Ulrich Bundles on Jacobian Variety of a Curve

Abstract

Let $C$ be a smooth complex projective curve of genus $g>1$ and $A=J(C)$ its Jacobian with principal polarization $Θ$. Starting from a semistable vector bundle $V$ on $C$ with $μ(V)>2g-2$, we consider the Fourier--Mukai transform $E=Φ_{\mathcal P}(a_*V)$. We prove that $E(Θ)$ satisfies the $\mathrm{IT}_0$ property. As a consequence, for every $m\ge g+1$, the polarized Jacobian $(A,mΘ)$ admits Ulrich bundles constructed functorially from $V$. Further we analyze stability and Chern classes of the resulting bundles. We show that the construction induces a natural morphism from the generically finite cover of moduli space of stable bundles on $C$ to the moduli space of stable bundles on $A$, producing positive-dimensional families of stable Ulrich bundles.

Theorems & Definitions (33)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Lemma 3.1: Uniform $H^1$-vanishing
• proof
• Remark 3.2
• Proposition 3.3: WIT$_0$ and local freeness
• proof
• Corollary 3.4: Rank formula