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Bounds for smooth Fano weighted complete intersections. (English) Zbl 1452.14043

Whereas the classification of smooth Fano varieties is achieved in dimension up to 3, there is no systematic approach to tackle the situation in which the dimension is greater than 3. In the paper under review, the authors examine the case of smooth Fano complete intersections \(X\) in weighted projective spaces.
When \(X\) is well-formed and is not an intersection with linear cones, sharp bounds on the weights of the corresponding ambient weighted projective spaces are obtained.
As a consequence of these bounds, the authors give a classification of smooth well-formed Fano weighted complete intersections of dimension 4 and 5. In addition, they deduce their Hodge numbers, together with the degrees of their defining equations, their anticanonical degrees \(K^4\) and \(K^5\) respectively, and the dimension \(h^0(-K)\) (content of Tables 1 and 3).

MSC:

14J45 Fano varieties
14M10 Complete intersections
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