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Tschirnhausen bundles of sextic covers of $\mathbb{P}^1$

Sam Frengley, Sameera Vemulapalli

Abstract

A degree $d$ genus $g$ cover of the complex projective line by a smooth irreducible curve $C$ yields a vector bundle on the projective line by pushforward of the structure sheaf. We classify the bundles that arise this way when $d = 6$. Interestingly, our methods show that all constraints on the pushforward are ``explained'' by multiplication in an algebra. Finally, we show that all possible pushforwards are realized by covers with a nontrivial proper subcover.

Tschirnhausen bundles of sextic covers of $\mathbb{P}^1$

Abstract

A degree genus cover of the complex projective line by a smooth irreducible curve yields a vector bundle on the projective line by pushforward of the structure sheaf. We classify the bundles that arise this way when . Interestingly, our methods show that all constraints on the pushforward are ``explained'' by multiplication in an algebra. Finally, we show that all possible pushforwards are realized by covers with a nontrivial proper subcover.

Paper Structure

This paper contains 20 sections, 22 theorems, 69 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Constraints on scrollar invariants
  3. Main ideas of the proof of \ref{['thm:main-thm']}
  4. Future directions
  5. Tschirnhausen bundles of primitive degree $6$ covers
  6. Maroni loci of larger than expected dimension
  7. Making a conjecture for the Tschirnhausen realization problem for all $d$
  8. Solving the Tschirnhausen realization problem in degrees $8$ and $9$
  9. History
  10. Conventions
  11. Acknowledgements
  12. Constraints on the Tschirnhausen bundle
  13. Proof of Proposition 2.3
  14. Double covers of triple covers
  15. Proof of Proposition \ref{['prop:main-cubic-prop']}
  16. ...and 5 more sections

Key Result

Theorem 1.3

A quintuple $\vec{e} \in \mathbb{Z}^5$ arises as the scrollar invariants of a smooth irreducible sextic cover if and only if $\vec{e} \in \mathcal{Q} \cup \mathcal{P}_3$. $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1: A cartoon drawing of the geography of degree $6$ covers of $\mathbb{P}^1$. Given a sextic cover $\pi \colon C \rightarrow \mathbb{P}^1$, its scrollar invariants $(e_1,\dots,e_5)$ yield an integer point of this region. We prove that every integer point of this region arises as the scrollar invariants of a sextic curve. The covers in the pink region $\mathcal{P}_3 \setminus \mathcal{P}_2$ necessarily factor through cubic subcovers. The covers in the blue and green regions $\mathcal{Q}$ necessarily factor through quadratic subcovers. The covers which do not factor through a nontrivial proper subcover are contained in the purple region $\mathcal{P}_2 \cap \mathcal{P}_3$.

Theorems & Definitions (48)

  • Definition 1.2
  • Theorem 1.3
  • Remark 1.4
  • Example 2.1
  • Theorem 2.2
  • Proposition 2.3
  • Lemma 2.4: Factoring through a quadratic subcover
  • proof
  • Lemma 2.5
  • Remark 2.6
  • ...and 38 more