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Let $X$ be a smooth projective curve of genus $g geq 2$ and $M$ be the moduli space of rank 2 stable vector bundles on $X$ whose determinants are isomorphic to a fixed odd degree line bundle $L$. There has been a lot of works studying the moduli and recently the bounded derived category of coherent sheaves on $M$ draws lots of attentions. It was proved that the derived category of $X$ can be embedded into the derived category of $M$ by the second named author and Fonarev-Kuznetsov. In this paper we prove that the derived category of the second symmetric product of $X$ can be embedded into derived category of $M$ when $X$ is non-hyperelliptic and $g geq 16$.
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Let $X$ be a smooth projective curve over the complex numbers. To every representation $rhocolon GL(r)lra GL(V)$ of the complex general linear group on the finite dimensional complex vector space $V$ which satisfies the assumption that there be an integer $alpha$ with $rho(z id_{C^r})=z^alpha id_V$ for all $zinC^*$ we associate the problem of classifying triples $(E,L,phi)$ where $E$ is a vector bundle of rank $r$ on $X$, $L$ is a line bundle on $X$, and $phicolon E_rholra L$ is a non trivial homomorphism. Here, $E_rho$ is the vector bundle of rank $dim V$ associated to $E$ via $rho$. If we take, for example, the standard representation of $GL(r)$ on $C^r$ we have to classify triples $(E,L,phi)$ consisting of $E$ as before and a non-zero homomorphism $phicolon Elra L$ which includes the so-called Bradlow pairs. For the representation of $GL(r)$ on $S^2C^3$ we find the conic bundles of Gomez and Sols. In the present paper, we will formulate a general semistability concept for the above triples which depends on a rational parameter $delta$ and establish the existence of moduli spaces of $delta$-(semi)stable triples of fixed topological type. The notion of semistability mimics the Hilbert-Mumford criterion for $SL(r)$ which is the main reason that such a general approach becomes feasible. In the known examples (the above, Higgs bundles, extension pairs, oriented framed bundles) we show how to recover the usual semistability concept. This process of simplification can also be formalized. Altogether, our results provide a unifying construction for the moduli spaces of most decorated vector bundle problems together with an automatism for finding the right notion of semistability and should therefore be of some interest.
We prove formulas for the rational Chow motives of moduli spaces of semistable vector bundles and Higgs bundles of rank 3 and coprime degree on a smooth projective curve. Our approach involves identifying criteria to lift identities in (a completion of) the Grothendieck group of effective Chow motives to isomorphisms in the category of Chow motives. For the Higgs moduli space, we use motivic Bialynicki-Birula decompositions associated to a scaling action with variation of stability and wall-crossing for moduli spaces of rank 2 pairs, which occur in the fixed locus of this action.
Let $C$ be an algebraic curve of genus $g$ and $L$ a line bundle over $C$. Let $mathcal{MS}_C(n,L)$ and $mathcal{MO}_C(n,L)$ be the moduli spaces of $L$-valued symplectic and orthogonal bundles respectively, over $C$ of rank $n$. We construct rational curves on these moduli spaces which generalize Hecke curves on the moduli space of vector bundles. As a main result, we show that these Hecke type curves have the minimal degree among the rational curves passing through a general point of the moduli spaces. As its byproducts, we show the non-abelian Torelli theorem and compute the automorphism group of moduli spaces.
We study the rational Chow motives of certain moduli spaces of vector bundles on a smooth projective curve with additional structure (such as a parabolic structure or Higgs field). In the parabolic case, these moduli spaces depend on a choice of stability condition given by weights; our approach is to use explicit descriptions of variation of this stability condition in terms of simple birational transformations (standard flips/flops and Mukai flops) for which we understand the variation of the Chow motives. For moduli spaces of parabolic vector bundles, we describe the change in motive under wall-crossings, and for moduli spaces of parabolic Higgs bundles, we show the motive does not change under wall-crossings. Furthermore, we prove a motivic analogue of a classical theorem of Harder and Narasimhan relating the rational cohomology of moduli spaces of vector bundles with and without fixed determinant. For rank 2 vector bundles of odd degree, we obtain formulas for the rational Chow motives of moduli spaces of semistable vector bundles, moduli spaces of Higgs bundles and moduli spaces of parabolic (Higgs) bundles that are semistable with respect to a generic weight (all with and without fixed determinant).
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 [phi:mathcal{O}(-d_1)oplus...oplusmathcal{O}(-d_n) tomathcal{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 $f_1,...,f_nin K[X_0,...,X_N]$ of the same degree $d$, for ${Nge2}$. The first goal is to prove that $mathrm{Syz}(f_1,...,f_n)$ is stable if [N+1le nletbinom{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{i}a Miro-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 ${ntbinom{d+N}{N}-n^2}$, if ${Nge3}$, and ${ntbinom{d+2}{2}+ntbinom{d-1}{2}-n^2}$, if ${N=2}$. The results in chapter 3, for $Nge3$, were obtained independently by Iustin Coandu{a} in arXiv:0909.4435.
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