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In this paper, we prove that the normal bundle of a general Brill-Noether space curve of degree $d$ and genus $g geq 2$ is stable if and only if $(d,g) otin { (5,2), (6,4) }$. When $gleq1$ and the characteristic of the ground field is zero, it is classical that the normal bundle is strictly semistable. We show that this fails in characteristic $2$ for all rational curves of even degree.
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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).
We show that given integers $N$, $d$ and $n$ such that ${Nge2}$, ${(N,d,n) e(2,2,5)}$, and ${N+1le nletbinom{d+N}{N}}$, 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 ${Nge3}$ was obtained independently by Coandv{a} 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.
We construct the Hilbert compactification of the universal moduli space of semistable vector bundles over smooth curves. The Hilbert compactification is the GIT quotient of some open part of an appropriate Hilbert scheme of curves in a Grassmannian. It has all the properties asked for by Teixidor.
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.
A projective algebraic surface which is homeomorphic to a ruled surface over a curve of genus $gge 1$ is itself a ruled surface over a curve of genus $g$. In this note, we prove the analogous result for projective algebraic manifolds of dimension 4 in case $gge 2$.
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