Narrow Results

In this paper, we represent vector bundles over a regular algebraic curve as pairs of lattices over the maximal orders of its function field and we give polynomial time algorithms for several tasks: computing determinants of vector bundles, kernels and images of global homomorphisms, isomorphisms between vector bundles, cohomology groups, extensions, and splitting into a direct sum of indecomposables. Most algorithms are deterministic except for computing isomorphisms when the base field is infinite. Some algorithms are only polynomial time if we may compute Hermite forms of pseudo-matrices in polynomial time. All algorithms rely exclusively on algebraic operations in function fields. For applications, we give an algorithm enumerating isomorphism classes of vector bundles on an elliptic curve, and to construct algebraic geometry codes over vector bundles. We implement all our algorithms into a SageMath package.

This article treats various aspects of the geometry of the moduli of r-spin curves and its compactification . Generalized spin curves, or r-spin curves, are a natural generalization of 2-spin curves (algebraic curves with a theta-characteristic), and have been of interest lately because of the similarities between the intersection theory of these moduli spaces and that of the moduli of stable maps. In particular, these spaces are the subject of a remarkable conjecture of E. Witten relating their intersection theory to the Gelfand–Dikii (KdV_r) heirarchy. There is also a W-algebra conjecture for these spaces [16] generalizing the Virasoro conjecture of quantum cohomology.

For any line bundle on the universal curve over the stack of stable curves, there is a smooth stack of triples (X, ℒ, b) of a smooth curve X, a line bundle ℒ on X, and an isomorphism . In the special case that is the relative dualizing sheaf, then is the stack of r-spin curves.

We construct a smooth compactification of the stack , describe the geometric meaning of its points, and prove that its coarse moduli is projective.

We also prove that when r is odd and g>1, the compactified stack of spin curves and its coarse moduli space are irreducible, and when r is even and is the disjoint union of two irreducible components. We give similar results for n-pointed spin curves, as required for Witten's conjecture, and also generalize to the n-pointed case the classical fact that when is the disjoint union of d(r) components, where d(r) is the number of positive divisors of r. These irreducibility properties are important in the study of the Picard group of [15], and also in the study of the cohomological field theory related to Witten's conjecture [16, 34].

We classify smooth projective algebraic curves C of genus g such that the variety of special linear systems has dimension g- 7. We first prove that if has dimension g-7≥0 then C is either trigonal, tetragonal, a double covering of a curve of genus 2 or a smooth plane sextic. This result establishes the next extension of dimension theorems of H. Martens and D. Mumford on the variety of special linear systems with the fullest possible generality. We then proceed to show that, under the assumption g≥11, has dimension g- 7 if and only if C is either a trigonal curve or a double covering of a curve of genus 2.

We give here a refinement of the classical Clifford's theorem for the upper bound of the number of independent global sections of a semistable vector bundle on a smooth curve. We also conjecture a new version of this theorem that takes into account the Clifford index of the curve. In the case of a bi-elliptic curve we obtain a very precise bound. Finally we study the case of rank 2 bundles.

Let X be a curve of genus g. A coherent system on X consists of a pair (E,V), where E is an algebraic vector bundle over X of rank n and degree d and V is a subspace of dimension k of the space of sections of E. The stability of the coherent system depends on a parameter α. We study the variation of the moduli space of coherent systems when we move the parameter. As an application, we analyze the cases k=1,2,3 and n=2 explicitly. For small values of α, the moduli spaces of coherent systems are related to the Brill–Noether loci, the subschemes of the moduli spaces of stable bundles consisting of those bundles with at least a prescribed number of independent sections. The study of coherent systems is applied to find the dimension, prove the irreducibility, and in some cases calculate the Picard groups of the Brill–Noether loci with k≤3.

Let C be an algebraic curve of genus g ≥ 2. A coherent system on C consists of a pair (E,V), where E is an algebraic vector bundle over C of rank n and degree d and V is a subspace of dimension k of the space of sections of E. The stability of the coherent system depends on a parameter α. We study the geometry of the moduli space of coherent systems for different values of α when k ≤ n and the variation of the moduli spaces when we vary α. As a consequence, for sufficiently large α, we compute the Picard groups and the first and second homotopy groups of the moduli spaces of coherent systems in almost all cases, describe the moduli space for the case k = n - 1 explicitly, and give the Poincaré polynomials for the case k = n - 2. In an appendix, we describe the geometry of the "flips" which take place at critical values of α in the simplest case, and include a proof of the existence of universal families of coherent systems when GCD(n,d,k) = 1.

This paper contains results on stable bundles of rank 2 with space of sections of dimension 4 on a smooth irreducible projective algebraic curve C. There is a known lower bound on the degree for the existence of such bundles; the main result of the paper is a geometric criterion for this bound to be attained. For a general curve C of genus 10, we show that the bound cannot be attained, but that there exist Petri curves of this genus for which the bound is sharp. We interpret the main results for various curves and in terms of Clifford indices and coherent systems. The results can also be expressed in terms of Koszul cohomology and the methods provide a useful tool for the study of the geometry of the moduli space of curves.

Let C be an algebraic smooth complex curve of genus g > 1. The object of this paper is the study of the birational structure of certain moduli spaces of vector bundles and of coherent systems on C and the comparison of different type of notions of stability arising in moduli theory. Notably we show that in certain cases these moduli spaces are birationally equivalent to fibrations over simple projective varieties, whose fibers are GIT quotients (ℙ^r-1)^rg//PGL(r), where r is the rank of the considered vector bundles. This allows us to compare different definitions of (semi-)stability (slope stability, α-stability, GIT stability) for vector bundles, coherent systems and point sets, and derive relations between them. In certain cases of vector bundles of low rank when C has small genus, our construction produces families of classical modular varieties contained in the Coble hypersurfaces.

Over the past 20 years, a great deal of work has been done on the moduli spaces of coherent systems on algebraic curves. Until recently, however, there has been very little work on the fixed determinant case, except for the special case of rank 2 and canonical determinant. This situation has changed due to two papers of Osserman, who has obtained lower bounds for the dimensions of the fixed determinant moduli spaces in some cases. Our object in this paper is to show that some of Osserman's bounds are sharp.

This paper derives several new results, as well as gives a partial survey of the recent development, on the value sharing for algebraic and holomorphic curves into ${\mathbb{P}}^n(ℂ)$.

The aim of this paper is to generalize the $m$-Segre invariant for vector bundles to coherent systems. Let $X$ be a non-singular irreducible complex projective curve of genus $g\ge 0$ and $G(\alpha ;n,d,k)$ be the moduli space of $\alpha$-stable coherent systems of type $(n,d,k)$ on $X$. For any pair of integers $(m,t)$ with $0<m<n$, $0\le t\le k$ we define the $(m,t)$-Segre invariant, and prove that it defines a lower semicontinuous function on the families of coherent systems. Thus, the $(m,t)$-Segre invariant induces a stratification of the moduli space $G(\alpha ;n,d,k)$ into locally closed subvarieties $G(\alpha ;n,d,k;m,t;s)$ according to the value $s$ of the function. We determine an above bound for the $(m,t)$-Segre invariant and compute a bound for the dimension of the different strata $G(\alpha ;n,d,k;m,t;s)$. Moreover, we give some conditions under which the different strata are nonempty. To prove the above results, we introduce the notion of coherent systems of subtype $(a)$.

We consider for structure groups $\mathrm{\text{SU}}(n)\subset \mathrm{\text{SL}}(n,ℂ)$ a densely defined toric structure on the moduli space of framed parabolic sheaves on a three-punctured sphere, which degenerates to an actual toric structure. In combination with previous degeneration results, these extend to similar moduli for arbitrary Riemann surfaces.

For any almost-simple group $G$ over an algebraically closed field $k$ of characteristic zero, we describe the automorphism group of the moduli space of semistable $G$-bundles over a connected smooth projective curve $C$ of genus at least $4$. 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.

Many free-form object boundaries can be modeled by quartics with bounded zero sets. The fact that any nondegenerate closed-bounded algebraic curve of even degree n=2p can be expressed as the product of p conics, which are real ellipses, plus a remaining polynomial of degree n-2,¹² can be utilized to express a nondegenerate quartic as the product of two leading ellipses plus a third conic which might be either a closed curve (an ellipse) or an open curve (a hyperbola). However, it can be shown that the leading ellipses can be modified with appropriate constants by constraining the third conic to be a circle, thus implying a 2-ellipse and 1-circle; i.e. an elliptical-circular(E²C)representation of the quartic. The use of such representations is to simplify the analysis of quartics by exploiting the well-known properties of conics and to develop a set of functionally independent geometric invariants for recognition purposes. Also, it is shown that the underlying Euclidean transformation between two configurations of the same quartic can be determined using the centers of the three conics.

A classical theorem of J. L. Burchnall and T. W. Chaundy shows that two commuting differential operators P and Q give rise, via a differential resultant, to a complex algebraic curve with equation F(x, y) = 0, such that formally inserting P and Q for x and y in F(x, y), gives identically zero. In addition, the points on this curve have coordinates which are exactly the eigenvalues associated with the operators P and Q (see the Introduction for a more precise statement). In this paper, we prove a generalization of this result using resultants in Ore extensions.

A Kuribayashi quartic curve

We denote by ${{\mathscr{H}}}_{d,g,r}$ the Hilbert scheme of smooth curves, which is the union of components whose general point corresponds to a smooth irreducible and non-degenerate curve of degree $d$ and genus $g$ in ${\mathbb{P}}^r$. In this paper, we show that for low genus $g$ outside the Brill–Noether range, the Hilbert scheme ${{\mathscr{H}}}_{g+1,g,4}$ is non-empty whenever $g\ge 9$ and irreducible whose only component generically consists of linearly normal curves unless $g=9$ or $g=12$. This complements the validity of the original assertion of Severi regarding the irreducibility of ${{\mathscr{H}}}_{d,g,r}$ outside the Brill–Nother range for $d=g+1$ and $r=4$.

An alternative proof of the necessary conditions on $A,B\in {𝔽}_{q^2}^{*}$ for $f(X)=X+A{X}^{1+q^2(q-1)/4}+B{X}^{1+3q^2(q-1)/4}$ to be a permutation polynomial in ${𝔽}_{q^2}$, $q$ even, is given. This proof involves standard arguments from algebraic geometry over finite fields and fast symbolic computations.

A classical theorem of Siegel asserts that the set of S-integral points of an algebraic curveC over a number field is finite unless C has genus 0 and at most two points at infinity. In this paper, we give necessary and sufficient conditions for C to have infinitely many S-integral points.

In this work we present an explicit relation between the number of points on a family of algebraic curves over 𝔽_q and sums of values of certain hypergeometric functions over 𝔽_q. Moreover, we show that these hypergeometric functions can be explicitly related to the roots of the zeta function of the curve over 𝔽_q in some particular cases. A general conjecture relating these last two is presented and advances toward its proof are shown in the last section.