arXiv subjects
Publications and source records attributed to William D. Montoya.
We study G-graded Artinian algebras having Poincaré duality, considering in particular their Lefschetz properties. We also prove a correspondence between the toric setup and the G-graded one, provide an application to toric geometry, and prove a Hessian criterion in the G-graded setup
We extend the definition of Noether-Leschetz components to quasi-smooth hypersurfaces in a projective simplicial toric variety of dimension 2k+1, and prove that asymptotically the components whose codimension is upper bounded by a suitable effective constant correspond to hypersurfaces with which one can associate a small degree k-dimensional subvariety. As a corollary we get an asymptotic characterization of the components with small codimension, generalizing Otwinowska's work for odd-dimensional projective spaces and Green and Voisin's for projective 3-space. Some tools developed in this paper are a generalization of Green's theorem for simplicial toric varieties, and an extension of the notion of artinian Gorenstein ideal for the Cox ring of a toric variety.
We take a first step towards the classification of singular Mori dream $K3$ surfaces. We prove that if the Picard lattice of a singular $K3$ surface is Mori dream, then the surface is Mori dream. Moreover, we show that for singular $K3$ surfaces, of Picard rank two, being Mori dream is equivalent to contain two negative curves intersecting each other, and apply this result to study Mori dreamness of $K3$ surfaces with a singular point of type $A_n$.
Firstly we show a generalization of the (1,1)-Lefschetz theorem for projective toric orbifolds and secondly we prove that on 2k-dimensional quasi-smooth hypersurfaces coming from quasi-smooth intersection surfaces, under the Cayley trick, every rational (k,k)-cohomology class is algebraic, i.e., the Hodge conjecture holds on them.
We continue our study of the Noether-Lefschetz loci in toric varieties and investigate deformation of pairs (V,X) where V is a complete intersection subvariety and X a quasi-smooth hypersurface in a odd dimensional simplicial projective toric variety, with V\subset X. Under some assumptions, we prove that the cohomological class in H^{k,k}(X) associated to V remains of type (k,k) under an infinitesimal deformation if and only if V remains algebraic. Actually we prove that locally the Noether-Lefschetz locus is an irreducible component of a suitable Hilbert scheme. This generalizes Theorem 4.2 in our previous work [4] and the main theorem proved by Dan in [10].
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 [3]. 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. This extendes to toric varieties Otwinowska's result in [15].
For a quasi-smooth hyper-surface $X$ in a projective simplicial toric variety $P$, the morphism $i:H^p(P) \to H^p(X)$ induced by the inclusion is injective for $p=d$ and an isomorphism for $p<d-1$, where $d=dim\ P$. This allows one to define the Noether-Lefschetz locus $NL_β$ as the locus of quasi-smooth hypersurfaces of degree $β$ such that $i$ acting on the middle algebraic cohomology is not an isomorphism. In this paper we prove that, under some assumptions, if $dim P =2k+1$ and $kβ-β_0=nη$ $(n\in\mathbb N)$, where $η$ is the class of a 0-regular ample divisor, and $β_0$ is the anticanonical class, then every irreducible component $V$ of the Noether-Lefschetz locus quasi-smooth hypersurfaces of degree $β$ satifies the bounds $n+1\leq codim\ V \leq h^{k-1,k+1}(X)$.