Projective Curve Research Articles

Stability approach to torsion pairs on abelian categories

In this paper we introduce a local-refinement procedure to investigate stability data on an abelian category, and provide a sufficient and necessary condition for a stability data to be finest. We classify all the finest stability data for the categories of coherent sheaves over certain weighted projective curves, including the classical projective line, smooth elliptic curves and certain weighted projective lines. As applications, we obtain a classification of torsion pairs for these categories via stability data approach. As a by-product, a new proof for the classification of torsion pairs in any tube category is also provided.

Degree zero Gromov–Witten invariants for smooth curves

AbstractFor a smooth projective curve, we derive a closed formula for the generating series of its Gromov–Witten invariants in genus and degree zero. It is known that the calculation of these invariants can be reduced to that of the and integrals on the moduli space of stable algebraic curves. The closed formula for the integrals is given by the conjecture, proved by Faber and Pandharipande. We compute in this paper the integrals via solving the degree zero limit of the loop equation associated to the complex projective line.

Essentially finite G-torsors

Let X be a smooth projective curve of genus g, defined over an algebraically closed field k, and let G be a connected reductive group over k. We say that a G-torsor is essentially finite if it admits a reduction to a finite group, generalising the notion of essentially finite vector bundles to arbitrary groups G. We give a Tannakian interpretation of such torsors, and we prove that all essentially finite G-torsors have torsion degree, and that the degree is 0 if X is an elliptic curve. We then study the density of the set of k-points of essentially finite G-torsors of degree 0, denoted MGef,0, inside MGss,0, the k-points of all semistable degree 0 G-torsors. We show that when g=1, MGef⊂MGss,0 is dense. When g>1 and when char(k)=0, we show that for any reductive group of semisimple rank 1, MGef,0⊂MGss,0 is not dense.

Castelnuovo–Mumford Regularity of Projective Monomial Curves via Sumsets

Let A={a_0,ldots ,a_{n-1}} be a finite set of nge 4 non-negative relatively prime integers, such that 0=a_0<a_1<cdots <a_{n-1}=d. The s-fold sumset of A is the set sA of integers that contains all the sums of s elements in A. On the other hand, given an infinite field k, one can associate with A the projective monomial curve mathcal {C}_A parametrized by A, CA={(vd:ua1vd-a1:⋯:uan-2vd-an-2:ud)∣(u:v)∈Pk1}⊂Pkn-1.\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\quad \\mathcal {C}_A=\\{(v^d:u^{a_1}v^{d-a_1}:\\cdots :u^{a_{n-2}}v^{d-a_{n-2}}:u^d) \\mid (u:v)\\in \\mathbb {P}^{1}_k\\}\\subset \\mathbb {P}^{n-1}_k. \\end{aligned}$$\\end{document}The exponents in the previous parametrization of mathcal {C}_A define a homogeneous semigroup mathcal {S}subset mathbb {N}^2. We provide several results relating the Castelnuovo–Mumford regularity of mathcal {C}_A to the behavior of the sumsets of A and to the combinatorics of the semigroup mathcal {S} that reveal a new interplay between commutative algebra and additive number theory.

Local constancy of pro‐unipotent Kummer maps

AbstractIt is a theorem of Kim–Tamagawa that the ‐pro‐unipotent Kummer map associated to a smooth projective curve over a finite extension of is locally constant when . This paper establishes two generalisations of this result. First, we extend the Kim–Tamagawa theorem to the case that is a smooth variety of any dimension. Second, we formulate and prove the analogue of the Kim–Tamagawa theorem in the case , again in arbitrary dimension. In the course of proving the latter, we give a proof of an étale–de Rham comparison theorem for pro‐unipotent fundamental groupoids using methods of Scholze and Diao–Lan–Liu–Zhu. This extends the comparison theorem proved by Vologodsky for certain truncations of the fundamental groupoids.

Siegel disks on rational surfaces

We show the existence of a rational surface automorphism of positive entropy with a given number of Siegel disks. Moreover, among automorphisms obtained from quadratic birational maps on the projective plane fixing irreducible cubic curves, we find out an automorphism of positive entropy with multiple Siegel disks.

Diagonal Dimension of Curves

Abstract We prove Conjecture 4.16 of the paper [ 2] of Elagin and Lunts; namely, that a smooth projective curve of genus at least $1$ over a field has diagonal dimension $2$.

On automorphisms of semistableG-bundles with decorations

AbstractWe prove a rigidity result for automorphisms of points of certain stacks admitting adequate moduli spaces. It encompasses as special cases variations of the moduli ofG-bundles on a smooth projective curve for a reductive algebraic groupG. For example, our result applies to the stack of semistableG-bundles, to stacks of semistable Hitchin pairs, and to stacks of semistable parabolicG-bundles. Similar arguments apply to Gieseker semistableG-bundles in higher dimensions. We present two applications of the main result. First, we show that in characteristic 0 every stack of semistable decoratedG-bundles admitting a quasiprojective good moduli space can be written naturally as aG-linearized global quotientY/G, so the moduli problem can be interpreted as a GIT problem. Secondly, we give a proof that the stack of semistable meromorphicG-Higgs bundles on a family of curves is smooth over any base in characteristic 0.

Global Critical Chart for Local Calabi–Yau Threefolds

Abstract In this paper, we investigate Keller’s deformed Calabi–Yau completion of the derived category of coherent sheaves on a smooth variety. In particular, for an $n$-dimensional smooth variety $Y$, we describe the derived category of the total space of an $\omega _{Y}$-torsor as a certain deformed $(n+1)$-Calabi–Yau completion of the derived category of $Y$. As an application, we investigate the geometry of the derived moduli stack of compactly supported coherent sheaves on a local curve, that is, a Calabi–Yau threefold of the form $\textrm{Tot}_{C}(N)$, where $C$ is a smooth projective curve and $N$ is a rank two vector bundle on $C$. We show that the derived moduli stack is equivalent to the derived critical locus of a function on a certain smooth moduli space. This result will be used by the first author and Naoki Koseki in their joint work on Higgs bundles and Gopakumar–Vafa invariants.

A 4-fold categorical equivalence

In this note, we will illuminate some immediate consequences of work done by Reineke in [Algebr. Represent. Theory 16 (2013), no. 5. 1313–1314] that may prove to be useful in the study of elliptic curves. In particular, we will construct an isomorphism between the category of smooth projective curves with a category of quiver Grassmannians. We will use this to provide a 4-fold categorical equivalence between a category of quiver Grassmannians, smooth projective curves, compact Riemann surfaces, and fields of transcendence degree 1 over C \mathbb {C} . We finish with noting that the category of elliptic curves is isomorphic to a category of quiver Grassmannians, whence providing an analytic group structure to a class of quiver Grassmannians.

Three instability stratifications of the stack of Higgs bundles on a smooth projective curve

Abstract We study three instability stratifications of the stack of twisted Higgs bundle of a fixed rank and degree on a smooth complex projective curve. The first is the Harder–Narasimhan (HN) stratification, defined by the instability type of the Higgs bundle. The second is the bundle Harder–Narasimhan (bHN) stratification, defined by the instability type of the underlying bundle. While an unstable HN stratum fibres over the stack parameterizing Higgs bundles which are isomorphic to their HN graded, this is not true for Higgs bundles of unstable bHN type. Obtaining such a fibration requires refining the bHN stratification; this is the third instability stratification. After introducing these three stratifications, we establish comparison results. In particular, we obtain explicit criteria for determining semistability of a Higgs bundle of low rank with unstable underlying bundle. Then we show how the HN and bHN stratifications can be used to filter the stack of Higgs bundles by global quotient stacks in two different ways. Finally, we use these filtrations to relate the HN and bHN stratifications to GIT instability stratifications and the refined bHN stratification to a Bialynicki–Birula stratification.

Massey products and elliptic curves

AbstractWe study the vanishing of Massey products of order at least 3 for absolutely irreducible smooth projective curves over a field with coefficients in . We mainly focus on elliptic curves, for which we obtain a complete characterization of when triple Massey products do not vanish.

Integral Fourier transforms and the integral Hodge conjecture for one-cycles on abelian varieties

We prove the integral Hodge conjecture for one-cycles on a principally polarized complex abelian variety whose minimal class is algebraic. In particular, the Jacobian of a smooth projective curve over the complex numbers satisfies the integral Hodge conjecture for one-cycles. The main ingredient is a lift of the Fourier transform to integral Chow groups. Similarly, we prove the integral Tate conjecture for one-cycles on the Jacobian of a smooth projective curve over the separable closure of a finitely generated field. Furthermore, abelian varieties satisfying such a conjecture are dense in their moduli space.

Elliptic surfaces and intersections of adelic $\mathbb{R}$-divisors

Suppose \mathcal{E} \to B is a non-isotrivial elliptic surface defined over a number field, for smooth projective curve B . Let k denote the function field \overline{\mathbb{Q}}(B) and E the associated elliptic curve over k . In this article, we construct adelically metrized \mathbb{R} -divisors \overline{D}_X on the base curve B over a number field, for each X \in E(k)\otimes \mathbb{R} . We prove non-degeneracy of the Arakelov–Zhang intersection numbers \overline{D}_X\cdot\overline{D}_Y , as a biquadratic form on E(k)\otimes \mathbb{R} . As a consequence, we have the following Bogomolov-type statement for the Néron–Tate height functions on the fibers E_t(\overline{\mathbb{Q}}) of \mathcal{E} over t \in B(\overline{\mathbb{Q}}) : given points P_1, \ldots, P_m \in E(k) with m\geq 2 , there exist an infinite sequence \{t_n\}\subset B(\overline{\mathbb{Q}}) and small-height perturbations P_{i,t_n}' \in E_{t_n}(\overline{\mathbb{Q}}) of specializations P_{i,t_n} such that the set \{P_{1, t_n}', \ldots, P_{m,t_n}'\} satisfies at least two independent linear relations for all n , if and only if the points P_1, \ldots, P_m are linearly dependent in E(k) . This gives a new proof of results of Masser and Zannier (2010, 2012) and of Barroero and Capuano (2016) and extends our earlier 2020 results. In the Appendix, we prove an equidistribution theorem for adelically metrized \mathbb{R} -divisors on projective varieties (over a number field) using results of Moriwaki (2016), extending the equidistribution theorem of Yuan (2012).

On some natural torsors over moduli spaces of parabolic bundles

A moduli space N of stable parabolic vector bundles, of rank r and parabolic degree zero, on a n-pointed smooth complex projective curve has two naturally occurring holomorphic T⁎N–torsors over it. One of them is given by the moduli space of pairs of the form (E⁎,D), where E⁎∈N and D is a connection on E⁎. This T⁎N–torsor has a C∞ section that sends any E⁎∈N to the unique connection on it with unitary monodromy. The other T⁎N–torsor is given by the sheaf of holomorphic connections on a theta line bundle over N. This T⁎N–torsor also has a C∞ section given by the Hermitian structure on the theta bundle constructed by Quillen. We prove that these two T⁎N–torsors are isomorphic by a canonical holomorphic map. This holomorphic isomorphism interchanges the above two C∞ sections.

Motives of moduli spaces of bundles on curves via variation of stability and flips

AbstractWe 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 formulae 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).

Algebraic subgroups of the group of birational transformations of ruled surfaces

We classify the maximal algebraic subgroups of Bir(CxPP^1), when C is a smooth projective curve of positive genus.

Stability of Tschirnhausen Bundles

Abstract Let $\alpha \colon X \to Y$ be a general degree $r$ primitive map of nonsingular, irreducible, projective curves over an algebraically closed field of characteristic zero or larger than $r$. We prove that the Tschirnhausen bundle of $\alpha $ is semistable if $g(Y) \geq 1$ and stable if $g(Y) \geq 2$.

On curves on Hirzebruch surfaces

We call a smooth irreducible projective curve a Castelnuovo curve if it admits a birational map into the projective r-space such that the image curve has degree at least 2r+1 and the maximum possible geometric genus (which one can calculate by a classical formula due to Castelnuovo). It is well known that a Castelnuovo curve must lie on a Hirzebruch surface (rational ruled surface). Conversely, making use of a result of W. Castryck and F. Cools concerning the scrollar invariants of curves on Hirzebruch surfaces we show that curves on Hirzebruch surfaces are Castelnuovo curves unless their genus becomes too small w.r.t. their gonality. We analyze the situation more closely, and we calculate the number of moduli of curves of fixed genus g and fixed gonality k lying on Hirzebruch surfaces, in terms of g and k.

Holomorphic differentials of Klein four covers

Let k be an algebraically closed field of characteristic two, and let G be isomorphic to Z/2×Z/2. Suppose X is a smooth projective irreducible curve over k with a faithful G-action, and assume that the cover X→X/G is totally ramified, in the sense that it is ramified and every branch point is totally ramified. We study to what extent the lower ramification groups of the closed points of X determine the isomorphism types of the indecomposable kG-modules and the multiplicities with which they occur as direct summands of the space H0(X,ΩX/k) of holomorphic differentials of X over k. In the case when X/G=Pk1, we completely determine the decomposition of H0(X,ΩX/k) into a direct sum of indecomposable kG-modules. Moreover, we show that the isomorphism classes of indecomposable kG-modules that actually occur as direct summands belong to an infinite list of non-isomorphic indecomposable kG-modules that contain modules of arbitrarily large k-dimension. In particular, our results show that [14, Theorem 6.4] is incorrect.