Smooth Plane Curves with Freely Acting Finite Groups

2020 ◽  
Author(s):  
Taro Hayashi
Keyword(s):  
Finite Groups ◽  
Plane Curves ◽  
Smooth Plane

scholarly journals Lower bounds for Clifford indices in rank three

2010 ◽  
Vol 150 (1) ◽  
pp. 23-33 ◽  
Author(s):  
H. LANGE ◽  
P. E. NEWSTEAD
Keyword(s):  
Lower Bounds ◽  
Vector Bundles ◽  
Plane Curves ◽  
Projective Curve ◽  
Smooth Projective Curve ◽  
Smooth Plane ◽  
Semistable Vector Bundles

AbstractClifford indices for semistable vector bundles on a smooth projective curve of genus at least 4 were defined in previous papers by the authors. In this paper, we establish lower bounds for the Clifford indices for rank 3 bundles. As a consequence we show that, on smooth plane curves of degree at least 10, there exist non-generated bundles of rank 3 computing one of the Clifford indices.

scholarly journals The kernel of the monodromy of the universal family of degree d smooth plane curves

2020 ◽  
pp. 1-15
Author(s):  
Reid Monroe Harris
Keyword(s):  
Plane Curve ◽  
Teichmüller Space ◽  
Mapping Class ◽  
Plane Curves ◽  
Teichmuller Space ◽  
Infinite Rank ◽  
Universal Family ◽  
Smooth Plane ◽  
Quartic Curves ◽  
Countably Infinite

We consider the parameter space [Formula: see text] of smooth plane curves of degree [Formula: see text]. The universal smooth plane curve of degree [Formula: see text] is a fiber bundle [Formula: see text] with fiber diffeomorphic to a surface [Formula: see text]. This bundle gives rise to a monodromy homomorphism [Formula: see text], where [Formula: see text] is the mapping class group of [Formula: see text]. The main result of this paper is that the kernel of [Formula: see text] is isomorphic to [Formula: see text], where [Formula: see text] is a free group of countably infinite rank. In the process of proving this theorem, we show that the complement [Formula: see text] of the hyperelliptic locus [Formula: see text] in Teichmüller space [Formula: see text] has the homotopy type of an infinite wedge of spheres. As a corollary, we obtain that the moduli space of plane quartic curves is aspherical. The proofs use results from the Weil–Petersson geometry of Teichmüller space together with results from algebraic geometry.

scholarly journals Sharp Bertini Theorem for Plane Curves over Finite Fields

2019 ◽  
Vol 62 (02) ◽  
pp. 223-230 ◽  
Author(s):  
Shamil Asgarli
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Plane Curve ◽  
Plane Curves ◽  
Smooth Plane ◽  
Curves Over Finite Fields

AbstractWe prove that if $C$ is a reflexive smooth plane curve of degree $d$ defined over a finite field $\mathbb{F}_{q}$ with $d\leqslant q+1$ , then there is an $\mathbb{F}_{q}$ -line $L$ that intersects $C$ transversely. We also prove the same result for non-reflexive curves of degree $p+1$ and $2p+1$ when $q=p^{r}$ .

scholarly journals The Picard group of Brauer-Severi varieties

2018 ◽  
Vol 16 (1) ◽  
pp. 1196-1203
Author(s):  
Eslam Badr ◽  
Francesc Bars ◽  
Elisa Lorenzo García
Keyword(s):  
Picard Group ◽  
Plane Curves ◽  
Base Field ◽  
Picard Groups ◽  
Severi Varieties ◽  
Smooth Plane

AbstractIn this paper, we provide explicit generators for the Picard groups of cyclic Brauer-Severi varieties defined over the base field. In particular,we provide such generators for all Brauer-Severi surfaces. To produce these generators we use the theory of twists of smooth plane curves.

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