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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.

##### MSC:
 14C05 Parametrization (Chow and Hilbert schemes) 14J10 Families, moduli, classification: algebraic theory 14J45 Fano varieties 14M25 Toric varieties, Newton polyhedra, Okounkov bodies
##### Software:
4ti2; Graded Ring Database; Macaulay2; TOPCOM; VersalDeformations
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