## 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)].
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 $$k>2$$).

### MSC:

 14M25 Toric varieties, Newton polyhedra, Okounkov bodies 14C20 Divisors, linear systems, invertible sheaves

### Citations:

Zbl 0659.14002; Zbl 1316.14001
Full Text:

### References:

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