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 case ${(N,n,d)=(2,5,2)}$. The second is to study moduli spaces of stable rank ${n-1}$ vector bundles on $\mathbb{P}^N$ containing syzygy bundles. In a joint work with Laura Costa and Rosa Mar{\'ı}a Miró-Roig, we prove that $N$, $d$ and $n$ are as above, then the syzygy bundle $\mathrm{Syz}(f_1,...,f_n)$ is unobstructed and it belongs to a generically smooth irreducible component of dimension ${n\tbinom{d+N}{N}-n^2}$, if ${N\ge3}$, and ${n\tbinom{d+2}{2}+n\tbinom{d-1}{2}-n^2}$, if ${N=2}$.
The results in chapter 3, for $N\ge3$, were obtained independently by Iustin Coandă in arXiv:0909.4435.
We study (slope-)stability properties of syzygy bundles on a projective space P^N given by ideal generators of a homogeneous primary ideal. In particular we give a combinatorial criterion for a monomial ideal to have a semistable syzygy bundle. Restriction theorems for semistable bundles yield the same stability results on the generic complete intersection curve. From this we deduce a numerical formula for the tight closure of an ideal generated by monomials or by generic homogeneous elements in a generic two-dimensional complete intersection ring.
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…
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{…
In this work we will prove results that ensure the simplicity and the exceptionality of vector bundles which are defined by the splitting of pure resolutions. We will call such objects syzygy bundles.
Let $(X, H)$ be a polarized smooth projective algebraic surface and $E$ is globally generated, stable vector bundle on $X$. Then the Syzygy bundle $M_E$ associated to it is defined as the kernel bundle corresponding to the evaluation map. In this article we will study the stability property of $M_E$ with respect to $H$.
Here we consider the set of bundles $\{V_n\}_{n\geq 1}$ associated to the plane trinomial curves $k[x,y,z]/(h)$. We prove that the Frobenius semistability behaviour of the reduction mod $p$ of $V_n$ is a function of the congruence class of $p$ modulo $2λ_h$ (an integer invariant associated to $h$). As one of the consequences of this, we prove that if $V_n$ is semistable in characteristic 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 $V_n$, there is a common Zariski dense set of such …
Let $L$ be a globally generated line bundle over a smooth irreducible complex projective surface $X$. The syzygy bundle $M_{L}$ is the kernel of the evaluation map $H^0(L)\otimes\mathcal O_X\to L$. We prove the $L$-stability of $M_L$ for Hirzebruch surfaces, del Pezzo surfaces and Enriques surfaces. The $(-K_X)$-stability of syzygy bundles $M_L$ over del Pezzo surfaces is also obtained.
Given a very ample line bundle L on a projective variety X, the syzygy bundle M_L associated to L is the kernel of the evaluation map on sections of L. Our main result is that if X is a smooth projective surface defined over an algebraically closed field, then M_L is slope-stable for any sufficiently positive L.
We prove that the kernel bundle of the evaluation morphism of global sections, namely the syzygy bundle, of a sufficiently ample line bundle on a smooth projective variety is slope stable with respect to any polarization. This settles a conjecture of Ein-Lazarsfeld-Mustopa.
Let K be an algebraically closed field of characteristic zero and let I=(f_1,...,f_n) be a homogeneous R_+-primary ideal in R:=K[X,Y,Z]. If the corresponding syzygy bundle Syz(f_1,...,f_n) on the projective plane is semistable, we show that the Artinian algebra R/I has the Weak Lefschetz property if and only if the syzygy bundle has a special generic splitting type. As a corollary we get the result of Harima et alt., that every Artinian complete intersection (n=3) has the Weak Lefschetz property. Furthermore, we show that an almost complete intersection (n=4) does not necessarily have the Weak…
We show that given integers $N$, $d$ and $n$ such that ${N\ge2}$, ${(N,d,n)\ne(2,2,5)}$, and ${N+1\le n\le\tbinom{d+N}{N}}$, there is a family of $n$ monomials in $K[X_0,\ldots,X_N]$ of degree $d$ such that their syzygy bundle is stable. Case ${N\ge3}$ was obtained independently by Coandǎ with a different choice of families of monomials [Coa09]. For ${(N,d,n)=(2,2,5)}$, there are $5$ monomials of degree~$2$ in $K[X_0,X_1,X_2]$ such that their syzygy bundle is semistable.
In this note, we prove that the syzygy bundle $M_L$ is cohomologically stable with respect to $L$ for any ample and globally generated line bundle $L$ on an Enriques (resp. bielliptic) surface over an algebraically closed field of characteristic $\neq 2$ (resp. $\neq 2,3$). In particular our result on complex Enriques surfaces improves a result of Torres-López and Zamora by removing a condition on Clifford index. Together with the results of Camere and Caucci--Lahoz, it implies that $M_L$ is stable with respect to $L$ for an ample and globally generated line bundle $L$ on any smooth minimal co…
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…
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 investigate degenerations of syzygy bundles on plane curves over $p$-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 fiber consists of multiple projective lines meeting in one point. On such models we investigate vector bundles whose generic fiber is a syzygy bundle and which become trivial when restricted to each projective line in the special fiber. Hence these syzygy bundles have strongly semistable reduction. This investigation is motivated by the fundamen…
We develop a semistability algorithm for vector bundles which are given as a kernel of a surjective morphism between splitting bundles on the projective space over an algebraically closed field K. This class of bundles is a generalization of syzygy bundles. We show how to implement this algorithm in a computer algebra system. Further we give applications, mainly concerning the computation of Tannaka dual groups of stable vector bundles of degree 0 on the projective space and on certain smooth complete intersection curves. We also use our algorithm to close an open case left in a recent work of…
In char $k = p >0$, A. Langer proved a strong restriction theorem (in the style of H. Flenner) for semistable sheaves to a very general hypersurface of degree $d$, on certain varieties, with the condition that `char $k > d$'. He remarked that to remove this condition, it is enough to answer either of the following questions affirmatively: {\it For the syzygy bundle $\sV_d$ of ${\mathcal O}(d)$, is $\sV_d$ semistable for arbitrary $n, d$ and $p = {char} k$?, or is there a good estimate on $μ_{max}(\sV_d^*)$?} Here we prove that (1) the bundle $\sV_d$ is semistable, for a certain infinite set of…
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 introduce a relativization of the secant sheaves used by Ein, Green and Lazarsfeld and apply this construction to the study of syzygies of canonical curves. As a first application, we give a simpler proof of Voisin's Theorem for general canonical curves. This completely determines the terms of the minimal free resolution of the coordinate ring of such curves. Secondly, in the case of curves of even genus, we enhance Voisin's Theorem by providing a structure theorem for the last syzygy space, resolving the Geometric Syzygy Conjecture in even genus.
We describe a sequence of effective divisors on the Hurwitz space $H_{d,g}$ for $d$ dividing $g-1$ and compute their cycle classes on a partial compactification. These divisors arise from vector bundles of syzygies canonically associated to a branched cover. We find that the cycle classes are all proportional to each other.
We investigate the resolution of a general branched cover $α: C \to {\bf P}^1$ in its relative canonical embedding $C \subset {\bf P} E$. 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 first 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 $g = 1 \mod d$.
We show that if $R$ is a two dimensional standard graded ring (with the graded maximal ideal ${\bf m}$) of characteristic $p>0$ and $I\subset R$ is a graded ideal with $\ell(R/I) <\infty$ then the $F$-threshold $c^I({\bf m})$ can be expressed in terms of a strong HN (Harder-Narasimahan) slope of the canonical syzygy bundle on $\mbox{Proj}~R$. Thus $c^I({\bf m})$ is a rational number. This gives us a well defined notion, of the $F$-threshold $c^I({\bf m})$ in characteristic $0$, in terms of a HN slope of the syzygy bundle on $\mbox{Proj}~R$. This generalizes our earlier result (in [TrW]) where …
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 p…
We introduce the problem of GIT stability for syzygy points of canonical curves with a view toward a GIT construction of the canonical model of the moduli space of stable curves. As the first step in this direction, we prove semi-stability of the first syzygy point for a general canonical curve of odd genus.
We give an explicit minimal graded free resolution, in terms of representations of the symmetric group $S_d$, of a Galois-theoretic configuration of $d$ points in $\mathbb{P}^{d-2}$ 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 $\mathbb{P}^1$ 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 cons…
This is an expository article on the recent developments of Hodge theory on moduli spaces of smooth projective varieties with semi-ample canonical line bundles.
In this paper, it is proved that certain stable rank-3 vector bundles can be written as extensions of line bundles and stable rank-2 bundles. As an application, we show the rationality of certain moduli spaces of stable rank-3 bundles over the projective plane P^2.
We prove that the normalized Poincaré bundle on the moduli space of stable rank $r$ vector bundles with a fixed determinant on a smooth projective curve $X$ induces a family of nef vector bundles on the moduli space. Two applications follow. We show that when the genus of $X$ is large, the derived category of $X$ is embedded into the derived category of the moduli space for arbitrary rank and coprime degree, which extends the results of Narasimhan, Fonarev-Kuznetsov, and Belmans-Mukhopadhyay. As the second application, we construct a family of ACM bundles on the moduli space. A key ingredient …
Recent work on the log minimal model program for the moduli space of curves, as well as past results of Caporaso, Pandharipande, and Simpson motivate an investigation of compactifications of the universal moduli space of slope semi-stable vector bundles over moduli spaces of curves arising in the Hassett-Keel program. Our main result is the construction of a universal moduli space of slope semi-stable sheaves which compactifies the moduli space of vector bundles over the moduli space of pseudo-stable curves.
We construct a compactification of the universal moduli space of semistable principal $G$-bundles over $\overline{\textrm{M}}_{g}$, the fibers of which over singular curves are the moduli spaces of $δ$-semistable singular principal G-bundles.
We discuss some relations of moduli of sheaves on rational surfaces by using universal extensions. These are a generalization of Maruyama's method to construct Uhlenbeck compactification of moduli of vector bundles.
We examine the moduli of framed holomorphic bundles over the blowup of a complex surface, by studying a filtration induced by the behavior of the bundles on a neighborhood of the exceptional divisor.
In this paper, we study rank 2 (quasi) parabolic bundles over the Riemann sphere with an effective divisor and these moduli spaces. First we consider a criterium when a parabolic bundle admits a unramified irregular singular parabolic connection. Second, to give a good compactification of the moduli space of semistable parabolic bundles, we introduce a generalization of parabolic bundles, which is called refined parabolic bundles. Third, we discuss a stability condition of refined parabolic bundles and define elementary transformations of the refined parabolic bundles. Finally, we describe the…
We study moduli of holomorphic vector bundles on non-compact varieties. We discuss filtrability and algebraicity of bundles and calculate dimensions of local moduli. As particularly interesting examples, we describe numerical invariants of bundles on some local Calabi-Yau threefolds.
We consider the moduli space of stable vector bundles on curves embedded in P_2 with Hilbert polynomial 3m+1 and construct a compactification of this space by vector bundles. The result is a blow up of the Simpson moduli space M_{3m+1}(P_2).
