Projective Curve Research Articles

On pseudo-effective cones of projective bundles and volume function

In this paper, we compute the pseudo-effective cones of various projective bundles [Formula: see text] over higher dimensional varieties [Formula: see text] under some assumptions on [Formula: see text] as well as on the vector bundle [Formula: see text]. We also compute the volume function on fiber product [Formula: see text] of two projective bundles over a smooth irreducible complex projective curve [Formula: see text]. In particular, we show that the volume function on the fiber product of two ruled surfaces is of polynomial type.

Torelli theorem for moduli stacks of vector bundles and principal G-bundles

Given any irreducible smooth complex projective curve X, of genus at least 2, consider the moduli stack of vector bundles on X of fixed rank and determinant. It is proved that the isomorphism class of the stack uniquely determines the isomorphism class of the curve X and the rank of the vector bundles. The case of trivial determinant, rank 2 and genus 2 is specially interesting: the curve can be recovered from the moduli stack, but not from the moduli space (since this moduli space is P3 thus independently of the curve).We also prove a Torelli theorem for moduli stacks of principal G-bundles on a curve of genus at least 3, where G is any non-abelian reductive group.

A functorial approach to the stability of vector bundles

On a normal projective variety the locus of μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document}-stable bundles that remain μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document}-stable on all Galois covers prime to the characteristic is open in the moduli space of Gieseker semistable sheaves. On a smooth projective curve of genus at least 2 this locus is big in the moduli space of stable bundles. As an application we obtain a very different behaviour of the étale fundamental group in positive vs. characteristic 0.

Counting maximal isotropic subbundles of orthogonal bundles over a curve

Let C be a smooth projective curve and V an orthogonal bundle over C. Let IQe(V) 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 IQe(V) whose generic element is saturated. As a special case, for g≥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 [20], and of the symplectic case studied in [8].

The Bourbaki degree of a plane projective curve

Bourbaki sequences and Bourbaki ideals have been studied by several authors since its inception sixty years ago circa. Generic Bourbaki sequences have been thoroughly examined by the senior author with B. Ulrich and W. Vasconcelos, but due to their nature, no numerical invariant was immediately available. Recently, J. Herzog, S. Kumashiro, and D. Stamate introduced the Bourbaki number in the category of graded modules as the shifted degree of a Bourbaki ideal corresponding to submodules generated in degree at least the maximal degree of a minimal generator of the given module. The present work introduces the Bourbaki degree as the algebraic multiplicity of a Bourbaki ideal corresponding to choices of minimal generators of minimal degree. The main intent is a study of plane curve singularities via this new numerical invariant. Accordingly, quite naturally, the focus is on the case where the standing graded module is the first syzygy module of the gradient ideal of a reduced form f ∈ k [ x , y , z ] f\in k[x,y,z] – i.e., the main component of the module of logarithmic derivations of the corresponding curve. The overall goal of this project is to allow for a facet of classification of projective plane curves based on the behavior of this new numerical invariant, with emphasis on results about its lower and upper bounds.

The Dirac–Higgs Complex and Categorification of (BBB)-Branes

Abstract Let ${\mathcal{M}}_{\operatorname{Dol}}(X,G)$ denote the hyperkähler moduli space of $G$-Higgs bundles over a smooth projective curve $X$. In the context of four dimensional supersymmetric Yang–Mills theory, Kapustin and Witten introduced the notion of (BBB)-brane: boundary conditions that are compatible with the B-model twist in every complex structure of ${\mathcal{M}}_{\operatorname{Dol}}(X,G)$. The geometry of such branes was initially proposed to be hyperkähler submanifolds that support a hyperholomorphic bundle. Gaiotto has suggested a more general type of (BBB)-brane defined by perfect analytic complexes on the Deligne–Hitchin twistor space $\operatorname{Tw}({\mathcal{M}}_{\operatorname{Dol}}(X,G))$. Following Gaiotto’s suggestion, this paper proposes a framework for the categorification of (BBB)-branes, both on the moduli spaces and on the corresponding derived moduli stacks. We do so by introducing the Deligne stack, a derived analytic stack with corresponding moduli space $\operatorname{Tw}({\mathcal{M}}_{\operatorname{Dol}}(X,G))$, defined as a gluing between two analytic Hodge stacks along the Riemann–Hilbert correspondence. We then construct a class of (BBB)-branes using integral functors that arise from higher non-abelian Hodge theory, before discussing their relation to the Wilson functors from the Dolbeault geometric Langlands correspondence.

The canonical representation of the Drinfeld curve

AbstractIf is a smooth projective curve over an algebraically closed field and is a group of automorphisms of , the canonical representation of is given by the induced ‐linear action of on the vector space of holomorphic differentials on . Computing it is still an open problem in general when the cover is wildly ramified. In this paper, we fix a prime power , we consider the Drinfeld curve, that is, the curve given by the equation over together with its standard action by , and decompose as a direct sum of indecomposable representations of , thus solving the aforementioned problem in this case.

Slope inequality for an arbitrary divisor

Let [Formula: see text] be a surjective morphism with connected fibers from a smooth complex projective surface [Formula: see text] to a smooth complex projective curve [Formula: see text] with general fiber [Formula: see text]. In this paper, we develop a more general version of the slope inequality for data [Formula: see text], where [Formula: see text] is an arbitrary relatively effective divisor on [Formula: see text] and [Formula: see text] is a locally free sub-sheaf of [Formula: see text]. We see how the speciality of [Formula: see text], restricted to the general fiber, plays a role in the results. Moreover, we compute some natural examples and provide applications.

Projective structures and Hodge theory

AbstractEvery compact Riemann surface X admits a natural projective structure $$p_u$$ p u as a consequence of the uniformization theorem. In this work we describe the construction of another natural projective structure on X, namely the Hodge projective structure $$p_h$$ p h , related to the second fundamental form of the period map. We then describe how projective structures correspond to (1, 1)-differential forms on the moduli space of projective curves and, from this correspondence, we deduce that $$p_u$$ p u and $$p_h$$ p h are not the same structure.

Extension groups of tautological bundles on punctual Quot schemes of curves

We prove formulas for the cohomology and the extension groups of tautological bundles on punctual Quot schemes over complex smooth projective curves. As a corollary, we show that the tautological bundle determines the isomorphism class of the original vector bundle on the curve. We also give a vanishing result for the push-forward along the Quot–Chow morphism of tensor and wedge products of duals of tautological bundles.

On the Algebraic Geometry of Multiview

We study the multiviews of algebraic space curves X from n pin-hole cameras of a real or complex projective space. We assume the pin-hole centers to be known, i.e., we do not reconstruct them. Our tools are algebro-geometric. We give some general theorems, e.g., we prove that a projective curve (over complex or real numbers) may be reconstructed using four general cameras. Several examples show that no number of badly placed cameras can make a reconstruction possible. The tools are powerful, but we warn the reader (with examples) that over real numbers, just using them correctly, but in a bad way, may give ghosts: real curves which are images of the emptyset. We prove that ghosts do not occur if the cameras are general. Most of this paper is devoted to three important cases of space curves: unions of a prescribed number of lines (using the Grassmannian of all lines in a 3-dimensional projective space), plane curves, and curves of low degree. In these cases, we also see when two cameras may reconstruct the curve, but different curves need different pairs of cameras.

Components of the Hilbert scheme of smooth projective curves using ruled surfaces II: existence of non-reduced components

Components of the Hilbert scheme of smooth projective curves using ruled surfaces II: existence of non-reduced components

The spectrum of Grothendieck monoid: Classifying Serre subcategories and reconstruction theorem

The Grothendieck monoid of an exact category is a monoid version of the Grothendieck group. We use it to classify Serre subcategories of an exact category and to reconstruct the topology of a noetherian scheme. We first construct bijections between (i) the set of Serre subcategories of an exact category, (ii) the set of faces of its Grothendieck monoid, and (iii) the monoid spectrum of its Grothendieck monoid. By using (ii), we classify Serre subcategories of exact categories related to a finite dimensional algebra and a smooth projective curve. For this, we determine the Grothendieck monoid of the category of coherent sheaves on a smooth projective curve. By using (iii), we introduce a topology on the set of Serre subcategories. As a consequence, we recover the topology of a noetherian scheme from the Grothendieck monoid.

Surface counterexamples to the Eisenbud-Goto conjecture

It is well known that the Eisenbud-Goto regularity conjecture is true for arithmetically Cohen-Macaulay varieties, projective curves, smooth surfaces, smooth threefolds in P 5 \mathbb {P}^5 , and toric varieties of codimension two. After J. McCullough and I. Peeva constructed counterexamples in 2018, it has been an interesting question to find the categories such that the Eisenbud-Goto conjecture holds. So far, surface counterexamples have not been found while counterexamples of any dimension greater or equal to 3 are known. In this paper, we construct counterexamples to the Eisenbud-Goto conjecture for projective surfaces in P 4 \mathbb {P}^4 and investigate projective invariants, cohomological properties, and geometric properties. The counterexamples are constructed via binomial rational maps between projective spaces.

Automorphisms of moduli spaces of principal bundles over a smooth curve

For any almost-simple group [Formula: see text] over an algebraically closed field [Formula: see text] of characteristic zero, we describe the automorphism group of the moduli space of semistable [Formula: see text]-bundles over a connected smooth projective curve [Formula: see text] of genus at least [Formula: see text]. The result is achieved by studying the singular fibers of the Hitchin fibration. As a byproduct, we provide a description of the irreducible components of two natural closed subsets in the Hitchin basis: the divisor of singular cameral curves and the divisor of singular Hitchin fibers.

On actions of Frobenius morphisms for moduli stacks of principal bundles over algebraic curves

We study the various arithmetic and geometric Frobenius morphisms on the moduli stack of principal bundles over a smooth projective algebraic curve and determine explicitly their actions on the l-adic cohomology of the moduli stack in terms of Chern classes.

Torsors on moduli spaces of principal G-bundles on curves

Let [Formula: see text] be a semisimple complex algebraic group with a simple Lie algebra [Formula: see text], and let [Formula: see text] denote the moduli stack of topologically trivial stable [Formula: see text]-bundles on a smooth projective curve [Formula: see text]. Fix a theta characteristic [Formula: see text] on [Formula: see text] which is even in case [Formula: see text] is odd. We show that there is a nonempty Zariski open substack [Formula: see text] of [Formula: see text] such that [Formula: see text], [Formula: see text], for all [Formula: see text]. It is shown that any such [Formula: see text] has a canonical connection. It is also shown that the tangent bundle [Formula: see text] has a natural splitting, where [Formula: see text] is the restriction of [Formula: see text] to the semi-stable locus. We also produce an isomorphism between two naturally occurring [Formula: see text]-torsors on the moduli space of regularly stable [Formula: see text].

Chen–Ruan cohomology and orbifold Euler characteristic of moduli spaces of parabolic bundles

We consider the moduli space of stable parabolic Higgs bundles of rank r and fixed determinant, and having full flag quasi-parabolic structures over an arbitrary parabolic divisor on a smooth connected complex projective curve X of genus g, with g≥2. The group Γ of r-torsion points of the Jacobian of X acts on this moduli space. We describe the connected components of the various fixed point loci of this moduli under non-trivial elements from Γ. When the Higgs field is zero, or in other words when we restrict ourselves to the moduli of stable parabolic bundles, we also compute the orbifold Euler characteristic of the corresponding global quotient orbifold. We also describe the Chen–Ruan cohomology groups of this orbifold under certain conditions on the rank and degree, and describe the Chen–Ruan product structure in special cases.

Lie algebroid connections, twisted Higgs bundles and motives of moduli spaces

Let L=(L,[⋅,⋅],δ) be an algebraic Lie algebroid over a smooth projective curve X of genus g≥2 such that L is a line bundle whose degree is less than 2−2g. Let r and d be coprime numbers. We prove that the motivic class of the moduli space of L-connections of rank r and degree d over X does not depend on the Lie algebroid structure [⋅,⋅] and δ of L and neither on the line bundle L itself, but only on the degree of L (and of course on r, d and X). In particular it is equal to the motivic class of the moduli space of KX(D)-twisted Higgs bundles of rank r and degree d, for D any effective divisor with the appropriate degree. As a consequence, similar results (actually slightly stronger) are obtained for the corresponding E-polynomials. Some applications of these results are then deduced.

Hurwitz moduli varieties parameterizing Galois covers of an algebraic curve

Given a smooth, projective curve \(Y\), a finite group \(G\) and a positive integer $n$ we study smooth, proper families \(X\to Y\times S\to S\) of Galois covers of \(Y\) with Galois group isomorphic to $G$ branched in \(n\) points, parameterized by algebraic varieties \(S\). When \(G\) is with trivial center we prove that the Hurwitz space \(H^G_n(Y)\) is a fine moduli variety for this moduli problem and construct explicitly the universal family. For arbitrary \(G\) we prove that \(H^G_n(Y)\) is a coarse moduli variety. For families of pointed Galois covers of \((Y,y_0)\) we prove that the Hurwitz space \(H^G_n(Y,y_0)\) is a fine moduli variety, and construct explicitly the universal family, for arbitrary group \(G\). We use classical tools of algebraic topology and of complex algebraic geometry.