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Abstract The resonance varieties are cohomological invariants that are studied in a variety of topological, combinatorial, and geometric contexts.We discuss their scheme structure in a general algebraic setting and introduce various properties that ensure the reducedness of the associated projective resonance scheme.We prove an asymptotic formula for the Hilbert series of the associated Koszul module, then discuss applications to vector bundles on algebraic curves and to Chen ranks formulas for finitely generated groups, with special emphasis on Kähler and right-angled Artin groups. 

Abstract We discuss various applications of a uniform vanishing result for the graded components of the finite length Koszul module associated to a subspace$$K\subseteq \bigwedge ^2 V$$ $K\subseteq {\bigwedge }^{2}V$, whereVis a vector space. Previously Koszul modules of finite length have been used to give a proof of Green’s Conjecture on syzygies of generic canonical curves. We now give applications to effective stabilization of cohomology of thickenings of algebraic varieties, divisors on moduli spaces of curves, enumerative geometry of curves onK3 surfaces and to skew-symmetric degeneracy loci. We also show that the instability of sufficiently positive rank 2 vector bundles on curves is governed by resonance and give a splitting criterion. 

For a finite dimensional vector space $V$of dimension $n$, we consider the incidence correspondence (or partial flag variety) $X\subset <\#comment/>\mathbb{P}V\times <\#comment/>\mathbb{P}{V}^{\vee <\#comment/>}$, parametrizing pairs consisting of a point and a hyperplane containing it. We completely characterize the vanishing and non-vanishing behavior of the cohomology groups of line bundles on $X$in characteristic $p>0$. If $n=3$then $X$is the full flag variety of $V$, and the characterization is contained in the thesis of Griffith from the 70s. In characteristic $0$, the cohomology groups are described for all $V$by the Borel–Weil–Bott theorem. Our strategy is to recast the problem in terms of computing cohomology of (twists of) divided powers of the cotangent sheaf on projective space, which we then study using natural truncations induced by Frobenius, along with careful estimates of Castelnuovo–Mumford regularity. When $n=3$, we recover the recursive description of characters from recent work of Linyuan Liu, while for general $n$we give character formulas for the cohomology of a restricted collection of line bundles. Our results suggest truncated Schur functions as the natural building blocks for the cohomology characters. 

Abstract Each connected graded, graded-commutative algebraAof finite type over a field$$\Bbbk $$ $k$of characteristic zero defines a complex of finitely generated, graded modules over a symmetric algebra, whose homology graded modules are called the(higher) Koszul modulesofA. In this note, we investigate the geometry of the support loci of these modules, called theresonance schemesof the algebra. When$$A=\Bbbk \langle \Delta \rangle $$ $A=k⟨\Delta ⟩$is the exterior Stanley–Reisner algebra associated to a finite simplicial complex$$\Delta $$ $\Delta$, we show that the resonance schemes are reduced. We also compute the Hilbert series of the Koszul modules and give bounds on the regularity and projective dimension of these graded modules. This leads to a relationship between resonance and Hilbert series that generalizes a known formula for the Chen ranks of a right-angled Artin group.