In previous works, the authors investigated the relationships between linear stability of a generated linear series $|V|$ on a curve $C$, and slope stabillity of the vector bundle $M_{V,L} := \ker (V \otimes \mathcal{O}_C \to L)$. In particular, the second named author and L. Stoppino conjecture that, for a complete linear system $|L|$, linear (semi)stability is equivalent to slope (semi)stability of $M_V$, and the first and third named authors proved that this conjecture holds for hyperelliptic and for generic curves.
In this work we provide a counterexample to this conjecture on any smooth plane curve of degree $7$.
In this paper we describe the Frobenius pull-backs of the syzygy bundles $Syz_C(X^a, Y^a, Z^a)$, $a \geq 1$, on the projective Fermat curve C of degree n in characteristics coprime to n, either by giving their strong Harder-Narasimhan Filtration if $Syz_C(X^a, Y^a, Z^a)$ 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 $Ω_{P^2}|_C$ of arbitrary length and a problem of Brenner regarding primes with strongly semistable reductio…
We prove the Geometric Syzygy Conjecture for generic canonical curves of even genus. This result extends Green's classical result on the generation of the ideal of a canonical curve by rank four quadrics to the highest linear syzygy group.
For a finite dimensional vector space G we define the k-th generic syzygy scheme Gensyz_k(G) by explicit equations. We show that the syzygy scheme Syz(f) of any syzygy in the linear strand of a projective variety X which is cut out by quadrics is a cone over a linear section of a corresponding generic syzygy scheme. We also give a geometric description of Gensyz_k(G) for k=0,1,2. In particular Gensyz_2(G) is the union of a Pl"ucker embedded Grassmannian and a linear space. From this we deduce that every smooth, non-degenerate projective curve C which is cut out by quadrics and has a p-th linea…
We prove the Green conjecture for generic curves of odd genus. That is we prove the vanishing $K_{k,1}(X,K_X)=0$ for $X$ generic of genus $2k+1$. The curves we consider are smooth curves $X$ on a K3 surface whose Picard group has rank 2. This completes our previous work, where the Green conjecture for generic curves of genus $g$ with fixed gonality $d$ was proved in the range $d\geq g/3$, with the possible exception of the generic curves of odd genus.
Let $C$ be a smooth complex irreducible projective curve of genus $g$ with general moduli, 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 $H^0(L)\otimes\mathcal O_C\to L$. In this work we have a double purpose. The first one is to give new examples of stable syzygy bundles admitting theta divisor over general curves. We prove that if $M_L$ is strictly semistable then $M_L$ admits reducible theta divisor. The second purpose is to study the cohomological semistability of $M_L$, and in…
We are concerned with the problem of the stability of the syzygy bundles associated to base point free vector spaces of forms of the same degree d on the projective space of dimension n. We deduce directly, from Mark Green's vanishing theorem for Koszul cohomology, that any such bundle is stable if his rank is sufficiently high. With a similar argument, we prove the semistability of a certain syzygy bundle on a general complete intersection of hypersurfaces of degree d in the projective space. This answers a question of H. Flenner (1984). We then give an elementary proof of H. Brenner's criter…
In this paper, we investigate linear systems on hyperelliptic varieties. We prove analogues of well-known theorems on abelian varieties, like Lefschetz's embedding theorem and higher k-jet embedding theorems. Syzygy or $N_p$-properties are also deduced for appropriate powers of ample line bundles.
It is a longstanding problem in Algebraic Geometry to determine whether the syzygy bundle $E_{d_1,...,d_n}$ on $\mathbb{P}^N$ defined as the kernel of a general epimorphism \[ϕ:\mathcal{O}(-d_1)\oplus...\oplus\mathcal{O}(-d_n) \to\mathcal{O}\] is (semi)stable. In this thesis, attention is restricted to the case of syzygy bundles $\mathrm{Syz}(f_1,...,f_n)$ on $\mathbb{P}^N$ associated to $n$ generic forms $f_1,...,f_n\in K[X_0,...,X_N]$ of the same degree $d$, for ${N\ge2}$. The first goal is to prove that $\mathrm{Syz}(f_1,...,f_n)$ is stable if \[N+1\le n\le\tbinom{d+N}{N},\] except for the …
It is a longstanding problem in Algebraic Geometry to determine whether the syzygy bundle $E_{d_1,..., d_n}$ on $\PP^N$ defined as the kernel of a general epimorphism $\xymatrix{ϕ:\cO(-d_1)\oplus...\oplus\cO(-d_n)\ar[r] &\cO}$ is (semi)stable. In this note, we restrict our attention to the case of syzygy bundles $E_{d,n}$ on $\PP^N$ associated to $n$ generic forms $f_1,...,f_n\in K[X_0,X_1,..., X_N]$ of the same degree $d$. Our first goal is to prove that $E_{d,n}$ is stable if $N+1\le n\le\tbinom{d+2}{2}+N-2$. This bound improves, in general, the bound $n\le d(N+1)$ given by G. Hein in \cite{…
Let $C:f=0$ be a reduced curve in the complex projective plane. The minimal degree $mdr(f)$ of a Jacobian syzygy for $f$, which is the same as the minimal degree of a derivation killing $f$, is an important invariant of the curve $C$, for instance it can be used to determined whether $C$ is free or nearly free. In this note we study the relations of this invariant $mdr(f)$ with a decomposition of $C$ as a union of two curves $C_1$ and $C_2$, without common irreducible components. When all the singularities that occur are quasihomogeneous, a result by Schenck, Terao and Yoshinaga yields finer i…
The Buchsbaum-Eisenbud-Horrocks rank conjecture proposes lower bounds for the Betti numbers of a graded module M based on the codimension of M. We prove a special case of this conjecture via Boij-Soederberg theory. More specifically, we show that the conjecture holds for graded modules where the regularity of M is small relative to the minimal degree of a first syzygy of M. Our approach also yields an asymptotic lower bound for the Betti numbers of powers of an ideal generated in a single degree.
We generalize the prior linked symplectic Grassmannian construction, applying it to to prove smoothing results for rank-2 limit linear series with fixed special determinant on chains of curves. We apply this general machinery to prove new results on nonemptiness and dimension of rank-2 Brill-Noether loci in a range of degrees.
In order to prove new existence results in Brill-Noether theory for rank-2 vector bundles with fixed special determinant, we develop foundational definitions and results for limit linear series of higher-rank vector bundles. These include two entirely new constructions of "linked linear series" generalizing earlier work of the author for the classical rank-1 case, as well as a new canonical stack structure for the previously developed theory due to Eisenbud, Harris and Teixidor i Bigas. This last structure is new even in the classical rank-1 case, and yields the first proper moduli space of Ei…
We propose a definition of tropical linear series that isolates some of the essential combinatorial properties of tropicalizations of not-necessarily-complete linear series on algebraic curves. The definition combines the Baker-Norine notion of rank with the notion of tropical independence and has the property that the restriction of a tropical linear series of rank r to a connected subgraph is a tropical linear series of rank r. We show that tropical linear series of rank 1 are finitely generated as tropical modules and state a number of open problems related to algebraic, combinatorial, and …
We develop a new technique for studying ranks of multiplication maps for linear series via limit linear series and degenerations to chains of genus-1 curves. We use this approach to prove a purely elementary criterion for proving cases of the Maximal Rank Conjecture, and then apply the criterion to several ranges of cases, giving a new proof of the case of quadrics, and also treating several families in the case of cubics. Our proofs do not require restrictions on direction of approach, so we recover new information on the locus in the moduli space of curves on which the maximal rank condition…
We construct a new infinite series of irreducible components of the Gieseker-Maruyama moduli scheme $\mathcal{M}(k), ~ k \geq 3$ of coherent semistable rank 2 sheaves with Chern classes $c_1=0,~ c_2=k,~ c_3=0$ on $\mathbb{P}^3$ whose general points are sheaves with singularities of mixed dimension. These sheaves are constructed by elementary transformations of stable and properly $μ$-semistable reflexive sheaves along disjoint union of collections of points and smooth irreducible curves which are rational or complete intersection curves. As a special member of this series we obtain a new compo…
We describe new components of the Gieseker--Maruyama moduli scheme $\mathcal{M}(n)$ of semistable rank 2 sheaves $E$ on $\mathbb{P}^3$ with $c_1(E)=0$, $c_2(E)=n$ and $c_3(E)=0$ whose generic point corresponds to non locally free sheaves. We show that such components grow in number as $n$ grows, and discuss how they intersect the instanton component. As an application, we prove that $\mathcal{M}(2)$ is connected, and identify a connected subscheme of $\mathcal{M}(3)$ consisting of 7 irreducible components.
In this article we study the Gieseker-Maruyama moduli spaces $\mathcal{B}(e,n)$ of stable rank 2 algebraic vector bundles with Chern classes $c_1=e\in\{-1,0\},\ c_2=n\ge1$ on the projective space $\mathbb{P}^3$. We construct two new infinite series $Σ_0$ and $Σ_1$ of irreducible components of the spaces $\mathcal{B}(e,n)$, for $e=0$ and $e=-1$, respectively. General bundles of these components are obtained as cohomology sheaves of monads, the middle term of which is a rank 4 symplectic instanton bundle in case $e=0$, respectively, twisted symplectic bundle in case $e=-1$. We show that the seri…
We develop new methods to study tropicalizations of linear series and show linear independence of sections. Using these methods, we prove two new cases of the strong maximal rank conjecture for linear series of degree 25 and 26 on curves of genus 22 and 23, respectively.
We prove the conjecture of Marian-Oprea-Pandharipande on the Segre series associated to a rank zero class. Hence the rank zero Segre integrals on the Hilbert schemes of points for all surfaces are determined.
Let C be a general curve of genus g, embedded in P^r via a general linear series of degree d. In this paper, we prove the Maximal Rank Conjecture, which determines the Hilbert function of C.
We study the dimension of loci of special line bundles on stable curves and for a fixed semistable multidegree. In case of total degree $d = g - 1$, we characterize when the effective locus gives a Theta divisor. In case of degree $g - 2$ and $g$, we show that the locus is either empty or has the expected dimension. This leads to a new characterization of semistability in these degrees. In the remaining cases, we show that the special locus has codimension at least $2$. If the multidegree in addition is non-negative on each irreducible component of the curve, we show that the special locus con…
This paper provides, over Henselian valued fields, some theorems on implicit function and of Artin--Mazur on algebraic power series. Also discussed are certain versions of the theorems of Abhyankar--Jung and Newton--Puiseux. The latter is used in analysis of functions of one variable, definable in the language of Denef--Pas, to obtain a theorem on existence of the limit, proven over rank one valued fields in one of our recent papers. This result along with the technique of fiber shrinking (developed there over rank one valued fields) were, in turn, two basic tools in the proof of the closednes…
We propose to study the tropical geometry specifically arising from convergent Hahn series in multiple indeterminates. One application is a new view on stable intersections of tropical hypersurfaces. Another one is perturbations of rank one tropical polytopes, which is beneficial for algorithmic purposes.
We investigate the smoothing problem of limit linear series of rank one on an enrichment of the notions of nodal curves and metrized complexes called saturated metrized complexes. We give a finitely verifiable full criterion for smoothability of a limit linear series of rank one on saturared metrized complexes, characterize the space of all such smoothings, and extend the criterion to metrized complexes. As applications, we prove that all limit linear series of rank one are smoothable on saturated metrized complexes corresponding to curves of compact-type, and prove an analogue for saturated m…
We prove a generalization of Gabrielov's rank theorem for families of rings of power series which we call W-temperate. Examples include the families of complex analytic functions and of Eisenstein series. As a Corollary, we provide rank Theorems for convergent series in general characteristic zero complete valued fields (not necessarily algebraically closed, nor archimedean).
In this paper we study the rank one discrete valuations of the field $k((X_1,..., X_n))$ whose center in $k\lcor\X\rcor$ is the maximal ideal. In sections 2 to 6 we give a construction of a system of parametric equations describing such valuations. This amounts to finding a parameter and a field of coefficients. We devote section 2 to finding an element of value 1, that is, a parameter. The field of coefficients is the residue field of the valuation, and it is given in section 5. The constructions given in these sections are not effective in the general case, because we need either to use the …
