Irreducible Projective Curve Of Genus Research Articles

New examples of theta divisors for some syzygy bundles

Let C be a smooth complex irreducible projective curve of genus g, and let $$(L,H^0(L))$$ be a generated complete linear series of type $$(d,r+1)$$ over C. The syzygy bundle, denoted by $$M_L$$ , is the kernel of the evaluation map . The aim of this paper is twofold. Firstly, to give new examples of stable syzygy bundles admitting a theta divisor over Petri curves. We prove that if $$M_L$$ is strictly semistable then $$M_L$$ admits a theta divisor. Secondly, to study the cohomological semistability of $$M_L$$ , and in this direction we give another proof of cohomological semistability of $$M_L$$ when L induces a birational map. This proof gives us precise conditions for the cohomological semistability of $$M_L$$ where such conditions agree with the semistability conditions for $$M_L$$ . Finally, we relate these two properties by showing that under certain conditions on Petri curves the cohomological semistability of $$M_L$$ implies the existence of reducible theta divisor for $$M_L$$ .

A note on theta divisors of stable bundles

Let C be a smooth complex irreducible projective curve of genus g \geq 3 . We show that if C is a Petri curve with g \geq 4 , a general stable vector bundle E on C , with integer slope, admits an irreducible and reduced theta divisor \Theta_E , whose singular locus has dimension g-4 . If C is non-hyperelliptic of genus 3 , then actually \Theta_E is smooth and irreducible for a general stable vector bundle E with integer slope on C .

On linear series and a conjecture of D. C. Butler

Let C be a smooth irreducible projective curve of genus g and L a line bundle of degree d generated by a linear subspace V of H0(L) of dimension n + 1. We prove a conjecture of D. C. Butler on the semistability of the kernel of the evaluation map V ⊗ 𝒪C → L and obtain new results on the stability of this kernel. The natural context for this problem is the theory of coherent systems on curves and our techniques involve wall crossing formulae in this theory.

Stratification of the Moduli Space of Four-Gonal Curves

AbstractLet X be a smooth irreducible projective curve of genus g and gonality 4. We show that the canonical model of X is contained in a uniquely defined surface, ruled by conics, whose geometry is deeply related to that of X. This surface allows us to define four invariants of X and, hence, to stratify the moduli space of four-gonal curves by means of closed irreducible subvarieties, whose dimensions we compute.

The minimal degree of plane models of algebraic curves and double coverings

Let C be a smooth irreducible projective curve of genus g and s(C, 2) (or simply s(2)) the minimal degree of plane models of C. We show the non-existence of curves with s(2) = g for g ≥ 10, g ≠ 11. Another main result is determining the value of s(2) for double coverings of hyperelliptic curves. We also give a criterion for a curve with big s(2) to be a double covering.

Curves without plane model of small degree

Let C be a smooth irreducible projective curve of genus g > 0 and sC (2) the minimal degree of plane models of C. Clearly, sC (2) ≤ g + 2. Our main result is: sC (2) = g + 2 – t (for some integer t ≥ 0) implies that C is a double cover of a curve of genus at most t provided that g is not too small with respect to t. For small t we can be more precise. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Relatively stable bundles over elliptic fibrations [Math. Nachr. 238, 23–36 (2002)

AbstractSee original Math. Nachr. 238, 23–36 (2002)

Solvable Points on Projective Algebraic Curves

AbstractWe examine the problem of finding rational points defined over solvable extensions on algebraic curves defined over general fields. We construct non-singular, geometrically irreducible projective curves without solvable points of genus g, when g is at least 40, over fields of arbitrary characteristic. We prove that every smooth, geometrically irreducible projective curve of genus 0, 2, 3 or 4 defined over any field has a solvable point. Finally we prove that every genus 1 curve defined over a local field of characteristic zero with residue field of characteristic p has a divisor of degree prime to 6p defined over a solvable extension.

Tables of curves with many points

These tables record results on curves with many points over finite fields. For relatively small genus (0 < g < 50) and q a small power of 2 or 3 we give in two tables the best presently known bounds for N q (g), the maximum number of rational points on a smooth absolutely irreducible projective curve of genus g over a field F q of cardinality q. In additional tables we list for a given pair (g, q) the type of construction of the best curve so far, and we give a reference to the literature where such a curve can be found.

Liftings of Galois Covers of Smooth Curves

Let (C,G) be a smooth irreducible projective curve of genus g over an algebraically closed field k of chararacteristic p&gt;0 and G be a finite group of automorphisms of C. It is well known that here, contrary to the characteristic 0 case, Hurwitz‘s bound |G|[les ] 84(g-1) doesn‘t hold in general; in such cases this gives an obstruction to obtaining a smooth galois lifting of (C,G) to characteristic 0. We shall give new obstructions of local nature to the lifting problem, even in the case where G is abelian. In the case where the inertia groups are p√1].

Moduli of Vector Bundles, Frobenius Splitting, and Invariant Theory

Let X be an irreducible projective curve of genus g over an algebraically closed field of positive characteristic + 2, 3. In Part I, we prove, adapting the degeneration arguments of (N-TR), that moduli spaces of rank- two (parabolic) bundles on X are Frobenius split (M-R) for generic smooth X. (A similar result holds for X nodal. A consequence is the Verlinde formula in positive characteristic.) In Part II, we give a direct proof of the fact that the local rings of the moduli spaces are F-split, and further, that they are Cohen- Macaulay. This involves showing that the ring of invariants of g copies of 2 x 2 matrices (under the adjoint action of SL(2)) is F-split and Cohen-Macaulay.