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Semiample hypersurfaces in toric varieties. (English) Zbl 1023.14027
The author investigates semiample (i.e., nef and big) hypersurfaces \(X\) of complete, simplicial toric varieties. This is a generalization of ample hypersurfaces treated by V. V. Batyrev and D. A. Cox [Duke Math. J. 75, 293-338 (1994; Zbl 0851.14021)] and J. A. Carlson and P. A. Griffiths [in: Journées de géométrie algébrique, Angers 1979, 51-76 (1980; Zbl 0479.14007)]. The semiample hypersurfaces arise from the ample ones via a modification of the ambient space obtained by a subdivision of the fan. An important example of where this generalization is needed is the mirror geometry construction via reflexive polytopes by Batyrev.
Let \(X\) be a regular hypersurface with respect to the fan, i.e., assume good intersection behavior of \(X\) with the torus orbits of the ambient space. Using Cox’s homogeneous coordinate ring and residues, the author constructs a map from certain homogeneous parts of the Jacobi ring to the middle cohomology of \(X\); different degrees yield different Hodge types. Moreover, the multiplication of the former ring gives the cup product in the latter.
While the kernel of this map is well understood, it is not surjective in the non-ample case. The author fills this gap by applying the Gysin map to the middle cohomology of some lower-dimensional strata. The behavior of these parts under the cup product is again investigated.
Finally, everything is demonstrated explicitly in the special case of \(\dim X =3\).

MSC:
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
14C30 Transcendental methods, Hodge theory (algebro-geometric aspects)
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