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Elliptic surfaces and linear systems with fat points

Adrian Zahariuc

Abstract

We investigate the expected dimensionality of linear systems with general fat points on certain surfaces using an approach by specialization to elliptic surfaces. For the projectivization of the Atiyah bundle over an elliptic curve with a certain polarization, we observe that the special case of only one fat point implies the general case of arbitrarily many fat points, as well as results concerning other surfaces. We conjecture that this special case holds in characteristic 0, but prove that it fails in any positive characteristic.

Elliptic surfaces and linear systems with fat points

Abstract

We investigate the expected dimensionality of linear systems with general fat points on certain surfaces using an approach by specialization to elliptic surfaces. For the projectivization of the Atiyah bundle over an elliptic curve with a certain polarization, we observe that the special case of only one fat point implies the general case of arbitrarily many fat points, as well as results concerning other surfaces. We conjecture that this special case holds in characteristic 0, but prove that it fails in any positive characteristic.

Paper Structure

This paper contains 8 sections, 11 theorems, 44 equations.

Table of Contents

  1. Clebsch-Gordan in characteristic $p$ and Atiyah's classification
  2. The case of the Atiyah ruled surface
  3. Generalities on the Atiyah ruled surface
  4. Statement of the main conjecture and some remarks
  5. Proof of the main results
  6. Deformation to an elliptic surface
  7. Some special cases
  8. Proof of Theorem \ref{['example theorem']}

Key Result

Theorem 2

Let $(S,{\mathcal{L}})$ be a general projective K3 surface of degree $2g-2$. If $m_1,m_2,...,m_n \leq 3$, then the linear system $|{\mathcal{L}}(m_1,m_2,...,m_n)|$ of curves in $|{\mathcal{L}}|$ with $n$ general fat points of multiplicities $m_1,m_2,...,m_n$ has the expected dimension (expdim). More

Theorems & Definitions (24)

  • Theorem 2
  • Lemma 1.1
  • proof
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Definition 2.4
  • ...and 14 more