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**Fujita’s freeness conjecture for \(T\)-varieties of complexity one.**
*(English)*
Zbl 1472.14054

For a not too singular projective variety \(X\) and an ample divisor \(H\) on it, T. Fujita conjectured in [Adv. Stud. Pure Math. 10, 167–178 (1987; Zbl 0659.14002)] that \(mH+K_X\) is basepoint free, if \(m > \dim(X)\).

This conjecture holds for curves by the Riemann-Roch theorem. If one asks only for nefness, then the conjecture holds for any \(X\) with at most rational Gorenstein singularities, as shown by Fujita in [loc. cit.]. As a nef divisor on a toric variety is automatically basepoint free, the original conjecture holds for Gorenstein toric varieties.

Based on these two results, it is quite natural to ask, whether the conjecture holds for \(T\)-varieties of complexity one with at most rational Gorenstein singularities. Such a variety \(X\) comes with an effectiv action of a torus \(T\) with \(\dim(T) = \dim(X)-1\), so the Chow quotient \(Y = X/T\) is a curve and a generic fiber a toric variety, see [K. Altmann et al., in: Contributions to algebraic geometry. Impanga lecture notes. Based on the Impanga conference on algebraic geometry, Banach Center, Bȩdlewo, Poland, July 4–10, 2010. Zürich: European Mathematical Society (EMS). 17–69 (2012; Zbl 1316.14001)].

The central result of this article is that in this case, Fujita’s Freeness Conjecture holds.

For the proof, note that for general \(T\)-varieties, nefness does not automatically imply basepoint freeness. Indeed the authors show that implication only for nef divisors of the form \(mH+K_X\), with \(H\) ample and \(m > \dim(X)\). Moreover, for \(k \in \mathbb N\), they give a sequence of smooth \(\mathbb K^*\)-surfaces \(X_k\) (the simplest \(T\)-varieties of complexity one) with ample divisor \(H_k\) such that \(kH_k\) is not basepoint free (but are nef as soon as \(k>2\)).

This conjecture holds for curves by the Riemann-Roch theorem. If one asks only for nefness, then the conjecture holds for any \(X\) with at most rational Gorenstein singularities, as shown by Fujita in [loc. cit.]. As a nef divisor on a toric variety is automatically basepoint free, the original conjecture holds for Gorenstein toric varieties.

Based on these two results, it is quite natural to ask, whether the conjecture holds for \(T\)-varieties of complexity one with at most rational Gorenstein singularities. Such a variety \(X\) comes with an effectiv action of a torus \(T\) with \(\dim(T) = \dim(X)-1\), so the Chow quotient \(Y = X/T\) is a curve and a generic fiber a toric variety, see [K. Altmann et al., in: Contributions to algebraic geometry. Impanga lecture notes. Based on the Impanga conference on algebraic geometry, Banach Center, Bȩdlewo, Poland, July 4–10, 2010. Zürich: European Mathematical Society (EMS). 17–69 (2012; Zbl 1316.14001)].

The central result of this article is that in this case, Fujita’s Freeness Conjecture holds.

For the proof, note that for general \(T\)-varieties, nefness does not automatically imply basepoint freeness. Indeed the authors show that implication only for nef divisors of the form \(mH+K_X\), with \(H\) ample and \(m > \dim(X)\). Moreover, for \(k \in \mathbb N\), they give a sequence of smooth \(\mathbb K^*\)-surfaces \(X_k\) (the simplest \(T\)-varieties of complexity one) with ample divisor \(H_k\) such that \(kH_k\) is not basepoint free (but are nef as soon as \(k>2\)).

Reviewer: Andreas Hochenegger (Milano)

### MSC:

14M25 | Toric varieties, Newton polyhedra, Okounkov bodies |

14C20 | Divisors, linear systems, invertible sheaves |

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\textit{K. Altmann} and \textit{N. Ilten}, Mich. Math. J. 69, No. 2, 323--340 (2020; Zbl 1472.14054)

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