Syzygy Bundle Research Articles

Matrices of linear forms of constant rank from vector bundles on projective spaces

We consider the problem of constructing matrices of linear forms of constant rank by focusing on the associated vector bundles on projective spaces. Important examples are given by the classical Steiner bundles, as well as some special (duals of) syzygy bundles that we call Drézet bundles. Using the classification of globally generated vector bundles with small first Chern class on projective spaces, we are able to describe completely the indecomposable matrices of constant rank up to six; some of them come from rigid homogeneous vector bundles, some other from Drézet bundles related either to plane quartics or to instanton bundles on P3.

Stability of Syzygy bundles corresponding to stable vector bundles on algebraic surfaces

Let (X,H) be a polarized smooth projective algebraic surface and E is globally generated, stable vector bundle on X. Then the Syzygy bundle ME associated to it is defined as the kernel bundle corresponding to the evaluation map. In this article we will study the stability property of ME with respect to H.

Ein–Lazarsfeld–Mustopa conjecture for the blow-up of a projective space

We solve the Ein–Lazarsfeld–Mustopa conjecture for the blow up of a projective space along a linear subspace. More precisely, let X be the blow up of {mathbb {P}}^n at a linear subspace and let L be any ample line bundle on X. We show that the syzygy bundle M_{L} defined as the kernel of the evalution map H^{0}(X,L)otimes {mathcal {O}}_{X}rightarrow L is L-stable. In the last part of this note we focus on the rigidness of M_{L} to study the local shape of the moduli space around the point [M_{L}].

Syzygy Bundles of Non-complete Linear Systems: Stability and Rigidness

Let (X, L) be a polarized smooth projective variety. For any basepoint-free linear system mathcal {L}_{V} with Vsubset {{,textrm{H},}}^{0}(X,mathcal {O}_{X}(L)), we consider the syzygy bundle M_{V} as the kernel of the evaluation map Votimes mathcal {O}_{X}rightarrow mathcal {O}_{X}(L). The purpose of this article is twofold. First, we assume that M_{V} is L-stable and prove that, in a wide family of projective varieties, it represents a smooth point [M_{V}] in the corresponding moduli space mathcal {M}. We compute the dimension of the irreducible component of mathcal {M} passing through [M_{V}] and whether it is an isolated point. It turns out that the rigidness of [M_{V}] is closely related to the completeness of the linear system mathcal {L}_{V}. In the second part of the paper, we address a question posed by Brenner regarding the stability of M_{V} when V is general enough. We answer this question for a large family of polarizations of X=mathbb {P}^{m}times mathbb {P}^{n}.

Scrollar invariants, syzygies and representations of the symmetric group

Abstract We give an explicit minimal graded free resolution, in terms of representations of the symmetric group S d {S_{d}} , of a Galois-theoretic configuration of d points in 𝐏 d - 2 {\mathbf{P}^{d-2}} that was studied by Bhargava in the context of ring parametrizations. When applied to the geometric generic fiber of a simply branched degree d cover of 𝐏 1 {\mathbf{P}^{1}} by a relatively canonically embedded curve C, our construction gives a new interpretation for the splitting types of the syzygy bundles appearing in its relative minimal resolution. Concretely, our work implies that all these splitting types consist of scrollar invariants of resolvent covers. This vastly generalizes a prior observation due to Casnati, namely that the first syzygy bundle of a degree 4 cover splits according to the scrollar invariants of its cubic resolvent. Our work also shows that the splitting types of the syzygy bundles, together with the multi-set of scrollar invariants, belong to a much larger class of multi-sets of invariants that can be attached to C → 𝐏 1 {C\to\mathbf{P}^{1}} : one for each irreducible representation of S d {S_{d}} , i.e., one for each partition of d.

A note on stability of syzygy bundles on Enriques and bielliptic surfaces

In this note, we prove that the syzygy bundle M L M_L is cohomologically stable with respect to L L for any ample and globally generated line bundle L L on an Enriques (resp. bielliptic) surface over an algebraically closed field of characteristic ≠ 2 \neq 2 (resp. ≠ 2 , 3 \neq 2,3 ). In particular our result on complex Enriques surfaces improves the result of Torres-López and Zamora [Beitr. Algebra Geom. (2021)] (Corollary 3.5) by removing the condition on Clifford index. Together with the results of Camere [Math. Z 271 (2012), pp. 499–507] and Caucci–Lahoz [Bull. Lond. Math. Soc. 53 (2021), pp. 1030–1036], it implies that M L M_L is stable with respect to L L for an ample and globally generated line bundle L L on any smooth minimal complex projective surface X X of Kodaira dimension zero.

F-thresholds c(m) for projective curves

We show that if R is a two dimensional standard graded domain (with the graded maximal ideal m ) of characteristic p > 0 and I ⊂ R is a graded ideal with ℓ ( R / I ) < ∞ , then the F -threshold c I ( m ) can be expressed in terms of a strong HN (Harder-Narasimhan) slope of the canonical syzygy bundle on Proj R . Thus c I ( m ) is a rational number. This gives us a well defined notion, of the F -threshold c I ( m ) in characteristic 0, in terms of a HN slope of the syzygy bundle on Proj R . This generalizes our earlier result (in [13] ) where we have shown that if I has homogeneous generators of the same degree, then the F -threshold c I ( m ) is expressed in terms of the minimal strong HN slope (in char p ) and in terms of the minimal HN slope (in char 0), respectively, of the canonical syzygy bundle on Proj R . In the present more general setting, the relevant slope may not be the minimal one. Here we also prove that, for a given pair ( R , I ) over a field of characteristic 0, if ( m p , I p ) is a reduction mod p of ( m , I ) then c I p ( m p ) ≠ c ∞ I ( m ) implies c I p ( m p ) has p in the denominator, for almost all p .

Strongly semistable reduction of syzygy bundles on plane curves

We investigate degenerations of syzygy bundles on plane curves over p $p$ -adic fields. We use Mustafin varieties which are degenerations of projective spaces to find a large family of models of plane curves over the ring of integers such that the special fibre consists of multiple projective lines meeting in one point. On such models we investigate vector bundles whose generic fibre is a syzygy bundle and which become trivial when restricted to each projective line in the special fibre. Hence these syzygy bundles have strongly semistable reduction. This investigation is motivated by the fundamental open problem in p $p$ -adic Simpson theory to determine the category of Higgs bundles corresponding to continuous representations of the étale fundamental group of a curve. Faltings' p $p$ -adic Simpson correspondence and work of Deninger and the second author shows that bundles with Higgs field zero and potentially strongly semistable reduction fall into this category. Hence the results in the present paper determine a class of syzygy bundles on plane curves giving rise to a p $p$ -adic local system. We apply our methods to a concrete example on the Fermat curve suggested by Brenner and prove that this bundle has potentially strongly semistable reduction.

Moduli of lattice polarized K3 surfaces via relative canonical resolutions

For a smooth canonically embedded curve C of genus 9 together with a pencil | L | of degree 6, we study the relative canonical resolution of C ⊂ X ⊂ P 8 , where X is the scroll swept out by the pencil | L | . We show that the second syzygy bundle in this resolution of C ⊂ X is unbalanced. The proof reveals a new geometric connection between the universal Brill–Noether variety W 9 , 6 1 and a moduli space F h of lattice polarized K 3 surfaces (for a certain rank 3 lattice h ). As a by-product we prove the unirationality of F h and show that W 9 , 6 1 is birational to a projective bundle over a moduli space of lattice polarized K 3 surfaces F h ′ for a certain rank 4 lattice h ′ which contains h as a sublattice .

H-stability of syzygy bundles on some regular algebraic surfaces

H-stability of syzygy bundles on some regular algebraic surfaces

Syzygy bundles and the weak Lefschetz property of monomial almost complete intersections

Deciding the presence of the weak Lefschetz property often is a challenging problem. Continuing studies of Brenner and Kaid (2007), Cook II and Nagel (2011) and Migliore, Miró-Roig, Murai and Nagel (2013) we carry out an in-depth study of Artinian monomial ideals with four generators in three variables. We use a connection to lozenge tilings to describe semistability of the syzygy bundle of such an ideal, to determine its generic splitting type, and to decide the presence of the weak Lefschetz property. We provide results in both characteristic zero and positive characteristic.

Stability of syzygy bundles on abelian varieties

We prove that the kernel of the evaluation morphism of global sections - namely the syzygy bundle - of a sufficiently ample line bundle on an abelian variety is stable. This settles a conjecture of Ein-Lazarsfeld-Mustopa, in the case of abelian varieties.

Hilbert-Kunz density functions and F-thresholds

The first author had shown earlier that for a standard graded ring R and a graded ideal I in characteristic p>0, with ℓ(R/I)<∞, there exists a compactly supported continuous function fR,I whose Riemann integral is the HK multiplicity eHK(R,I). We explore further some other invariants, namely the shape of the graph of fR,m (where m is the graded maximal ideal of R) and the maximum support (denoted as α(R,I)) of fR,I.In case R is a domain of dimension d≥2, we prove that (R,m) is a regular ring if and only if fR,m has a symmetry fR,m(x)=fR,m(d−x), for all x.If R is strongly F-regular on the punctured spectrum then we prove that the F-threshold cI(m) coincides with α(R,I).As a consequence, if R is a two dimensional domain and I is generated by homogeneous elements of the same degree, then we have (1) a formula for the F-threshold cI(m) in terms of the minimum strong Harder-Narasimhan slope of the syzygy bundle and (2) a well defined notion of the F-threshold cI(m) in characteristic 0.This characterisation readily computes cI(n)(m), for the set of all irreducible plane trinomials k[x,y,z]/(h), where m=(x,y,z) and I(n)=(xn,yn,zn).

Asymptotic socle behaviors for cones over curves in positive characteristic

This paper studies the distribution of socle degrees of $R/I^{[p^e]}$ when $e$ is large, for a homogeneous ideal $I$ in a two-dimensional standard-graded normal domain $R$ in positive characteristic $p$. We prove that the distribution is very much related to the asymptotic slopes of the syzygy bundle Syz$(I)$, which have been known to determine the Hilbert-Kunz multiplicity of $I$.

Invariants of a General Branched Cover of P1

Abstract We investigate the resolution of a general branched cover $\alpha \colon C \to \mathbf{P}^1$ in its relative canonical embedding $C \subset \mathbf{P} E$. We conjecture that the syzygy bundles appearing in the resolution are balanced for a general cover, provided that the genus is sufficiently large compared to the degree. We prove this for the Casnati–Ekedahl bundle, or bundle of quadrics $F$—the 1st bundle appearing in the resolution of the ideal of the relative canonical embedding. Furthermore, we prove the conjecture for all syzygy bundles in the resolution when the genus satisfies $g = 1 \mod d$.

New examples of theta divisors for some syzygy bundles

Let C be a smooth complex irreducible projective curve of genus g, and let $$(L,H^0(L))$$ be a generated complete linear series of type $$(d,r+1)$$ over C. The syzygy bundle, denoted by $$M_L$$ , is the kernel of the evaluation map . The aim of this paper is twofold. Firstly, to give new examples of stable syzygy bundles admitting a theta divisor over Petri curves. We prove that if $$M_L$$ is strictly semistable then $$M_L$$ admits a theta divisor. Secondly, to study the cohomological semistability of $$M_L$$ , and in this direction we give another proof of cohomological semistability of $$M_L$$ when L induces a birational map. This proof gives us precise conditions for the cohomological semistability of $$M_L$$ where such conditions agree with the semistability conditions for $$M_L$$ . Finally, we relate these two properties by showing that under certain conditions on Petri curves the cohomological semistability of $$M_L$$ implies the existence of reducible theta divisor for $$M_L$$ .

Linear stability and stability of syzygy bundles

Let [Formula: see text] be a smooth irreducible projective curve and let [Formula: see text] be a complete and generated linear series on [Formula: see text]. Denote by [Formula: see text] the kernel of the evaluation map [Formula: see text]. The exact sequence [Formula: see text] fits into a commutative diagram that we call the Butler’s diagram. This diagram induces in a natural way a multiplication map on global sections [Formula: see text], where [Formula: see text] is a subspace and [Formula: see text] is the dual of a subbundle [Formula: see text]. When the subbundle [Formula: see text] is a stable bundle, we show that the map [Formula: see text] is surjective. When [Formula: see text] is a Brill–Noether general curve, we use the surjectivity of [Formula: see text] to give another proof of the semistability of [Formula: see text], moreover, we fill up a gap in some incomplete argument by Butler: With the surjectivity of [Formula: see text] we give conditions to determine the stability of [Formula: see text], and such conditions imply the well-known stability conditions for [Formula: see text] stated precisely by Butler. Finally we obtain the equivalence between the (semi)stability of [Formula: see text] and the linear (semi)stability of [Formula: see text] on [Formula: see text]-gonal curves.

Arithmetic behaviour of Frobenius semistability of syzygy bundles for plane trinomial curves

Here we consider the set of bundles {Vn}n∈N associated to the plane trinomial curves k[x,y,z]/(h). We prove that the Frobenius semistability behaviour of the reduction mod p of Vn is a function of the congruence class of p modulo 2λh (an integer invariant associated to h).As one of the consequences of this, we prove that if Vn is semistable in char 0 then its reduction mod p is strongly semistable, for p in a Zariski dense set of primes. Moreover, for any given finitely many such semistable bundles Vn, there is a common Zariski dense set of such primes.

The minimal number of generators of a Togliatti system

We compute the minimal and the maximal bound on the number of generators of a minimal smooth monomial Togliatti system of forms of degree d in $$n+1$$ variables, for any $$d\ge 2$$ and $$n\ge 2$$ . We classify the Togliatti systems with number of generators reaching the lower bound or close to the lower bound. We then prove that if $$n=2$$ (resp. $$n=2,3$$ ) all range between the lower and upper bound is covered, while if $$n\ge 3$$ (resp. $$n\ge 4$$ ) there are gaps if we only consider smooth minimal Togliatti systems (resp. if we avoid the smoothness hypothesis). We finally analyze for $$n=2$$ the Mumford–Takemoto stability of the syzygy bundle associated with smooth monomial Togliatti systems.

Rank- $$2$$ 2 syzygy bundles on Fermat curves and an application to Hilbert–Kunz functions

In this paper we describe the Frobenius pull-backs of the syzygy bundles $${{\mathrm{Syz}}}_C(X^a,Y^a,Z^a)$$ , $$a\ge 1$$ , on the projective Fermat curve $$C$$ of degree $$n$$ in characteristics coprime to $$n$$ , either by giving their strong Harder–Narasimhan filtration if $${{\mathrm{Syz}}}_C(X^a,Y^a,Z^a)$$ is not strongly semistable or in the strongly semistable case by their periodicity behavior. Moreover, we apply these results to Hilbert–Kunz functions, to find Frobenius periodicities of the restricted cotangent bundle $$\Omega _{\mathbb {P}^2}|_C$$ of arbitrary length and a problem of Brenner regarding primes with strongly semistable reduction.