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Hilbert schemes and toric degenerations for low degree Fano threefolds. (English) Zbl 1445.14006
Smooth Fano threefolds \(V\) with very ample anticanonical divisor \(-K_V\) lead to points \([V]\in\mathcal H_d\) in the Hilbert scheme of degree \(d:=-K_V^3\) subvarieties of \(\mathbb{P}(|-K_V|)\). These point lie on unique irreducible components. Maximal families of those Fanos, i.e. the set of those components of \(\mathcal H_d\) are completely classified. For small degree \(d\), these families can be distinguished by \((d,b_2,b_3)\) where \(b_i\) denote the \(i\)-th Betti number of \(V\). The dimensions of the corresponding components, i.e. \(h^0(\mathcal N_{V})\), are known, too.
On the other hand, toric Fano threefolds \(X\) of degree \(d\) with Gorenstein singularities correspond to three-dimensional reflexive polytopes. They are classified, too. They do also provide points \([X]\in\mathcal H_d\) which might, however, sit in several components. Presenting this incidence for \(d\leq 12\) is the main point of the paper. Thus, it provides a complete overview about toric degenerations of the above \(V\) which is important from the view point of mirror symmetry.
One of the tools is the identification of certain Stanley-Reisner schemes \(S\) wich provide points \([S]\in\mathcal H_d\), too. These are the most degenerate versions. Degenerations from \(X\) to \(S\) are understood by unimodular triangulations of the reflexive polytopes. Moreover, deformation theory of \(X\) is used because the components of \(\mathcal H_d\) containing \([X]\) are identified by the components of the tangent cone of \(\mathcal H_d\) in \([X]\). The equations of the latter are obtained via explicit computeralgebra computations.

14C05 Parametrization (Chow and Hilbert schemes)
14J10 Families, moduli, classification: algebraic theory
14J45 Fano varieties
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
Full Text: DOI
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