Birational Geometry of Toric Varieties

Bondal's conjecture states that the Frobenius push-forward of the structure sheaf 𝒪X generates the derived category Db(X) for smooth projective toric varieties X. Bernardi and Tirabassi exhibit a full strong exceptional collection consisting of line bundles on smooth toric Fano 3-folds assuming Bondal's conjecture. In this paper, we prove Bondal's conjecture for smooth toric Fano 3-folds and improve upon their result using birational geometry.

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Codimension bounds for the Noether–Lefschetz components for toric varieties

European Journal of Mathematics ◽

10.1007/s40879-021-00461-0 ◽

2021 ◽

Author(s):

Ugo Bruzzo ◽

William D. Montoya

Keyword(s):

Irreducible Component ◽

Toric Variety ◽

Toric Varieties ◽

Ample Divisor ◽

Smooth Hypersurface

AbstractFor a quasi-smooth hypersurface X in a projective simplicial toric variety $$\mathbb {P}_{\Sigma }$$ P Σ , the morphism $$i^*:H^p(\mathbb {P}_{\Sigma })\rightarrow H^p(X)$$ i ∗ : H p ( P Σ ) → H p ( X ) induced by the inclusion is injective for $$p=\dim X$$ p = dim X and an isomorphism for $$p<\dim X-1$$ p < dim X - 1 . This allows one to define the Noether–Lefschetz locus $$\mathrm{NL}_{\beta }$$ NL β as the locus of quasi-smooth hypersurfaces of degree $$\beta $$ β such that $$i^*$$ i ∗ acting on the middle algebraic cohomology is not an isomorphism. We prove that, under some assumptions, if $$\dim \mathbb {P}_{\Sigma }=2k+1$$ dim P Σ = 2 k + 1 and $$k\beta -\beta _0=n\eta $$ k β - β 0 = n η , $$n\in \mathbb {N}$$ n ∈ N , where $$\eta $$ η is the class of a 0-regular ample divisor, and $$\beta _0$$ β 0 is the anticanonical class, every irreducible component V of the Noether–Lefschetz locus quasi-smooth hypersurfaces of degree $$\beta $$ β satisfies the bounds $$n+1\leqslant \mathrm{codim}\,Z \leqslant h^{k-1,\,k+1}(X)$$ n + 1 ⩽ codim Z ⩽ h k - 1 , k + 1 ( X ) .

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On the Hodge conjecture for quasi-smooth intersections in toric varieties

São Paulo Journal of Mathematical Sciences ◽

10.1007/s40863-021-00247-y ◽

2021 ◽

Author(s):

Ugo Bruzzo ◽

William Montoya

Keyword(s):

Complete Intersection ◽

Toric Variety ◽

Toric Varieties ◽

Picard Group ◽

Hodge Conjecture ◽

Smooth Hypersurface ◽

Suitable Notion

AbstractWe establish the Hodge conjecture for some subvarieties of a class of toric varieties. First we study quasi-smooth intersections in a projective simplicial toric variety, which is a suitable notion to generalize smooth complete intersection subvarieties in the toric environment, and in particular quasi-smooth hypersurfaces. We show that under appropriate conditions, the Hodge conjecture holds for a very general quasi-smooth intersection subvariety, generalizing the work on quasi-smooth hypersurfaces of the first author and Grassi in Bruzzo and Grassi (Commun Anal Geom 28: 1773–1786, 2020). We also show that the Hodge Conjecture holds asymptotically for suitable quasi-smooth hypersurface in the Noether–Lefschetz locus, where “asymptotically” means that the degree of the hypersurface is big enough, under the assumption that the ambient variety $${{\mathbb {P}}}_\Sigma ^{2k+1}$$ P Σ 2 k + 1 has Picard group $${\mathbb {Z}}$$ Z . This extends to a class of toric varieties Otwinowska’s result in Otwinowska (J Alg Geom 12: 307–320, 2003).

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Embedding the Picard group inside the class group: the case of $$\mathbb {Q}$$-factorial complete toric varieties

Journal of Algebraic Combinatorics ◽

10.1007/s10801-021-01025-x ◽

2021 ◽

Author(s):

Michele Rossi ◽

Lea Terracini

Keyword(s):

Free Group ◽

Toric Variety ◽

Class Group ◽

Abelian Groups ◽

Toric Varieties ◽

Picard Group ◽

Closed Field ◽

Finitely Generated ◽

Free Part ◽

Canonical Injection

AbstractLet X be a $$\mathbb {Q}$$ Q -factorial complete toric variety over an algebraic closed field of characteristic 0. There is a canonical injection of the Picard group $$\mathrm{Pic}(X)$$ Pic ( X ) in the group $$\mathrm{Cl}(X)$$ Cl ( X ) of classes of Weil divisors. These two groups are finitely generated abelian groups; while the first one is a free group, the second one may have torsion. We investigate algebraic and geometrical conditions under which the image of $$\mathrm{Pic}(X)$$ Pic ( X ) in $$\mathrm{Cl}(X)$$ Cl ( X ) is contained in a free part of the latter group.

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