arXiv subjects
Publications and source records attributed to George H. Hitching.
Let $X$ be a projective bundle over a smooth curve $C$ of genus $g \ge 3$, and consider the relative hyperplane bundle ${\mathcal O}_X (1) \to X$. Let $V \subseteq H^0 ( X , {\mathcal O}_X (1) )$ be a generating subspace. We prove that when $C$, $X$ and $V$ are general in moduli and ${\mathcal O}_X (1)$ is sufficiently ample, the kernel bundle of the system $({\mathcal O}_X (1) , V)$ is ${\mathcal O}_X (1)$-stable.
For a vector bundle $V$ over a curve $X$, the Segre invariant $s_n (V)$ encodes the maximal degree attained by rank $n$ subbundles of $V$. The functions $s_n$ define stratifications on moduli of $V$ which are well studied. Let $G$ be a connected reductive algebraic group, and $E \to X$ a principal $G$-bundle. For each parabolic subgroup $P \subset G$ there is a Segre number $s_P (E)$, generalising $s_n (V)$. We show that $s_P$ is semicontinuous in families of $G$-bundles, and thus defines stratifications on moduli spaces of $G$-bundles over $X$. We study the invariance properties of $s_P$, relating the behaviour of $s_P$ and $s_{ϕ(P)}$ for a surjective homomorphism $ϕ\colon G \to H$ and allowing us to compare the Segre stratifications for $G$ and $H$. Finally, we analyse the stratification for the Borel subgroup $B$ of ${\rm GL}_3$, identifying patterns in the geometry and proving, in particular, a sharp Hirschowitz-type bound on $s_B (E)$ for certain topological types.
Let $E$ be a vector bundle over a smooth curve $C$, and $V$ a generating space of sections of $E$. We characterise Mumford linear stability of the associated projective model of $\mathbb{P} E^\vee$ in $\mathbb{P} V^\vee$ in terms of geometric and cohomological properties of the coherent system $(E, V)$, and give some applications. We show that any $\mathbb{P}^{r-1}$-bundle over $C$ has a linearly stable model in $\mathbb{P}^{n-1}$ for any $n \ge r+2$. Furthermore; linear stability of $(E, V)$ is a necessary condition for stability of the kernel bundle $M_{E, V}$ of $(E, V)$, which is predicted by Butler's conjecture for general $C$ and $(E, V)$. We give new examples showing that it is not in general sufficient; in particular, a general bundle $E$ of large degree fits into a linearly stable coherent system $(E, V)$ with nonsemistable kernel bundle. Finally, we use these ideas to show the stability of $M_{E, V}$ for certain $(E, V)$ of type $(r, d, r+2)$ where $E$ is not necessarily stable.
There is a well studied notion of GIT-stability for coherent systems over curves, which depends on a real parameter $α$. For generated coherent systems, there is a further notion of stability derived from Mumford's definition of linear stability for varieties in projective space. Let $α_S$ be close to zero and $α_L \gg 0$. We show that a generated coherent system which is $α_S$-stable and linearly stable is $α_L$-stable, and give examples showing that without further assumptions, there are no other implications between these three types of stability. We observe that several of the systems constructed have stable dual span bundle, including systems which are not $α$-semistable for any value of $α$. We use this to prove a case of Butler's conjecture for systems of type $(2, d, 5)$.
Generalizing the Martens theorem for line bundles over a curve $C$, we obtain upper bounds on the dimension of the Brill--Noether locus $B^k_{n, d}$ parametrizing stable bundles of rank $n \ge 2$ and degree $d$ over $C$ with at least $k$ independent sections. This proves a conjecture of the second author and generalizes bounds obtained by him in the rank two case. We give more refined results for some values of $d$, including a generalized Mumford theorem for $n \ge 2$ when $d \le g - 1$. The statements are obtained chiefly by analysis of the tangent spaces of $B^k_{n, d}$. As an application, we show that for $n \ge 5$ the locus $B^2_{n, n(g-1)}$ is irreducible and reduced for any $C$.
The notion of linear stability of a variety in projective space was introduced by Mumford in the context of GIT. It has subsequently been applied by Mistretta and others to Butler's conjecture on stability of the dual span bundle (DSB) $M_{V, E}$ of a general generated coherent system $( E, V )$. We survey recent progress in this direction on rank one coherent systems, prove a new result for hyperelliptic curves, and state some open questions. We then extend the definition of linear stability to generated coherent systems of higher rank. We show that various coherent systems with unstable DSB studied by Brambila-Paz, Mata-Gutierrez, Newstead and Ortega are also linearly unstable. We show that linearly stable coherent systems of type $(2, d, 4)$ for low enough $d$ have stable DSB, and use this to prove a particular case of a generalized Butler conjecture. We then exhibit a linearly stable generated coherent system with unstable DSB, confirming that linear stability of $( E, V )$ in general remains weaker than semistability of $M_{V, E}$ in higher rank. We end with a list of open questions on the higher rank case.
Let $C$ be a smooth projective curve and $V$ an orthogonal bundle over $C$. Let $\IQeV$ be the isotropic Quot scheme parameterizing degree $e$ isotropic subsheaves of maximal rank in $V$. We give a closed formula for intersection numbers on components of $\IQeV$ whose generic element is saturated. As a special case, for $g \ge 2$, we compute the number of isotropic subbundles of maximal rank and degree of a general stable orthogonal bundle in most cases when this is finite. This is an orthogonal analogue of Holla's enumeration of maximal subbundles in \cite{Ho}, and of the symplectic case studied in \cite{CCH1}.
Let $C$ be a curve and $V \to C$ an orthogonal vector bundle of rank $r$. For $r \le 6$, the structure of $V$ can be described using tensor, symmetric and exterior products of bundles of lower rank, essentially due to the existence of exceptional isomorphisms between $\mathrm{Spin} (r , \mathbb{C})$ and other groups for these $r$. We analyze these structures in detail, and in particular use them to describe moduli spaces of orthogonal bundles. Furthermore, the locus of isotropic vectors in $V$ defines a quadric subfibration $Q_V \subset \mathbb{P} V$. Using familiar results on quadrics of low dimension, we exhibit isomorphisms between isotropic Quot schemes of $V$ and certain ordinary Quot schemes of line subbundles. In particular, for $r \le 6$ this gives a method for enumerating the isotropic subbundles of maximal degree of a general $V$, when there are finitely many.
Given a curve $C$ and a linear system $\ell$ on $C$, the secant locus $V_e^{e-f}( \ell )$ parametrises effective divisors of degree $e$ which impose at most $e-f$ conditions on $\ell$. For $E \to C$ a vector bundle of rank $r$, we define determinantal subschemes $H_e^{e-f} ( \ell ) \subseteq \mathrm{Hilb}^e ( \mathbb{P} E )$ and $Q_e^{e-f} (V) \subseteq \mathrm{Quot}^{0, e} ( E^* )$ which generalise $V_e^{e-f} ( \ell )$, giving several examples. We describe the Zariski tangent spaces of $Q_e^{e-f} (V)$, and give examples showing that smoothness of $Q_e^{e-f} (V)$ is not necessarily controlled by injectivity of a Petri map. We generalise the Abel--Jacobi map and the notion of linear series to the context of Quot schemes. We give some sufficient conditions for nonemptiness of generalised secant loci, and a criterion in the complete case when $f = 1$ in terms of the Segre invariant $s_1 (E)$. This leads to a geometric characterisation of semistability similar to that in arxiv:1812.00706. Using these ideas, we also give a partial answer to a question of Lange on very ampleness of ${\mathcal O}_{\mathbb{P} E} (1)$, and show that for any curve, $Q_e^{e-1} (V)$ is either empty or of the expected dimension for sufficiently general $E$ and $V$. When $Q_e^{e-1} (V)$ has and attains expected dimension zero, we use formulas of Oprea--Pandharipande and Stark to enumerate $Q_e^{e-1} (V)$. We mention several possible avenues of further investigation.
The variety of minimal rational tangents associated to Hecke curves was used by J.-M.Hwang [8] to prove the simplicity of the tangent bundle on the moduli of vector bundles over a curve. In this paper, we use the tangent maps of the symplectic and orthogonal Hecke curves to prove an analogous result for symplectic and orthogonal bundles. In particular, we show the nondegeneracy of the associated variety of minimal rational tangents, which implies the simplicity of the tangent bundle on the moduli spaces of symplectic and orthogonal bundles over a curve. We also show that for large enough genus, the tangent map is an embedding for a general symplectic or orthogonal bundle.
Let $C$ be a complex projective smooth curve and $W$ a symplectic vector bundle of rank $2n$ over $C$. The Lagrangian Quot scheme $LQ_{-e}(W)$ parameterizes subsheaves of rank $n$ and degree $-e$ which are isotropic with respect to the symplectic form. We prove that $LQ_{-e}(W)$ is irreducible and generically smooth of the expected dimension for all large $e$, and that a generic element is saturated and stable.
We study the isotropic Quot schemes $IQ_e (V)$ parameterizing degree $e$ isotropic subsheaves of maximal rank of an orthogonal bundle $V$ over a curve. The scheme $IQ_e (V)$ contains a compactification of the space $IQ^o_e (V)$ of degree $e$ maximal isotropic subbundles, but behaves quite differently from the classical Quot scheme, and the Lagrangian Quot scheme in [6]. We observe that for certain topological types of $V$, the scheme $IQ_e (V)$ is empty for all $e$. In the remaining cases, for infinitely many $e$ there are irreducible components of $IQ_e (V)$ consisting entirely of nonsaturated subsheaves, and so $IQ_e (V)$ is strictly larger than the closure of $IQ^o_e (V)$. As our main result, we prove that for any orthogonal bundle $V$ and for $e \ll 0$, the closure $\overline{IQ^o_e (V)}$ of $IQ^o_e (V)$ is either empty or consists of one or two irreducible connected components, depending on $°(V)$ and $e$. In so doing, we also characterize the nonsaturated part of $\overline{IQ^o_e (V)}$ when $V$ has even rank.
The symplectic Brill--Noether locus ${\mathcal S}_{2n, K}^k$ associated to a curve $C$ parametrises stable rank $2n$ bundles over $C$ with at least $k$ sections and which carry a nondegenerate skewsymmetric bilinear form with values in the canonical bundle. This is a symmetric determinantal variety whose tangent spaces are defined by a symmetrised Petri map. We obtain upper bounds on the dimensions of various components of ${\mathcal S}_{2n, K}^k$. We show the nonemptiness of several ${\mathcal S}_{2n, K}^k$, and in most of these cases also the existence of a component which is generically smooth and of the expected dimension. As an application, for certain values of $n$ and $k$ we exhibit components of excess dimension of the standard Brill--Noether locus $B^k_{2n, 2n(g-1)}$ over any curve of genus $g \ge 122$. We obtain similar results for moduli spaces of coherent systems.
Let $V$ be a vector bundle over a smooth curve $C$. In this paper, we study twisted Brill--Noether loci parametrising stable bundles $E$ of rank $n$ and degree $e$ with the property that $h^0 (C, V \otimes E) \ge k$. We prove that, under conditions similar to those of Teixidor i Bigas and of Mercat, the Brill-Noether loci are nonempty, and in many cases have a component which is generically smooth and of the expected dimension. Along the way, we prove the irreducibility of certain components of both twisted and "nontwisted" Brill--Noether loci. We describe the tangent cones to the twisted Brill-Noether loci. We end with an example of a general bundle over a general curve having positive-dimensional twisted Brill--Noether loci with negative expected dimension.
Let $C$ be a smooth projective curve and $W$ a symplectic bundle over $C$. Let $LQ_e (W)$ be the Lagrangian Quot scheme parametrizing Lagrangian subsheaves $E \subset W$ of degree $e$. We give a closed formula for intersection numbers on $LQ_e (W)$. As a special case, for $g \ge 2$, we compute the number of Lagrangian subbundles of maximal degree of a general stable symplectic bundle, when this is finite. This is a symplectic analogue of Holla's enumeration of maximal subbundles in [13].
Let $E$ be a vector bundle over a smooth curve $C$, and $S = \mathbb{P} E$ the associated projective bundle. We describe the inflectional loci of certain projective models $ψ\colon S \dashrightarrow \mathbb{P}^n$ in terms of Quot schemes of $E$. This gives a geometric characterisation of the Segre invariant $s_1 (E)$, which leads to new geometric criteria for semistability and cohomological stability of bundles over $C$. We also use these ideas to show that for general enough $S$ and $ψ$, the inflectional loci are all of the expected dimension. An auxiliary result, valid for a general subvariety of $\mathbb{P}^n$, is that under mild hypotheses, the inflectional loci associated to a projection from a general centre are of the expected dimension.
Given a vector bundle $V$ over a curve $X$, we define and study a surjective rational map $\mathrm{Hilb}^d (\mathbb{P} V ) - \mathrm{Quot}^{0, d} ( V^* )$ generalising the natural map $\mathrm{Sym}^d X \to \mathrm{Quot}^{0, d} ({\mathcal O}_X)$. We then give a generalisation of the geometric Riemann--Roch theorem to vector bundles of higher rank over $X$. We use this to give a geometric description of the tangent cone to the Brill--Noether locus $B^r_{r, d}$ at a suitable bundle $E$ with $h^0 (E) = r+n$. This gives a generalisation of the Riemann--Kempf singularity theorem. As a corollary, we show that the $n$th secant variety of the rank one locus of $\mathbb{P} \mathrm{End} E$ is contained in the tangent cone.
Let $C$ be a Petri general curve of genus $g$ and $E$ a general stable vector bundle of rank $r$ and slope $g-1$ over $C$ with $h^0 (C, E) = r+1$. For $g > (2r+2)(2r+1)$, we show how the bundle $E$ can be recovered from the tangent cone to the theta divisor $Θ_E$ at ${\mathcal O}_C$. We use this to give a constructive proof and a sharpening of Brivio and Verra's theorem that the theta map $SU_C (r) -rightarrow |r Θ|$ is generically injective for large values of $g$.