Nonsingular Projective Curve Research Articles

Decomposed Richelot isogenies of Jacobian varieties of curves of genus 3

For a non-singular projective curve C of genus 3 defined over an algebraically closed field of characteristic p≠2, we give a necessary and sufficient condition that the Jacobian variety J(C) has a decomposed Richelot isogeny outgoing from it and we determine the structures of decomposed ones.

Chow group of 1-cycles of the moduli of parabolic bundles over a curve

We study the Chow group of 1-cycles of the moduli space of semistable parabolic vector bundles of fixed rank, determinant and a generic weight over a nonsingular projective curve over \({\mathbb {C}}\) of genus at least 3. We show that, the Chow group of 1-cycles remains isomorphic as we vary the generic weight. As a consequence, we can give an explicit description of the Chow group in the case of rank 2 and determinant \({\mathcal {O}}(x)\), where \(x\in X\) is a fixed point, which extends the earlier result of Choe and Hwang (Math. Z. 253 (2006) 253–281, Main theorem).

Projective plane curves whose automorphism groups are simple and primitive

We study complex projective plane curves with a given group of automorphisms. Let $G$ be a simple primitive subgroup of $\mathrm{PGL}(3, \mathbf{C})$, which is isomorphic to $\mathfrak{A}_{6}$, $\mathfrak{A}_{5}$ or $\mathrm{PSL}(2, \mathbf{F}_{7})$. We obtain a necessary and sufficient condition on $d$ for the existence of a nonsingular projective plane curve of degree $d$ invariant under $G$. We also study an analogous problem on integral curves.

Global Prym–Torelli theorem for double coverings of elliptic curves

The Prym variety for a branched double covering of a nonsingular projective curve is defined as a polarized abelian variety. We prove that any double covering of an elliptic curve which has more than $4$ branch points is recovered from its Prym variety.

Singularities and syzygies of secant varieties of nonsingular projective curves

In recent years, the equations defining secant varieties and their syzygies have attracted considerable attention. The purpose of the present paper is to conduct a thorough study on secant varieties of curves by settling several conjectures and revealing interaction between singularities and syzygies. The main results assert that if the degree of the embedding line bundle of a nonsingular curve of genus $g$ is greater than $2g+2k+p$ for nonnegative integers $k$ and $p$, then the $k$-th secant variety of the curve has normal Du Bois singularities, is arithmetically Cohen--Macaulay, and satisfies the property $N_{k+2, p}$. In addition, the singularities of the secant varieties are further classified according to the genus of the curve, and the Castelnuovo--Mumford regularities are also obtained as well. As one of the main technical ingredients, we establish a vanishing theorem on the Cartesian products of the curve, which may have independent interests and may find applications elsewhere.

On the disposition of cubic and pair of conics in a real projective plane

In the first part of the 16th Hilbert problem the question about the topology of nonsingular projective algebraic curves and surfaces was formulated. The problem on topology of algebraic manifolds with singularities belong to this subject too. The particular case of this problem is the study of curves that are decompozable into the product of curves in a general position. This paper deals with the problem of topological classification of mutual positions of a nonsingular curve of degree three and two nonsingular curves of degree two in the real projective plane. Additiolal conditions for this problem include general position of the curves and its maximality; in particular, the number of common points for each pair of curves-factors reaches its maximum. It is proved that the classification contains no more than six specific types of positions of the species under study. Four position types are built, and the question of realizability of the two remaining ones is open.

The combinatorics of Lehn's conjecture

Let $S$ be a nonsingular projective surface equipped with a line bundle $H$. Lehn's conjecture is a formula for the top Segre class of the tautological bundle associated to $H$ on the Hilbert scheme of points of $S$. Voisin has recently reduced Lehn's conjecture to the vanishing of certain coefficients of special power series. The first result here is a proof of the vanishings required by Voisin by residue calculations (A. Szenes and M. Vergne have independently found the same proof). Our second result is an elementary solution of the parallel question for the top Segre class on the symmetric power of a nonsingular projective curve $C$ associated to a higher rank vector bundle $V$ on $C$. Finally, we propose a complete conjecture for the top Segre class on the Hilbert scheme of points of $S$ associated to a higher rank vector bundle on $S$ in the $K$-trivial case.

Chern slopes of surfaces of general type in positive characteristic

Let k be an algebraically closed field of characteristic p>0, and let C be a nonsingular projective curve over k. We prove that for any real number x≥2, there are minimal surfaces of general type X over k such that (a) c12(X)>0, c2(X)>0, (b) π1e´t(X)≃π1e´t(C), and (c) c12(X)/c2(X) is arbitrarily close to x. In particular, we show the density of Chern slopes in the pathological Bogomolov–Miyaoka–Yau interval (3,∞) for any given p. Moreover, we prove that for C=P1 there exist surfaces X as above with H1(X,OX)=0, that is, with Picard scheme equal to a reduced point. In this way, we show that even surfaces with reduced Picard scheme are densely persistent in [2,∞) for any given p.

Twists of non-hyperelliptic curves

In this paper we present a method for computing the set of twists of a non-singular projective curve defined over an arbitrary (perfect) field k . The method is based on a correspondence between twists and solutions to a Galois embedding problem. When in addition, this curve is non-hyperelliptic we show how to compute equations for the twists. If k=\mathbb{F}_q the method then becomes an algorithm, since in this case, it is known how to solve the Galois embedding problems that appear. As an example we compute the set of twists of the non-hyperelliptic genus 6 curve x^7-y^3-1=0 when we consider it defined over a number field such that [k(\zeta_{21}):k]=12 . For each twist equations are exhibited.

A compactification of the space of maps from curves

We construct a new compactification of the moduli space of maps from pointed nonsingular projective stable curves to a nonsingular projective variety with prescribed ramification indices at the points. It is shown to be a proper Deligne-Mumford stack equipped with a natural virtual fundamental class.

CERTAIN VALUES OF GAUSSIAN HYPERGEOMETRIC SERIES AND A FAMILY OF ALGEBRAIC CURVES

Let λ ∈ ℚ\{0, -1} and l ≥ 2. Denote by Cl, λ the nonsingular projective algebraic curve over ℚ with affine equation given by [Formula: see text] In this paper, we give a relation between the number of points on Cl, λ over a finite field and Gaussian hypergeometric series. We also give an alternate proof of a result of [D. McCarthy, 3F2 Hypergeometric series and periods of elliptic curves, Int. J. Number Theory6(3) (2010) 461–470]. We find some special values of 3F2 and 2F1 Gaussian hypergeometric series. Finally we evaluate the value of 3F2(4) which extends a result of [K. Ono, Values of Gaussian hypergeometric series, Trans. Amer. Math. Soc.350(3) (1998) 1205–1223].

Hypergeometric functions and a family of algebraic curves

Let λ∈ℚ∖{0,1} and l≥2, and denote by Cl,λ the nonsingular projective algebraic curve over ℚ with affine equation given by $$y^l=x(x-1)(x-\lambda).$$ In this paper, we define Ω(Cl,λ) analogous to the real periods of elliptic curves and find a relation with ordinary hypergeometric series. We also give a relation between the number of points on Cl,λ over a finite field and Gaussian hypergeometric series. Finally, we give an alternate proof of a result of Rouse (Ramanujan J. 12(2):197–205, 2006).

Galois morphism computing the gonality of a nonsingular projective curve on a Hirzebruch surface

Let Π : X e → P 1 ( e ≥ 0 ) be the rational ruled complex surface defined by O P 1 ⊕ O P 1 ( − e ) on P 1 , i.e., the e th Hirzebruch surface. Let C be a nonsingular projective curve on X e , and π : C → P 1 the restriction of Π to C . We assume that C is not rational nor elliptic nor hyperelliptic. Then, we consider the question: when is the function field extension C ( C ) / C ( P 1 ) induced by π Galois? We determine the defining equation of C and the Galois group when the function field extension is Galois. We also prove the following theorem: if C is not isomorphic to a nonsingular plane curve, then every automorphism of C can be extended to an automorphism of X e .

On Weierstraß semigroups at one and two points and their corresponding Poincaré series

The aim of this paper is to introduce and investigate the Poincar\'e series associated with the Weierstra{\ss} semigroup of one and two rational points at a (not necessarily irreducible) non-singular projective algebraic curve defined over a finite field, as well as to describe their functional equations in the case of an affine complete intersection.

Canonical fixed parts of fibred algebraic surfaces

It is shown that the fixed part of the canonical linear system of a fibre in a relatively minimal fibred surface supports at most exceptional sets of weakly elliptic singular- ities. Introduction. Let S be a non-singular projective surface and f : S → C a surjective morphism of S onto a non-singular projective curve C with connected fibres. We call f a relatively minimal fibration of genus g if a general fibre is a non-singular projective curve of genus g and there are no (−1)-curves contained in fibres. We assume that g ≥ 2 throughout the paper. Let F be afi bre off . Then the intersection form is negative semi-definite on Supp(F ) by Zariski's lemma. Furthermore, there exist a positive integer m and a 1-connected curve D such that F = mD .W henm is strictly greater than one, F is called a multiple fibre of multiplicity m and OD(D) is a torsion of order m. In (8), we considered the canonical linear system on the minimal resolution of a normal surface singularity and showed that the fixed part supports at most exceptional sets of rational singular points (cf. (1) and (2)). The present article is an extension of it to the semi-global case and we study the fixed part of the canonical linear system |KF | which we call the canonical fixed part in this paper. Recall that the canonical fixed part is closely related to the Horikawa index (see (3, p. 12)), an analytic invariant of a singular fibre germ. In fact, according to (6, Lemma 10 and Theorem 3), if g = 2, the canonical fixed part is a chain of (−2)-curves (of type A) and the Horikawa index is almost equivalent to the number of its irreducible components.

Generalizations of rigid analytic Picard theorems

Berkovich's Picard theorem states that there are no non-constant analytic maps from the affine line to the complement of two points on a non-singular projective curve. The purpose of this article is to find generalizations of this result in higher dimensional varieties.

Smooth curves having a large automorphismp-group in characteristicp >0

Let k be an algebraically closed field of characteristic p > 0 and C a connected nonsingular projective curve over k with genus g ≥ 2. This paper continues the work begun in [LM05], namely the study of big actions, that is, pairs (C, G) where G is a p-subgroup of the k-automorphism group of C such that |G| > 2 p p−1 g. If G2 denotes the second ramification group of G at the unique ramification point of the cover C → C/G, we display necessary conditions on G2 for (C, G) to be a big action, which allows us to pursue the classification of big actions. Our main source of examples comes from the construction of curves with many rational points using ray class field theory for global function fields, as initiated by J-P. Serre and continued by Lauter ([Lau99]) and by Auer ([Au99]). In particular, we obtain explicit examples of big actions with G2 abelian of large exponent.

Large p-group actions with a p-elementary abelian derived group

Let k be an algebraically closed field of characteristic p > 0 and C a connected nonsingular projective curve over k with genus g ⩾ 2 . Let ( C , G ) be a “big action,” i.e. a pair ( C , G ) where G is a p-subgroup of the k-automorphism group of C such that | G | g > 2 p p − 1 . The aim of this paper is to describe the big actions whose derived group G ′ is p-elementary abelian. In particular, we obtain a structure theorem for the functions parametrizing the Artin–Schreier cover C → C / G ′ . Using Artin–Schreier duality, we shift to a group-theoretic point of view to characterize relevant cases. Then, we display universal families and discuss the corresponding deformation space for p = 5 .

On [formula omitted]-Weierstrass sets and gaps

Let X be a nonsingular projective curve of genus g ⩾ 1 defined over a field F which is assumed to be the full field of constants of F ( X ) , and let V be a linear system of divisors on X. We define the set G ( V ; P 1 , … , P n ) of V -Weierstrass gaps at the rational points P 1 , … , P n of X as being formed by the n-tuples α = ( α 1 , … , α n ) such that V ( α ) ⊊ V ( α − e j ) for some j ∈ { 1 , … , n } (here V ( α ) : = { D ∈ V | D ⩾ ∑ i = 1 n α i P i } and e j = ( δ j i ) 1 ⩽ i ⩽ n ∈ N n ). We call N n ∖ G ( V ; P 1 , … , P n ) the V -Weierstrass set associated to V and P 1 , … P n . The main result of this paper proves a sharp lower bound for # G ( V ; P 1 , … , P n ) .

SIGNATURES OF COLORED LINKS WITH APPLICATION TO REAL ALGEBRAIC CURVES

We construct the signature of a μ-colored oriented link, as a locally constant integer valued function with domain (S1 - {1})μ. It restricts to the Tristram–Levine's signature on the diagonal and the discontinuities can occur only at the zeros of the colored Alexander polynomial. Moreover, the signature and the related nullity verify the Murasugi–Tristram inequality. This gives a new necessary condition for a link to bound a smoothly and properly embedded surface in B4, with given Betti numbers. As an application, we achieve the classification of the complex orientations of maximal plane non-singular projective algebraic curves of degree 7, up to isotopy.