On the cone of effective surfaces on \(\overline{\mathcal{A}}_3\). (English) Zbl 1505.14100
In this nice article the authors study the cone of effective surfaces on the toroidal compactification of the moduli space of complex principally polarized abelian threefolds. The main result determines five extremal effective rays of the cone by constructing five geometrically meaningful families of abelian threefolds over surface bases. In order to prove the result, the authors study in detail the intersection theory and boundary strata of the moduli space as well as the cycle classes of these extremal surfaces. They further conjecture that these extremal rays generate the cone of effective surfaces in this case and more generally generate the cone of effective surfaces on the perfect cone compactification for any dimension bigger than or equal to three.
Reviewer: Dawei Chen (Chestnut Hill)
MSC:
| 14K10 | Algebraic moduli of abelian varieties, classification |
| 14E30 | Minimal model program (Mori theory, extremal rays) |
| 14C25 | Algebraic cycles |
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