Fujita freeness for projectivized toric vector bundles
The paper resolves Fujita's freeness conjecture for projectivized toric vector bundles, a significant result in algebraic geometry with applications to adjoint symmetric powers of vector bundles.
The paper proves that K_Y + mA is globally generated for m ≥ n+1 and ma ≥ r, where Y is the projective bundle of one-dimensional quotients of a toric vector bundle E over a smooth projective toric variety X of dimension n. The result shows that every projectivized toric vector bundle satisfies Fujita's freeness conjecture, with a uniform bound that is sharp. The authors also relate their findings to Seshadri-constant results from Hering--Mustaţă--Payne and Fulger--Murayama.
Let $X$ be a smooth projective toric variety of dimension $n\geq1$ over an algebraically closed field of characteristic zero, let $\mathcal E$ be a toric vector bundle of rank $r\geq2$, and let $π\colon Y=\mathbb P_X(\mathcal E)\to X$ be the projective bundle of one-dimensional quotients. Write an ample line bundle on $Y$ as $A=\mathcal O_Y(a)\otimesπ^*L$, with $a\geq1$. We record a blow-up argument proving that $K_Y+mA$ is globally generated whenever an integer $m$ satisfies $ma\geq r$ and $mδ(A)>n$, where $δ(A)$ is a positive integer obtained from the degrees of $A$ on the invariant quotient sections over the torus-invariant curves of $X$. In particular, $K_Y+mA$ is globally generated for $m\geq n+1$ and $ma\geq r$. Consequently every projectivized toric vector bundle satisfies Fujita's freeness conjecture. The uniform bound is sharp. We also formulate the result as a global-generation theorem for adjoint symmetric powers of $\mathcal E$ and explain its relation with the Seshadri-constant results of Hering--Mustaţă--Payne and Fulger--Murayama. ChatGPT (OpenAI) was used to assist with mathematical discussion, language, and bibliographic searches.
