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Conic linear series and pencils of plane quartics

Riccardo Moschetti, Gian Pietro Pirola, Lidia Stoppino

TL;DR

The paper develops a comprehensive framework for conic linear systems on a smooth curve $C\subset\mathbb{P}^3$ cut out by cones of fixed degree, and studies their limits as the vertex approaches a point of $C$ via both geometric and analytic methods. It constructs a projective model of the blow-up of $\mathbb{P}^3$ along $C$ and analyzes the cone map and its differential, establishing conditions for generic injectivity and maximal rank, and shows that the cone-induced divisors on $C$ form a rich structure controlled by a vector-bundle $\mathcal{E}$ over an open set $U$. A key structural result is that the cone map and its extensions are dominant exactly when the cone degree $d\le4$, with precise rank statements for $d=3,4$, and a detailed description of the limits of conic linear series. As an application, the authors construct a non-isotrivial pencil of plane quartics with base point one, irreducible members, and smooth general member, using a degree-4 elliptic normal curve and a 1-parameter family of degree-4 cones, thereby linking cone geometry to explicit plane curve pencils and opening avenues for further geometric and arithmetic explorations in higher degrees and ambient spaces.

Abstract

We study linear systems cut out by cones of fixed degree on a smooth complex curve $C\subset\mathbb{P}^{3}$. We develop a systematic study of the families of such systems, considering their limits, their infinitesimal behaviour and some associated geometric structures. As an application, we prove the existence of a non-isotrivial pencil of quartics with only one base point, all whose members are irreducible and whose general member is smooth.

Conic linear series and pencils of plane quartics

TL;DR

The paper develops a comprehensive framework for conic linear systems on a smooth curve cut out by cones of fixed degree, and studies their limits as the vertex approaches a point of via both geometric and analytic methods. It constructs a projective model of the blow-up of along and analyzes the cone map and its differential, establishing conditions for generic injectivity and maximal rank, and shows that the cone-induced divisors on form a rich structure controlled by a vector-bundle over an open set . A key structural result is that the cone map and its extensions are dominant exactly when the cone degree , with precise rank statements for , and a detailed description of the limits of conic linear series. As an application, the authors construct a non-isotrivial pencil of plane quartics with base point one, irreducible members, and smooth general member, using a degree-4 elliptic normal curve and a 1-parameter family of degree-4 cones, thereby linking cone geometry to explicit plane curve pencils and opening avenues for further geometric and arithmetic explorations in higher degrees and ambient spaces.

Abstract

We study linear systems cut out by cones of fixed degree on a smooth complex curve . We develop a systematic study of the families of such systems, considering their limits, their infinitesimal behaviour and some associated geometric structures. As an application, we prove the existence of a non-isotrivial pencil of quartics with only one base point, all whose members are irreducible and whose general member is smooth.

Paper Structure

This paper contains 10 sections, 27 theorems, 106 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Linear limits of cones
  4. Analytical proof of \ref{['lemma:limitconesexplicit']}
  5. A projective model of the blow up of ${\mathbb P}^3$ along $C$
  6. Conic linear systems
  7. The cone map and its differential
  8. Structural results on conic linear series
  9. An application: some special pencils of plane quartics
  10. Conclusions and open problems

Key Result

Proposition 1.1

Let $p\in C \subset \mathbb P^3$ and let $t_p$ be the projective tangent line to $C$ at $p$. Consider a line $\ell \neq t_p$ through $p$. Then, the limit of a cone of degree $d$ over $C$ as its vertex specializes to $p$ along $\ell$ is the union of the cone over $C$ with vertex in $p$ and the plane

Figures (1)

  • Figure 1: Configuration of points in $z=0$

Theorems & Definitions (78)

  • Proposition 1.1: \ref{['lemma:limitconesexplicit']}
  • Theorem 1.3: \ref{['thm:diffmaxrank']}
  • Corollary 1.4: \ref{['cor: dominant']}, \ref{['do']}
  • Theorem 1.5
  • Theorem 1.6: \ref{['thm: pencil']}
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • proof
  • ...and 68 more