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Ulrich bundles on smooth toric threefolds with Picard number $2$

Debojyoti Bhattacharya, Francesco Malaspina

Abstract

In this paper, we study Ulrich bundles on smooth toric threefolds with Picard number$~2$, namely $\mathbb P(\mathcal O_{\mathbb P^{2}}(a_0) \oplus \mathcal O_{\mathbb P^{2}}(a_1))$. We construct resolutions and monads for Ulrich bundles of arbitrary rank, and provide explicit examples together with a complete classification of those arising as pullbacks from $\mathbb{P}^2$. As a consequence, we also show that these varieties are Ulrich wild.

Ulrich bundles on smooth toric threefolds with Picard number $2$

Abstract

In this paper, we study Ulrich bundles on smooth toric threefolds with Picard number, namely . We construct resolutions and monads for Ulrich bundles of arbitrary rank, and provide explicit examples together with a complete classification of those arising as pullbacks from . As a consequence, we also show that these varieties are Ulrich wild.
Paper Structure (7 sections, 12 theorems, 39 equations)

This paper contains 7 sections, 12 theorems, 39 equations.

Table of Contents

  1. Preliminaries
  2. Cohomology of Ulrich bundles on O(P2)(a0) + O(P2)(a1)
  3. Smooth toric threefolds with Picard number 2
  4. Cohomological vansihing properties
  5. Resolutions of Ulrich bundles
  6. Examples and Ulrich Wildness
  7. Acknowledgements

Key Result

Proposition 1.3

Let $(X, \mathcal{O}_X(1))$ be a $n$-dimensional smooth projective variety and $\mathcal{E}$ be an initialized vector bundle on $X$. Then the following conditions are equivalent: $(i)$$\mathcal{E}$ is Ulrich. $(ii)$$H^i(\mathcal{E}(-t))=0$ for $i \geq 0$ and $1 \leq t \leq n$. $(iii)$$H^i(\mathcal{E

Theorems & Definitions (31)

  • Definition 1.1
  • Definition 1.2
  • Proposition 1.3
  • proof
  • Definition 1.4
  • Theorem 1.5: Beilinson spectral sequence
  • Theorem 1.6
  • Remark 1.7
  • Lemma 2.1
  • proof
  • ...and 21 more