dots,\ell_k)$4

(&&&^^^^4all:\4^^^^&&&). 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 $\Sigma=(\ell_1,\ell_2,\dots,\ell_k)$5-fold arrangement case, the proof uses an exact complex built from circuit data. Writing $\Sigma=(\ell_1,\ell_2,\dots,\ell_k)$6 and $\Sigma=(\ell_1,\ell_2,\dots,\ell_k)$7, one constructs maps $\Sigma=(\ell_1,\ell_2,\dots,\ell_k)$8 and $\Sigma=(\ell_1,\ell_2,\dots,\ell_k)$9 such that $S=\mathbb{k}[x_0,x_1,\dots,x_n]$^^^^4query4^^^^, while $S=\mathbb{k}[x_0,x_1,\dots,x_n]$^^^^4all:\4^^^^ records the linear relations arising from unique $S=\mathbb{k}[x_0,x_1,\dots,x_n]$^^^^4 OR all:\4^^^^-dependencies $S=\mathbb{k}[x_0,x_1,\dots,x_n]$^^^^4 OR ti:\4^^^^ (&&&^^^^4 OR all:\4^^^^&&&). The identity $S=\mathbb{k}[x_0,x_1,\dots,x_n]$4 shows that the only linear relations among the $S=\mathbb{k}[x_0,x_1,\dots,x_n]$5 come from $S=\mathbb{k}[x_0,x_1,\dots,x_n]$6-circuits (&&&^^^^4 OR all:\4^^^^&&&). Lifting the resulting complex by tensoring with Koszul resolutions and taking mapping cones yields the minimal linear resolution (&&&^^^^4 OR all:\4^^^^&&&).

For the generic Betti-number theorem, (&&&^^^^4query4^^^^&&&) states a sketch based on linear powers, purity, and combinatorial enumeration. One first shows, by induction or by deletion–contraction, that $S=\mathbb{k}[x_0,x_1,\dots,x_n]$7 has linear powers, so each power $S=\mathbb{k}[x_0,x_1,\dots,x_n]$8 has a linear resolution. By Eisenbud–Goto or Herzog–Kühl one then deduces that the resolution of $S=\mathbb{k}[x_0,x_1,\dots,x_n]$9 is pure of degree PRESERVED_PLACEHOLDER_^^^^4all:\4query4query4^^^^, and a combinatorial count or direct mapping-cone argument yields the explicit binomial formula for PRESERVED_PLACEHOLDER_^^^^4all:\4query4all:\4^^^^ (&&&^^^^4query4^^^^&&&).

5. Examples and limiting cases

Two worked examples from (&&&^^^^4query4^^^^&&&) illustrate the generic formulas.

Case	Ideal	Minimal free resolution
PRESERVED_PLACEHOLDER_^^^^4all:\4query4 OR all:\4^^^^	PRESERVED_PLACEHOLDER_^^^^4all:\4query4 OR ti:\4^^^^	PRESERVED_PLACEHOLDER_^^^^4all:\4query44^^^^
PRESERVED_PLACEHOLDER_^^^^4all:\4query4^^^^5	PRESERVED_PLACEHOLDER_^^^^4all:\4query4^^^^6	PRESERVED_PLACEHOLDER_^^^^4all:\4query4^^^^7

For PRESERVED_PLACEHOLDER_^^^^4all:\4query4^^^^8, one has

PRESERVED_PLACEHOLDER_^^^^4all:\4query4^^^^9

and the ideal is the classical star configuration of three lines in PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4query4^^^^ (&&&^^^^4query4^^^^&&&). For PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4all:\4^^^^, the six quadrics generate PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4 OR all:\4^^^^, and the Betti numbers are

PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4 OR ti:\4^^^^

(&&&^^^^4query4^^^^&&&).

A basic non-generic example appears in PRESERVED_PLACEHOLDER_^^^^4all:\4all:\44^^^^ with PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4^^^^5 and PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4^^^^6. The standard generators are PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4^^^^7, PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4^^^^8, and PRESERVED_PLACEHOLDER_^^^^4all:\4all:\4^^^^9, and one checks directly that

PRESERVED_PLACEHOLDER_^^^^4all:\4 OR all:\4query4^^^^

(&&&^^^^4all:\4^^^^&&&). Since PRESERVED_PLACEHOLDER_^^^^4all:\4 OR all:\4all:\4^^^^ has the linear resolution

PRESERVED_PLACEHOLDER_^^^^4all:\4 OR all:\4 OR all:\4^^^^

this exhibits linearity concretely (&&&^^^^4all:\4^^^^&&&).

The limiting cases organize the construction. If PRESERVED_PLACEHOLDER_^^^^4all:\4 OR all:\4 OR ti:\4^^^^, then PRESERVED_PLACEHOLDER_^^^^4all:\4 OR all:\44^^^^, while if PRESERVED_PLACEHOLDER_^^^^4all:\4 OR all:\4^^^^5, then PRESERVED_PLACEHOLDER_^^^^4all:\4 OR all:\4^^^^6 is principal (&&&^^^^4query4^^^^&&&). The first is described in (&&&^^^^4query4^^^^&&&) as a complete intersection of PRESERVED_PLACEHOLDER_^^^^4all:\4 OR all:\4^^^^7 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 PRESERVED_PLACEHOLDER_^^^^4all:\4 OR all:\4^^^^8 is the defining linear forms of an essential arrangement of PRESERVED_PLACEHOLDER_^^^^4all:\4 OR all:\4^^^^9 hyperplanes in PRESERVED_PLACEHOLDER_^^^^4all:\4 OR ti:\4query4^^^^, then the ideal

PRESERVED_PLACEHOLDER_^^^^4all:\4 OR ti:\4all:\4^^^^

cuts out the classical codimension-PRESERVED_PLACEHOLDER_^^^^4all:\4 OR ti:\4 OR all:\4^^^^ star configuration of linear subspaces (&&&^^^^4query4^^^^&&&). Thus star-configuration ideals arise as a distinguished family of fold-product ideals.

If PRESERVED_PLACEHOLDER_^^^^4all:\4 OR ti:\4 OR ti:\4^^^^ consists of PRESERVED_PLACEHOLDER_^^^^4all:\4 OR ti:\44^^^^ copies of the coordinate form PRESERVED_PLACEHOLDER_^^^^4all:\4 OR ti:\4^^^^5, then

PRESERVED_PLACEHOLDER_^^^^4all:\4 OR ti:\4^^^^6

which is exactly a Veronese-type ideal in the sense of Abdollmaleki–Zaare-Nahandi (&&&^^^^4query4^^^^&&&). The Betti-number formula in (&&&^^^^4query4^^^^&&&) specializes to their known formulas in that case.

The PRESERVED_PLACEHOLDER_^^^^4all:\4 OR ti:\4^^^^7-fold case is also tied to Orlik–Terao theory. For a hyperplane arrangement PRESERVED_PLACEHOLDER_^^^^4all:\4 OR ti:\4^^^^8, the second-order Orlik–Terao algebra is

PRESERVED_PLACEHOLDER_^^^^4all:\4 OR ti:\4^^^^9

and it is isomorphic to the special fiber PRESERVED_PLACEHOLDER_^^^^4all:\44query4^^^^ (&&&^^^^4 OR all:\4^^^^&&&). The presentation ideal PRESERVED_PLACEHOLDER_^^^^4all:\44all:\4^^^^ contains linear generators coming from PRESERVED_PLACEHOLDER_^^^^4all:\44 OR all:\4^^^^-circuits and quadratic Plücker-type generators indexed by four distinct indices (&&&^^^^4 OR all:\4^^^^&&&). Via Sylvester forms, one can recover generators of PRESERVED_PLACEHOLDER_^^^^4all:\44 OR ti:\4^^^^ from the linear relations in the symmetric ideal PRESERVED_PLACEHOLDER_^^^^4all:\444^^^^ (&&&^^^^4 OR all:\4^^^^&&&).

The ^^^^4 OR all:\4query4 OR all:\4query4^^^^ paper further states that when PRESERVED_PLACEHOLDER_^^^^4all:\44^^^^5 defines a line arrangement PRESERVED_PLACEHOLDER_^^^^4all:\44^^^^6 and PRESERVED_PLACEHOLDER_^^^^4all:\44^^^^7, explicit generators of the defining ideal of the special fiber PRESERVED_PLACEHOLDER_^^^^4all:\44^^^^8 are determined, recovering and extending conjectures in the literature (&&&^^^^4all:\4^^^^&&&). In the case PRESERVED_PLACEHOLDER_^^^^4all:\44^^^^9 and PRESERVED_PLACEHOLDER_^^^^4all:\4max_results4query4^^^^, PRESERVED_PLACEHOLDER_^^^^4all:\4max_results4all:\4^^^^ is shown to be of fiber type, meaning that the Rees ideal is generated by the linear syzygies plus the special-fiber equations (&&&^^^^4all:\4^^^^&&&).

7. Scope, applications, and adjacent notions

The applications described in (&&&^^^^4all:\4^^^^&&&) extend beyond free resolutions. For a collection of PRESERVED_PLACEHOLDER_^^^^4all:\4max_results4 OR all:\4^^^^ hyperplanes in PRESERVED_PLACEHOLDER_^^^^4all:\4max_results4 OR ti:\4^^^^ meeting properly, with defining ideal PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^54 of the codimension-PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^55 star configuration PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^56, the paper proves containments of symbolic and ordinary powers predicted by Harbourne–Huneke and verifies Chudnovsky’s and Demailly’s conjectures on PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^57-invariants (&&&^^^^4all:\4^^^^&&&). 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

PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^58

where the field norm factors as

PRESERVED_PLACEHOLDER_^^^^4all:\4^^^^59

after choosing an integral basis and embeddings PRESERVED_PLACEHOLDER_^^^^4all:\4sort_by4query4^^^^ (Browning et al., 2013). 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 (Browning et al., 2013). The common phrase “product of linear forms” therefore refers to different mathematical structures in the commutative-algebraic and arithmetic contexts.

The paper “Betti numbers of fold products of linear forms” (&&&^^^^4query4^^^^&&&) indicates that the subject has moved from proving linearity toward finer homological invariants. Combined with the structural theorem of (&&&^^^^4all:\4^^^^&&&) and the arrangement-theoretic analysis of (&&&^^^^4 OR all:\4^^^^&&&), 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.