Complete Algebraic Curve Research Articles

Quantum parameters of the geometric Langlands theory

Fix a smooth, complete algebraic curve X over an algebraically closed field k of characteristic zero. To a reductive group G over k, we associate an algebraic stack {text {Par}}_G of quantum parameters for the geometric Langlands theory. Then we construct a family of (quasi-)twistings parametrized by {text {Par}}_G, whose module categories give rise to twisted {mathcal {D}}-modules on {text {Bun}}_G as well as quasi-coherent sheaves on the DG stack {text {LocSys}}_G.

Complete curves in the strata of differentials

Gendron proved that the strata of holomorphic differentials with prescribed orders of zeros do not contain complete algebraic curves by applying the maximum modulus principle to saddle connections. Here we provide an alternative proof for this result by using positivity of divisor classes on moduli spaces of curves.

Arithmeticity of the monodromy of some Kodaira fibrations

A question of Griffiths–Schmid asks when the monodromy group of an algebraic family of complex varieties is arithmetic. We resolve this in the affirmative for a class of algebraic surfaces known as Atiyah–Kodaira manifolds, which have base and fibers equal to complete algebraic curves. Our methods are topological in nature and involve an analysis of the ‘geometric’ monodromy, valued in the mapping class group of the fiber.

AFFINE GEOMETRY OF STRATA OF DIFFERENTIALS

Affine varieties among all algebraic varieties have simple structures. For example, an affine variety does not contain any complete algebraic curve. In this paper, we study affine-related properties of strata of $k$ -differentials on smooth curves which parameterize sections of the $k$ th power of the canonical line bundle with prescribed orders of zeros and poles. We show that if there is a prescribed pole of order at least $k$ , then the corresponding stratum does not contain any complete curve. Moreover, we explore the amusing question whether affine invariant manifolds arising from Teichmüller dynamics are affine varieties, and confirm the answer for Teichmüller curves, Hurwitz spaces of torus coverings, hyperelliptic strata as well as some low genus strata.

Adelic descent theory

A result of André Weil allows one to describe rank$n$vector bundles on a smooth complete algebraic curve up to isomorphism via a double quotient of the set$\text{GL}_{n}(\mathbb{A})$of regular matrices over the ring of adèles (over algebraically closed fields, this result is also known to extend to$G$-torsors for a reductive algebraic group $G$). In the present paper we develop analogous adelic descriptions for vector and principal bundles on arbitrary Noetherian schemes, by proving an adelic descent theorem for perfect complexes. We show that for Beilinson’s co-simplicial ring of adèles$\mathbb{A}_{X}^{\bullet }$, we have an equivalence$\mathsf{Perf}(X)\simeq |\mathsf{Perf}(\mathbb{A}_{X}^{\bullet })|$between perfect complexes on$X$and cartesian perfect complexes for$\mathbb{A}_{X}^{\bullet }$. Using the Tannakian formalism for symmetric monoidal$\infty$-categories, we conclude that a Noetherian scheme can be reconstructed from the co-simplicial ring of adèles. We view this statement as a scheme-theoretic analogue of Gelfand–Naimark’s reconstruction theorem for locally compact topological spaces from their ring of continuous functions. Several results for categories of perfect complexes over (a strong form of) flasque sheaves of algebras are established, which might be of independent interest.

Subcanonical points on algebraic curves

If C C is a smooth, complete algebraic curve of genus g ≥ 2 g\geq 2 over the complex numbers, a point p p of C C is subcanonical if K C ≅ O C ( ( 2 g − 2 ) p ) K_C \cong \mathcal {O}_C\big ((2g-2)p\big ) . We study the locus G g ⊆ M g , 1 \mathcal {G}_g\subseteq \mathcal {M}_{g,1} of pointed curves ( C , p ) (C,p) , where p p is a subcanonical point of C C . Subcanonical points are Weierstrass points, and we study their associated Weierstrass gap sequences. In particular, we find the Weierstrass gap sequence at a general point of each component of G g \mathcal {G}_g and construct subcanonical points with other gap sequences as ramification points of certain cyclic covers and describe all possible gap sequences for g ≤ 6 g\leq 6 .

Algebraic and hamiltonian approaches to isostokes deformations

We study a generalization of the isomonodromic deformation to the case of connections with irregular singularities. We call this generalization Isostokes Deformation. A new deformation parameter arises: one can deform the formal normal forms of connections at irregular points. We study this part of the deformation, giving an algebraic description. Then we show how to use loop groups and hypercohomology to write explicit hamiltonians. We work on an arbitrary complete algebraic curve, the structure group is an arbitrary semisimiple group.

A Note on Steinberg Symbols on Algebraic Curves

The aim of this paper is to provide a method for defining Steinberg symbols on a complete algebraic curve over a perfect field k from the commutator of a certain extension of groups. This extension is associated with a group morphism ϕ : k* → G. With this definition the reciprocity law is a consequence of the finiteness of the cohomology groups H 0(C, 𝒪 C ) and H 1(C, 𝒪 C ). Using this method, Hilbert's norm residue symbol on an algebraic curve and the symbol (a, b) v for the field ℚ p (n = 2) can be defined.

Arithmetically Cohen Macaulay embeddings of reducible curves

Here we study the arithmetically Cohen—Macaulay non-special embeddings of reducible connected complete algebraic curves. We allow to take “general” the irreducible components and the line bundle (with fixed multidegree) and in this case we obtain sharp results.

Equations of the moduli of pointed curves in the infinite Grassmannian

The main result of this paper is the explicit computa- tion of the equations defining the moduli space of triples (C,p,�), where C is an integral and complete algebraic curve, p a smooth rational point anda certain isomorphism. This is achieved by introducing algebraically infinite Grassmannians, tau and Baker- Ahkiezer functions and by proving an Addition Formula for tau functions.

Groups of equivariant loops in SL(n+1,C) and soliton equations

This article studies a generalization of the Drinfel’d and Sokolov hierarchies for sln+1 and shows how the dressing transformation [as interpreted by Segal and Wilson, Publ. Math. IHES 61, 5 (1985)] produces a large class of almost globally analytic solutions. Many of these solutions are ‘‘of finite type,’’ i.e., related to the Jacobian of a complete algebraic curve.

A note on algebraic curves in characteristic 2

Let X be a curve of genus g ≥ 3 over an algebraically closed field of characteristic 2. Let rx be the rank of the Cartier operator acting on thespace of regular differentials. We show that if rx ≤, (2g-4)/3 then g is odd and X is a double cover of a curve of genus at most rx-(g-l)/2. We also study the rank of the Cartier operator on double covers Let X be a non-singular, irreducible complete algebraic curve defined over an algebraicallyclosed field k of characteristic 2. The Cartier operator Cacts on differential forms of X by (See [3] for the proprerties of the Cartier operator on curves)C is a 1/2-linear map and it is well known that Cacts on the space of regular differentials of X. Let rx be the rank of C on this space. We shall prove the following result.

Finiteness of the Family of Rational and Meromorphic Mappings Into Algebraic Varieties

Introduction. In this paper we deal with the higher dimensional version of de Franchis' theorem which says that there are only a finite number of nonconstant separable rational mappings of an algebraic curve into a complete algebraic curve with genus greater than 1. KobayashiOchiai [11] proved that there are only a finite number of dominant meromorphic mappings onto a complex space of general type, and its algebraic proof which covers the positive characteristic case was given by [14] (cf. also [23]). Here we shall consider the case where the mappings are not necessarily dominant. Let k be an algebraically closed field with arbitrary characteristic. All algebraic varieties and mappings (morphisms) in the following will be defined over k. Let R be an algebraic variety, V a smooth complete algebraic variety, and T(V) denote the tangent bundle over V. Set rank f = sup {dim R - dimtf-1(f (t)); t E R } for rational mappings

On the Geometry of a Theorem of Riemann

Let C be a smooth complete algebraic curve. Let I: C-+J be an universal abelian integral of C into its Jacobian J. Furthermore, let I(i): C(i) J be the mapping sending a point c1 + *-+ ci in the ith symmetric product C(i) to I(c1) + *-+ I(ci). Let Wi be the image of I(i) when i is less than the dimension of J. My purpose is to describe the tangent cone of Wi at any point. Let L(w) be the inverse image by I(i) of a point w in J. L(w) is a projective space representing a complete linear system of effective divisors on C. Geometrically, my main result is