Toric Ulrich-wildness of higher-degree Veronese surfaces

Establish that the Veronese surface \(\bigl(\mathbb{P}^2,\mathcal{O}_{\mathbb{P}^2}(d)\bigr)\) is toric Ulrich-wild for every integer \(d>4\), by constructing an appropriate equivariant Ulrich bundle or representation embedding into the category of toric Ulrich bundles.

Background

The paper proves toric Ulrich-wildness for the Veronese surfaces of degrees $d=3$ and $d=4$. The proof begins with stable toric $d$-arithmetically Cohen–Macaulay bundles associated with Schur representations of star-shaped quivers, twists these bundles to obtain Ulrich bundles, and then applies an exact representation embedding from finite-dimensional modules over the free algebra in two generators.

For degrees $d>4$, the bundles constructed for the toric-wildness result are not Ulrich: their Euler polynomial has roots whose difference is exactly $d$ only in the cases $d=3,4$. Consequently, resolving the conjecture requires constructing a new equivariant Ulrich bundle or developing a different representation-embedding construction rather than simply rescaling the existing bundles.