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Counting $h^0(D)$ on primary Burniat surfaces

Yonghwa Cho

TL;DR

The paper develops a complete algorithm to compute the cohomology of arbitrary divisors on a primary Burniat surface with $K_X^2=6$, incorporating a detailed analysis of the Burniat configuration, effective divisors, and torsion in $\operatorname{Pic} X$. It classifies numerical divisor classes, reduces computations to nef reduced forms, and handles torsion twists to obtain exact $h^p(D)$ values. The authors apply these methods to Ulrich bundle questions, proving there are no Ulrich line bundles and producing a rank-2 Ulrich bundle for $3K_X$ that cannot be obtained by Casnati’s construction, thus enriching the landscape of Ulrich phenomena on Burniat surfaces. Overall, the work provides explicit, algorithmic tools for divisor cohomology and has meaningful implications for Ulrich theory on surfaces of general type.

Abstract

We study the cohomology of divisors on a Burniat surface $X$ with $K_X^2=6$. We provide an algorithm for computing the cohomology groups of arbitrary divisors on $X$. As an application, we prove that there are no Ulrich line bundles\,(with respect to an arbitrary polarization), and that there exists an Ulrich vector bundle of rank 2 with respect to $3K_X$. The existence of Ulrich vector bundle of rank 2 was previously established by Casnati, but our construction yields one that cannot be obtained by his method.

Counting $h^0(D)$ on primary Burniat surfaces

TL;DR

The paper develops a complete algorithm to compute the cohomology of arbitrary divisors on a primary Burniat surface with , incorporating a detailed analysis of the Burniat configuration, effective divisors, and torsion in . It classifies numerical divisor classes, reduces computations to nef reduced forms, and handles torsion twists to obtain exact values. The authors apply these methods to Ulrich bundle questions, proving there are no Ulrich line bundles and producing a rank-2 Ulrich bundle for that cannot be obtained by Casnati’s construction, thus enriching the landscape of Ulrich phenomena on Burniat surfaces. Overall, the work provides explicit, algorithmic tools for divisor cohomology and has meaningful implications for Ulrich theory on surfaces of general type.

Abstract

We study the cohomology of divisors on a Burniat surface with . We provide an algorithm for computing the cohomology groups of arbitrary divisors on . As an application, we prove that there are no Ulrich line bundles\,(with respect to an arbitrary polarization), and that there exists an Ulrich vector bundle of rank 2 with respect to . The existence of Ulrich vector bundle of rank 2 was previously established by Casnati, but our construction yields one that cannot be obtained by his method.

Paper Structure

This paper contains 11 sections, 34 theorems, 138 equations, 1 figure, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Burniat configuration
  3. Effective divisors and numerical equivalences
  4. Effectiveness test
  5. Numerical equivalences and ${\sf e}$-numbers
  6. Main algorithm and its verification
  7. Reduced forms
  8. Torsions in $\operatorname{Pic} X$
  9. Classifying numerical classes
  10. Explicit computations for the remaining cases
  11. Application to the study of Ulrich bundles on $X$

Key Result

Theorem 1.1

Let $X$ be a primary Burniat surface with the canonical divisor $K_X$. Given $D \in \operatorname{Pic} X$, the Algorithm Main Algorithm computes the dimension of $H^p(D)$ for each $p=0,1,2$, and its time complexity is $O(d)$ where $d = (D. K_X)$.

Figures (1)

  • Figure 2.1: Burniat configuration

Theorems & Definitions (71)

  • Theorem 1.1
  • Theorem 1.2: see Theorem \ref{['thm: no Ulrich line bundles']}
  • Theorem 2.1: see Alexeev:Burniat and AlexeevOrlov:Burniat
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Example 3.1
  • Definition 3.2
  • Lemma 3.3
  • proof
  • ...and 61 more