Curves Of Gonality Research Articles

Characterizing gonality for two-component stable curves

It is a well-known result that a stable curve of compact type over \({\mathbb {C}}\) having two components is hyperelliptic if and only if both components are hyperelliptic and the point of intersection is a Weierstrass point for each of them. With the use of admissible covers, we generalize this characterization in two ways: for stable curves of higher gonality having two smooth components and one node; and for hyperelliptic and trigonal stable curves having two smooth non-rational components and any number of nodes.

The unirational components of the strata of genus $11$ curves with several pencils of degree $6$ in $\mathcal{M}_{11}$

We show that the strata \mathcal{M}_{11,6}(k)\subset \mathcal{M}_{11,6} of 6 -gonal curves of genus 11 , equipped with k mutually independent and type I pencils of degree six, have a unirational irreducible component for 5\leq k\leq 9 . The unirational families arise from degree 9 plane curves with 4 ordinary triple and 5 ordinary double points that dominate an irreducible component of expected dimension. We will further show that the family of degree 8 plane curves with 10 ordinary double points covers an irreducible component of excess dimension in \mathcal{M}_{11,6}(10) .

Projecting syzygies of curves

We explore the concept of projections of syzygies and prove two new technical results; we firstly give a precise characterization of syzygy schemes in terms of their projections, secondly, we prove a converse to Aprodu's Projection Theorem. Applying these results, we prove that extremal syzygies of general curves of non-maximal gonality embedded by a linear system of sufficiently high degree arise from scrolls. Lastly, we prove Green's Conjecture for general covers of elliptic curves (of arbitrary degree) as well as proving a new result for curves of even genus and maximal gonality.

Algebro-Geometric Integration of the Modified Belov—Chaltikian Lattice Hierarchy

Using the Lenard recurrence relations and the zero-curvature equation, we derive the modified Belov—Chaltikian lattice hierarchy associated with a discrete 3×3 matrix spectral problem. Using the characteristic polynomial of the Lax matrix for the hierarchy, we introduce a tri gonal curve Km−2 of arithmetic genus m−2. We study the asymptotic properties of the Baker—Akhiezer function and the algebraic function carrying the data of the divisor near \(P_{\infty_{1}}\), \(P_{\infty_{2}}\), \(P_{\infty_{3}}\), and P0 on Km−2. Based on the theory of trigonal curves, we obtain the explicit theta-function representations of the algebraic function, the Baker—Akhiezer function, and, in particular, solutions of the entire modified Belov—Chaltikian lattice hierarchy.

Linear stability and stability of syzygy bundles

Let [Formula: see text] be a smooth irreducible projective curve and let [Formula: see text] be a complete and generated linear series on [Formula: see text]. Denote by [Formula: see text] the kernel of the evaluation map [Formula: see text]. The exact sequence [Formula: see text] fits into a commutative diagram that we call the Butler’s diagram. This diagram induces in a natural way a multiplication map on global sections [Formula: see text], where [Formula: see text] is a subspace and [Formula: see text] is the dual of a subbundle [Formula: see text]. When the subbundle [Formula: see text] is a stable bundle, we show that the map [Formula: see text] is surjective. When [Formula: see text] is a Brill–Noether general curve, we use the surjectivity of [Formula: see text] to give another proof of the semistability of [Formula: see text], moreover, we fill up a gap in some incomplete argument by Butler: With the surjectivity of [Formula: see text] we give conditions to determine the stability of [Formula: see text], and such conditions imply the well-known stability conditions for [Formula: see text] stated precisely by Butler. Finally we obtain the equivalence between the (semi)stability of [Formula: see text] and the linear (semi)stability of [Formula: see text] on [Formula: see text]-gonal curves.

Moduli spaces of bundles and Hilbert schemes of scrolls over $$\nu $$ ν -gonal curves

The aim of this paper is twofold. We first strongly improve our previous main result Choi et al. (Proc Am Math Soc 146(8):3233–3248, 2018, Theorem 3.1), concerning classification of irreducible components of the Brill–Noether locus parametrizing rank 2 semistable vector bundles of suitable degrees d, with at least $$d-2g+4$$ independent global sections, on a general $$\nu $$ -gonal curve C of genus g. We then uses this classification to study several properties of the Hilbert scheme of suitable surface scrolls in projective space, which turn out to be special and stable.

The geometric torsion conjecture for abelian varieties with real multiplication

The geometric torsion conjecture asserts that the torsion part of the Mordell–Weil group of a family of abelian varieties over a complex quasi-projective curve is uniformly bounded in terms of the genus of the curve. We prove the conjecture for abelian varieties with real multiplication, uniformly in the field of multiplication. Fixing the field, we, furthermore, show that the torsion is bounded in terms of the gonality of the base curve, which is the closer analog of the arithmetic conjecture. The proof is a hybrid technique employing both the hyperbolic and algebraic geometry of the toroidal compactifications of the Hilbert modular varieties $\overline{X}(1)$ parameterizing such abelian varieties. We show that only finitely many torsion covers $\overline{X}_1 (\mathfrak{n})$ contain $d$-gonal curves outside of the boundary for any fixed $d$; the same is true for entire curves $\mathbb{C} \to \overline{X}_1 (\mathfrak{n})$. We also deduce some results about the birational geometry of Hilbert modular varieties.

Brill–Noether loci of rank 2 vector bundles on a general $\nu $-gonal curve

In this paper we study the Brill Noether locus of rank 2, (semi)stable vector bundles with at least two sections and of suitable degrees on a general ν-gonal curve. We classify its irreducible components having at least expected dimension. We moreover describe the general member F of such components just in terms of extensions of line bundles with suitable minimality properties, providing information on the birational geometry of such components as well as on the very-ampleness of F.

Point counting on curves using a gonality preserving lift

We study the problem of lifting curves from finite fields to number fields in a genus and gonality preserving way. More precisely, we sketch how this can be done efficiently for curves of gonality at most four, with an in-depth treatment of curves of genus at most five over finite fields of odd characteristic, including an implementation in Magma. We then use such a lift as input to an algorithm due to the second author for computing zeta functions of curves over finite fields using p-adic cohomology.

The extremal secant conjecture for curves of arbitrary gonality

We prove the Green–Lazarsfeld secant conjecture [Green and Lazarsfeld, On the projective normality of complete linear series on an algebraic curve, Invent. Math. 83 (1986), 73–90; Conjecture (3.4)] for extremal line bundles on curves of arbitrary gonality, subject to explicit genericity assumptions.

A LOWER BOUND FOR THE GONALITY CONJECTURE

For every integer we construct a -gonal curve along with a very ample divisor of degree (where is the genus of ) to which the vanishing statement from the Green–Lazarsfeld gonality conjecture does not apply.

Computing gonal maps of algebraic curves

The gonality of an algebraic curve C is the smallest integer N such that there exists a degree N morphism from C to the projective line. For instance curves of gonality 2 are hyperelliptic curves. As for the genus, the gonality measures the non rationality of the curve. Computing the gonality and a gonal map is important in order to represent function fields as small extension of K(t), with various applications (parametrization by radicals for instance). In that talk, I will prove the existence of an algorithm that, given a curve C, computes the gonality and a gonal map. The proof is based on the study of the syzygies of a canonical model. This is a joint work with J. Schicho and F-O. Schreyer.

A note on Harris Morrison sweeping families

Harris and Morrison (Invent. Math. 99:321–355, 1990, Theorem 2.5), constructed certain semistable fibrations \(f:F\rightarrow Y\) in \(k\)-gonal curves of genus \(g\), such that for every \(k\) the corresponding modular curves give a sweeping family in the \(k\)-gonal locus \(\overline{\mathcal {M}^{k}_{g}}\). Their construction depends on the choice of a smooth curve \(X\). We show that if the genus \(g(X)\) is sufficiently high with respect to \(g\), then the ratio \(\frac{K^{2}_{F}}{\chi (\mathcal {O}_{F})}\) is \(8\) asymptotically with respect to \(g(X)\). Moreover, if the conjectured estimates given in Harris and Morrison (Invent. Math. 99:321–355, 1990, pp. 351–352) hold, we show that if \(g\) is big enough, then \(F\) is a surface of positive index.

Computational aspects of gonal maps and radical parametrization of curves

We develop in this article an algorithm that, given a projective curve $$C$$ , computes a gonal map, that is, a finite morphism from $$C$$ to $$\mathbb P ^1$$ of minimal degree. Our method is based on the computation of scrollar syzygies of canonical curves. We develop an improved version of our algorithm for curves with a unique gonal map and we discuss a characterization of such curves in terms of Betti numbers. Finally, we derive an efficient algorithm for radical parametrization of curves of gonality $$\le 4$$ .

Linear pencils on real algebraic curves

Our knowledge of linear series on real algebraic curves is still very incomplete. In this paper we restrict to pencils (complete linear series of dimension one). Let X denote a real curve of genus g with real points and let k ( R ) be the smallest degree of a pencil on X (the real gonality of X ). Then we can find on X a base point free pencil of degree g + 1 (resp. g if X is not hyperelliptic, i.e. if k ( R ) > 2 ) with an assigned geometric behaviour w.r.t. the real components of X , and if g = 2 n − 2 ( n ≥ 1 ) we prove that k ( R ) ≤ g 2 + 1 which is the same bound as for the gonality of a complex curve of even genus g . Furthermore, if the complexification of X is a k -gonal curve ( k ≥ 2 ) one knows that k ≤ k ( R ) ≤ 2 k − 2 , and we show that for any two integers k ≥ 2 and 0 ≤ n ≤ k − 2 there is a real curve with real points and k -gonal complexification such that its real gonality is k + n .

Correction to the Automorphism Group of a Cyclic $p$-gonal Curve

Correction to the Automorphism Group of a Cyclic $p$-gonal Curve

The automorphism group of a cyclic $p$-gonal curve

Let $M$ be a cyclic $p$-gonal curve with a positive prime number $p$, and let $V$ be the automorphism of order $p$ satisfying $M/ \lt V) \simeq \bf{P}^{1}$. It is well-known that finite subgroups $H$ of $\operatorname{Aut}(\bf{P}^{1})$ are classified into five types. In this paper, we determine the defining equation of $M$ with $H \subset \operatorname{Aut}(M/ \lt V \gt)$ for each type of $H$, and we make a list of hyperelliptic curves of genus 2 and cyclic trigonal curves of genus 5, 7, 9 with $H = Aut(M/ \lt V \gt)$.

Regular Components of Moduli Spaces of Stable Maps and $K$-Gonal Curves

Regular Components of Moduli Spaces of Stable Maps and $K$-Gonal Curves

BRILL-NOETHER DIVISORS ON THE MODULI SPACE OF CURVES AND APPLICATIONS

Here we generalize previous work by Eisenbud-Harris and Farkas in order to prove that certain Brill-Noether divisors on the moduli space of curves have distinct supports. From this fact we deduce non-trivial regularity results for a higher codimensional Brill-Noether locus and for the general g+1 2 -gonal curve of odd genus g.

Remarks on syzygies of $d$-gonal curves

We apply a degenerate version of a result due to Hirschowitz, Ramanan and Voisin to verify Green and Green-Lazarsfeld conjectures over explicit open sets inside each $d$-gonal stratum of curves $X$ with $d<[g_X/2]+2$. By the same method, we verify the Green-Lazarsfeld conjecture for any curve of odd genus and maximal gonality. The proof invokes Voisin's solution to the generic Green conjecture as a key argument.