Equivariant stable sheaves and toric GIT

For $(X,\,L)$ a polarized toric variety and $G\subset \mathrm {Aut}(X,\,L)$ a torus, denote by $Y$ the GIT quotient $X/\!\!/G$ . We define a family of fully faithful functors from the category of torus equivariant reflexive sheaves on $Y$ to the category of torus equivariant reflexive sheaves on $X$ . We show, under a genericity assumption on $G$ , that slope stability is preserved by these functors if and only if the pair $((X,\,L),\,G)$ satisfies a combinatorial criterion. As an application, when $(X,\,L)$ is a polarized toric orbifold of dimension $n$ , we relate stable equivariant reflexive sheaves on certain $(n-1)$ -dimensional weighted projective spaces to stable equivariant reflexive sheaves on $(X,\,L)$ .

Multigraded Castelnuovo–Mumford regularity via Klyachko filtrations

In this paper, we consider Z r \mathbb{Z}^{r} -graded modules on the Cl ⁡ ( X ) \operatorname{Cl}(X) -graded Cox ring C ⁢ [ x 1 , … , x r ] \mathbb{C}[x_{1},\dotsc,x_{r}] of a smooth complete toric variety 𝑋. Using the theory of Klyachko filtrations in the reflexive case, we construct a collection of lattice polytopes codifying the multigraded Hilbert function of the module. We apply this approach to reflexive Z s + r + 2 \mathbb{Z}^{s+r+2} -graded modules over any non-standard bigraded polynomial ring C ⁢ [ x 0 , … , x s , y 0 , … , y r ] \mathbb{C}[x_{0},\dotsc,x_{s},\allowbreak y_{0},\dotsc,y_{r}] . In this case, we give sharp bounds for the multigraded regularity index of their multigraded Hilbert function, and a method to compute their Hilbert polynomial.

What makes a complex a virtual resolution?

Virtual resolutions are homological representations of finitely generated Pic ( X ) \text {Pic}(X) -graded modules over the Cox ring of a smooth projective toric variety. In this paper, we identify two algebraic conditions that characterize when a chain complex of graded free modules over the Cox ring is a virtual resolution. We then turn our attention to the saturation of Fitting ideals by the irrelevant ideal of the Cox ring and prove some results that mirror the classical theory of Fitting ideals for Noetherian rings.

Fano schemes of complete intersections in toric varieties

We study Fano schemes $$\mathrm{F}_k(X)$$ F k ( X ) for complete intersections X in a projective toric variety $$Y\subset \mathbb {P}^n$$ Y ⊂ P n . Our strategy is to decompose $$\mathrm{F}_k(X)$$ F k ( X ) into closed subschemes based on the irreducible decomposition of $$\mathrm{F}_k(Y)$$ F k ( Y ) as studied by Ilten and Zotine. We define the “expected dimension” for these subschemes, which always gives a lower bound on the actual dimension. Under additional assumptions, we show that these subschemes are non-empty and smooth of the expected dimension. Using tools from intersection theory, we can apply these results to count the number of linear subspaces in X when the expected dimension of $$\mathrm{F}_k(X)$$ F k ( X ) is zero.

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

We 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).

Codimension bounds for the Noether–Lefschetz components for toric varieties

For 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 ) .

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

Let 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.

A comparison of positivity in complex and tropical toric geometry

Given a smooth complex toric variety we will compare real Lagerberg forms and currents on its tropicalization with invariant complex forms and currents on the toric variety. Our main result is a correspondence theorem which identifies the cone of invariant closed positive currents on the complex toric variety with closed positive currents on the tropicalization. In a subsequent paper, this correspondence will be used to develop a Bedford–Taylor theory of plurisubharmonic functions on the tropicalization.