Homogeneous deformations of toric pairs

We extend the Altmann–Mavlyutov construction of homogeneous deformations of affine toric varieties to the case of toric pairs (X,∂X), where X is an affine or projective toric variety and ∂X is its toric boundary. As an application, we generalise a result due to Ilten to the case of Fano toric pairs

Some examples of non-smoothable Gorenstein Fano toric threefolds

We present a combinatorial criterion on reflexive polytopes of dimension 3 which gives a local-to-global obstruction for the smoothability of the corresponding Fano toric threefolds. As a result, we show examples of singular Gorenstein Fano toric threefolds which have compound Du Val, hence smoothable, singularities but are not smoothable.Comment: 11 pages, 2 figure

On Mirror Symmetry for Fano varieties and for singularities

In this thesis we discuss some aspects of Mirror Symmetry for Fano varieties and toric singularities. We formulate a conjecture that relates the quantum cohomology of orbifold del Pezzo surfaces to a power series that comes from Fano polygons. We verify this conjecture in some cases, in joint work with A. Oneto. We generalise the Altmann–Mavlyutov construction of deformations of toric singularities: from Minkowski sums of polyhedra we construct deformations of affine toric pairs. Moreover, we propose an approach to the study of deformations of Gorenstein toric singularities of dimension 3 in the context of the Gross–Siebert program. We construct deformations of polarised projective toric varieties by deforming their affine cones. This method is explicit in terms of Cox coordinates and it allows us to give explicit equations for a construction, due to Ilten, which produces a deformation between two toric Fano varieties when their corresponding polytopes are mutation equivalent. We also provide examples of Gorenstein toric Fano 3-folds which are locally smoothable, but not globally smoothable.Open Acces

Deformations of log Calabi–Yau pairs can be obstructed

We exhibit examples of pairs (X,D) where X is a smooth projective variety and D is an anticanonical reduced simple normal crossing divisor such that the deformations of (X,D) are obstructed. These examples are constructed via toric geometry

On deformations of toric Fano varieties

In this note we collect some results on the deformation theory of toric Fano varieties.Comment: 24 pages, 3 figure

Mirror Symmetry and smoothing Gorenstein toric affine 3-folds

We state two conjectures that together allow one to describe the set of smoothing components of a Gorenstein toric affine 3-fold in terms of a combinatorially defined and easily studied set of Laurent polynomials called 0-mutable polynomials. We explain the origin of the conjectures in mirror symmetry and present some of the evidence.Comment: 23 pages, 6 figures. The paper has been submitted to "Facets of Algebraic Geometry: A Volume in Honour of William Fulton's 80th Birthday

Reflexive polygons and rational elliptic surfaces

In this note we study in detail the geometry of eight rational elliptic surfaces naturally associated to the sixteen reflexive polygons. The elliptic fibrations supported by these surfaces correspond under mirror symmetry to the eight families of smooth del Pezzo surfaces with very ample anticanonical bundle

On K-moduli of quartic threefolds

The family of smooth Fano 3-folds with Picard rank 1 and anticanonical volume 4 consists of quartic 3-folds and of double covers of the 3-dimensional quadric branched along an octic surface. They can all be parametrised as complete intersections of a quadric and a quartic in the weighted projective space $\mathbb{P}(1,1,1,1,1,2)$, denoted by $X_{2,4} \subset \mathbb{P}(1^5,2)$; all such smooth complete intersections are K-stable. With the aim of investigating the compactification of the moduli space of quartic 3-folds given by K-stability, we exhibit three phenomena: (i) there exist K-polystable complete intersection $X_{2,2,4} \subset \mathbb{P}(1^5,2^2)$ Fano 3-folds which deform to quartic 3-folds and are neither quartic 3-folds nor double covers of quadric 3-folds - in other words, the closure of the locus parametrising complete intersections $X_{2,4}\subset \mathbb{P}(1^5,2)$ in the K-moduli contains elements that are not of this type; (ii) any quasi-smooth $X_{2,2,4} \subset \mathbb{P}(1^5,2^2)$ is K-polystable; (iii) the closure in the K-moduli space of the locus parametrising complete intersections $X_{2,2,4} \subset \mathbb{P}(1^5,2^2)$ which are not complete intersections $X_{2,4} \subset \mathbb{P}(1^5,2)$ contains only points which correspond to complete intersections $X_{2,2,4} \subset \mathbb{P}(1^5,2^2)$.Comment: 27 pages. Exposition improved and results on wall-crossing strengthened. Final version to appear in Algebraic Geometr

Reproductive aspects and habitat use by Upland Goose (Chloephaca Picta) and Ashyheaded Goose (Chloephaga Poliocephala) in Staten Island, Argentina

Durante las últimas décadas, las poblaciones de cauquenes (Chloephaga sp.) en Argentina han disminuido drásticamente. En Bahía Franklin, Isla de los Estados, se registraron y estudiaron 21 nidos de Cauquén Común (Chloephaga picta) y 4 de Cauquén Real (Chloephaga poliocephala). Todos los nidos estaban ubicados en el suelo en sectores de pendiente muy baja. La distancia entre nidos fue de 216.6 y 973 m, su densidad de 0.10 y 0.02 nidos/ha y el tamaño promedio de puesta de 5.4 y 4.7 huevos para el Cauquén Común y el Cauquén Real, respectivamente. Los nidos estaban ubicados principalmente en el turbal graminoideo, seguido por el pastizal y el arbustal. Las plantas predominantes en las inmediaciones del nido fueron Marsippospermum grandiflorum, Poa flabellata, Empetrum rubrum y Chiliotrichum diffusum, con una altura promedio de la vegetación de 72 cm en los nidos del Cauquén Común y 97 cm en los del Cauquén Real, y una baja cobertura. Se observó una asociación espacial interespecífica, posiblemente de tipo “nidificación protectora”, entre ambos cauquenes con el Pingüino Penacho Amarillo (Eudyptes chrysocome) y el Pingüino Patagónico (Spheniscus magellanicus). En el sur de la Patagonia, los perros (Canis lupus familiaris) y el visón americano (Neovison vison), entre otras especies nativas y exóticas, están provocando consecuencias negativas para la nidificación de los cauquenes. La Isla de los Estados aún no ha sido colonizada por estos predadores, lo que la hace un sitio de importancia para la reproducción y conservación de estas aves amenazadas.During the last decades wild geese (Chloephaga sp.) populations in Argentina have decreased dramatically. We recorded and studied 21 nests from the Upland Goose (Chloephaga picta) and 4 from the Ashy-headed Goose (Chloephaga poliocephala) in Bahía Franklin, Staten Island. All the nests were located on the ground in low slope areas. Distances between nests were 216.6 and 973 m, nest densities were 0.10 and 0.02 nests/ha, and clutch size was 5.4 and 4.7 eggs for the Upland Goose and the Ashyheaded Goose, respectively. Nests were located mainly on the graminoid bog, followed by grasslands and shrubby areas. Predominant plants near the nest were Marsippospermum grandiflorum, Poa flabellata, Empetrum rubrum and Chiliotrichum diffusum, with an average vegetation height of 72 cm for nests of the Upland Goose and 97 cm for nests of the Ashy-headed Goose, and low cover. We observed an interspecific spatial association between both species and the Rockhopper Penguin (Eudyptes chrysocome) and the Magellanic Penguin (Spheniscus magellanicus), possibly from the type “protective nesting”. In southern Patagonia dogs (Canis lupus familiaris) and the American mink (Neovison vison), among other native and exotic predators, have negative consequences for the nesting geese. Staten Island has not been yet colonized by these predators, thus highlighting the relevance of this site for the reproduction and conservation of these endangered species