本文利用有限长度 Koszul 模分次分量的一致消失结果，推动 Green 猜想相关方法在代数簇加厚、曲线模空间、K3 曲线枚举及退化轨迹中的应用，并证明秩 2 正向量丛稳定性由共振决定。
摘要：
我们讨论一个一致消失结果的多种应用；该结果涉及与向量空间第二外积中的一个子空间相关联的有限长度 Koszul 模的分次分量。此前，有限长度 Koszul 模已被用于证明关于一般典范曲线合冲的 Green 猜想。我们现在给出其在代数簇加厚的上同调有效稳定化、曲线模空间上的除子、K3 曲面上曲线的枚举几何，以及斜对称退化轨迹方面的应用。我们还表明，曲线上充分正的秩 2 向量丛的稳定性由共振支配。
英文摘要：
We discuss various applications of a uniform vanishing result for the graded components of the finite length Koszul module associated to a subspace in the second wedge product of 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 on K3 surfaces and to skew-symmetric degeneracy loci. We also show that the stability of sufficiently positive rank 2 vector bundles on curves is governed by resonance.
We study the deformation-obstruction theory of Koszul cohomology groups of $g^r_d$'s on singular nodal curves. We compute the obstruction classes for Koszul cohomology classes on singular curves to deform to a smooth one. In the case the obstructions are nontrivial, we obtain some partial results for generic vanishing of Koszul cohomology groups.
Let C be a semidualizing complex over a noetherian local ring A. If there exists a local homomorphism with source A satisfying some homological properties, then C is dualizing.
In this Note we prove the vanishing of (twisted) Koszul cohomology groups $K_{p,1}$ of an abelian variety with values in powers of an ample line bundle. It complements the work of G. Pareschi on the property $(N_p)$.
We give a sufficient condition to study the vanishing of certain Koszul cohomology groups for general pairs $(X,L)\in W^r_{g,d}$ by induction. As an application, we show that to prove the Maximal Rank Conjecture (for quadrics), it suffices to check all cases with the Brill-Noether number $ρ=0$.
We completely revised the paper after the referee's comments. In the new version, we replaced two erroneous examples, studied a link with earlier work of Koh and Stilmann, and strengthened the main theorem.
We prove a strong vanishing result for finite length Koszul modules, and use it to derive Green's conjecture for every g-cuspidal rational curve over an algebraically closed field k with char(k) = 0 or char(k) >= (g+2)/2. As a consequence, we deduce that the general canonical curve of genus g satisfies Green's conjecture in this range. Our results are new in positive characteristic, whereas in characteristic zero they provide a different proof for theorems first obtained in two landmark papers by Voisin. Our strategy involves establishing two key results of independent interest: (1) we describ…
We extend the theory of Koszul modules to the bi-graded case, and prove a vanishing theorem that allows us to show that the Canonical Ribbon Conjecture of Bayer and Eisenbud holds over a field of characteristic zero or at least equal to the Clifford index. Our results confirm a conjecture of Eisenbud and Schreyer regarding the characteristics where the generic statement of Green's conjecture holds. They also recover and extend to positive characteristics results due to Aprodu and Voisin asserting that Green's Conjecture holds for generic curves of each gonality.
One describes generators of disguised residual intersections in any commutative Noetherian rings. It is shown that, over Cohen-Macaulay rings, the disguised residual intersections and algebraic residual intersections are the same, for ideals with sliding depth. This coincidence provides structural results for algebraic residual intersections in quite general setting. It is shown how the DG-algebra structure of Koszul homologies affects the determination of generators of residual intersections. In the midway it is shown that the Buchsbaum-Eisenbud family of complexes can be derived from the Kos…
We give examples of Koszul rings that arise naturally in algebraic geometry. In the first part, we prove a general result on Koszul property associated to an ample line bundle on a projective variety. Specifically, we show how Koszul property of multiples of a base point free ample line bundle depends on its Castelnuovo-Mumford regularity. In the second part, we give examples of Koszul rings that come from adjoint line bundles on irregular surfaces of general type.
In this article we construct a Koszul-type resolution of the p-th exterior power of the sheaf of holomorphic differential forms on smooth toric varieties and use this to prove a Nadel-type vanishing theorem for Hodge ideals associated to effective \mathbb{Q}-divisors on smooth projective toric varieties. This extends earlier results of Mustaţă and Popa.
Let $R$ be a noetherian ring and $M$ a finite $R$-module. With a linear form $χ$ on $M$ one associates the Koszul complex $K(χ)$. If $M$ is a free module, then the homology of $K(χ)$ is well-understood, and in particular it is grade sensitive with respect to $\Imχ$. In this note we investigate the case of a module $M$ of projective dimension 1 (more precisely, $M$ has a free resolution of length 1) for which the first non-vanishing Fitting ideal $\I_M$ has the maximally possible grade $r+1$, $r=\rank M$. Then $h=\grade \Imχ\le r+1$ for all linear forms $χ$ on $M$, and it turns out that $H_{r-i…
We consider moduli spaces of Azumaya algebras on K3 surfaces and construct an example. In some cases we show a derived equivalence which corresponds to a derived equivalence between twisted sheaves. We prove if $A$ and $A'$ are Morita equivalent Azumaya algebras of degree $r$ then $2r$ divides $c_2(A) - c_2(A')$. In particular this implies that if $A$ is an Azumaya algebra on a K3 surface and $c_2(A)$ is within $2r$ of its minimal bound then the moduli stack of Azumaya algebras with the same underlying gerbe, if non empty, is a proper algebraic space.
The purpose of this paper is to prove Ein--Lazarsfeld's conjecture on asymptotic vanishing of syzygies of algebraic varieties. This result, together with Ein--Lazarsfeld's asymptotic nonvanishing theorem, describes the overall picture of asymptotic behaviors of the minimal free resolutions of the graded section rings of line bundles on a projective variety as the positivity of the line bundles grows. Previously, Raicu reduced the problem to the case of products of three projective spaces, and we resolve this case here.
The Euler-Koszul complex is the fundamental tool in the homological study of A-hypergeometric differential systems and functions. We compare Euler-Koszul homology with D-module direct images from the torus to the base space through orbits in the corresponding toric variety. Our approach generalizes a result by Gel'fand et al. and yields a simpler, more algebraic proof. In the process we extend the Euler-Koszul functor a category of infinite toric modules and describe multigraded localizations of Euler-Koszul homology.
The main purpose of this paper is computing higher algebraic $K$-theory of Koszul complexes over principal ideal domains. The second purpose of this paper is giving examples of comparison techniques on algebraic $K$-theory for Waldhausen categories without the factorization axiom.
We prove that the algebra of closed differential forms in an (algebraic, formal, or analytic) disk with logarithmic singularities along several coordinate hyperplanes is (both nontopologically and topologically) Koszul. The connection with variations of mixed Hodge-Tate structures, based on a preprint by Andrey Levin, is discussed in the introduction.
Let K be a finite field and let X be a subset of a projective space, over the field K, which is parameterized by monomials arising from the edges of a clutter. We show some estimates for the degree-complexity, with respect to the revlex order, of the vanishing ideal I(X) of X. If the clutter is uniform, we classify the complete intersection property of I(X) using linear algebra. We show an upper bound for the minimum distance of certain parameterized linear codes along with certain estimates for the algebraic invariants of I(X).
We give a theorem on the effective non-vanishing problem for algebraic surfaces in positive characteristic. For the Kawamata-Viehweg vanishing, the logarithmic Kollar vanishing and the logarithmic semipositivity, we give their counterexamples on ruled surfaces in positive characteristic.
Let K be a finite field with q elements and let X be a subset of a projective space P^{s-1}, over the field K, which is parameterized by Laurent monomials. Let I(X) be the vanishing ideal of X. Some of the main contributions of this paper are in determining the structure of I(X) and some of their invariants. It is shown that I(X) is a lattice ideal. We introduce the notion of a parameterized code arising from X and present algebraic methods to compute and study its dimension, length and minimum distance. For a parameterized code arising from a connected graph we are able to compute its length …
We develop in this paper some methods for studying the implicitization problem for a rational map $ϕ: \mathbb{P}^n \to (\mathbb{P}^1)^{n+1}$ defining a hypersurface in $(\mathbb{P}^1)^{n+1}$, based on computing the determinant of a graded strand of a Koszul complex. We show that the classical study of Macaulay Resultants and Koszul complexes coincides, in this case, with the approach of approximation complexes and we study and give a geometric interpretation for the acyclicity conditions. Under suitable hypotheses, these techniques enable us to obtain the implicit equation, up to a power, and …
We show that the homotopy invariant algebraic K-theory of Weibel vanishes below the negative of the Krull dimension of a noetherian scheme. This gives evidence for a conjecture of Weibel about vanishing of negative algebraic K-groups.
For a local system and a function on a smooth complex algebraic variety, we give a proof of a conjecture of M. Kontsevich on a formula for the vanishing cycles using the twisted de Rham complex of the formal microlocalization of the corresponding locally free sheaf with integrable connection having regular singularity at infinity. We also prove its local version, which may be viewed as a natural generalization of a result of E. Brieskorn in the isolated singularity case. We then generalize these to the case of the de Rham complexes of regular holonomic D-modules where we have to use the tensor…
Bott proved a strong vanishing theorem for sheaf cohomology on projective space. It holds for toric varieties, but not for most other varieties. We prove Bott vanishing for the quintic del Pezzo surface, also known as the moduli space M_{0,5}^bar of 5-pointed stable curves of genus zero. This is the first non-toric Fano variety for which Bott vanishing has been shown, answering a question by Achinger, Witaszek, and Zdanowicz. In another direction, we prove sharp results on which K3 surfaces satisfy Bott vanishing. For example, a K3 surface with Picard number 1 satisfies Bott vanishing if and o…
Given any Koszul algebra of finite global dimension one can define a new algebra, which we call a higher zigzag algebra, as a twisted trivial extension of the Koszul dual of our original algebra. If our original algebra is the path algebra of a quiver whose underlying graph is a tree, this construction recovers the zigzag algebras of Huerfano and Khovanov. We study examples of higher zigzag algebras coming from Iyama's iterative construction of type A higher representation finite algebras. We give presentations of these algebras by quivers and relations, and describe relations between spherica…
We investigate compact complex manifolds of dimension three and second Betti number $b_2(X) = 0$. We are interested in the algebraic dimension $a(X)$, which is by definition the transcendence degree of the field of meromorphic functions over the field of complex numbers. The topological Euler characteristic $χ_{\mathrm{ top}}(X) $ equals the third Chern class $c_3(X)$ by a theorem of Hopf. Our main result is that, if $X$ is a compact 3-dimensional complex manifold with $b_2(X) = 0$ and $a(X) > 0$, then $c_3(X) = χ_{\rm top}(X) = 0$, that is, we either have $b_1(X) = 0, \ b_3(X) = 2$ or $b_1(X)…
Let X be a homogeneous space of a quasi-trivial k-group G, with geometric stabilizer H, over a number field k. We prove that under certain conditions on the character group of H, certain algebraic Brauer-Manin obstructions to the Hasse principle and weak approximation vanish, because the abelian groups where they take values vanish. When H is connected or abelian, these algebraic Brauer-Manin obstructions to the Hasse principle and weak approximation are the only ones, so we prove the Hasse principle and weak approximation for X under certain conditions. As an application, we obtain new suffic…
We give a reduction of the irregular case for the effective non-vanishing conjecture by virtue of the Fourier-Mukai transform. As a consequence, we reprove that the effective non-vanishing conjecture holds on algebraic surfaces.
We establish the analogue of the Friedlander-Mazur conjecture for Teh's reduced Lawson homology groups of real varieties, which says that the reduced Lawson homology of a real quasi-projective variety $X$ vanishes in homological degrees larger than the dimension of $X$ in all weights. As an application we obtain a vanishing of homotopy groups of the mod-2 topological groups of averaged cycles and a characterization in a range of indices of the motivic cohomology of a real variety as homotopy groups of the complex of averaged equidimensional cycles. We also establish an equivariant Poincare dua…
We study the regularity and the algebraic properties of certain lattice ideals. We establish a map I --> I\~ between the family of graded lattice ideals in an N-graded polynomial ring over a field K and the family of graded lattice ideals in a polynomial ring with the standard grading. This map is shown to preserve the complete intersection property and the regularity of I but not the degree. We relate the Hilbert series and the generators of I and I\~. If dim(I)=1, we relate the degrees of I and I\~. It is shown that the regularity of certain lattice ideals is additive in a certain sense. The…
For perverse sheaves K on abelian varieties X defined over a finitely generated field F we prove that the Euler-Poincare characteristic (defined for the extension of K to the algebraic closure of F) is non-negative.
The purpose of this note is to give a survey of the algebraic properties of multiplier ideals, and illustrate some of their applications to classical projective geometry.
The notes start with an elementary introduction to a few important analytic techniques of algebraic geometry: closed positive currents, $L^2$ estimates for the $\dbar$-operator on positive vector bundles, Nadel's vanishing theorem for multiplier ideal sheaves (a generalization of the well-known Kawamata-Viehweg vanishing theorem). Applications to adjoint line bundles are then discussed. T.~Fujita conjectured in 1987 that $K_X+(n+2)L$ is very ample for every ample line bundle $L$ on a non singular projective variety $X$ with $\dim X=n$. The answer is known only for $n\le 2$ (I.~Reider, 1988). I…
Let $X$ be a compact Kähler manifold with vanishing Riemann curvature. We prove that there exists a manifold $X'$, deformation equivalent to $X$, which is not an analytification of any projective variety, if and only if $H^0(X, Ω^2) \neq 0$. Using this, we recover a recent theorem of Catanese and Demleitner, which states that a rigid smooth quotient of a complex torus is always projective. We also produce many examples of non-algebraic flat Kähler manifolds with vanishing first Betti number.
