# Toric degenerations of Fano threefolds giving weak Landau–Ginzburg models

@article{Ilten2013ToricDO, title={Toric degenerations of Fano threefolds giving weak Landau–Ginzburg models}, author={Nathan Owen Ilten and Jacob Lewis and Victor Przyjalkowski}, journal={Journal of Algebra}, year={2013}, volume={374}, pages={104-121} }

## 32 Citations

### Calabi-Yau compactifications of toric Landau-Ginzburg models for smooth Fano threefolds

- Mathematics
- 2016

We prove that smooth Fano threefolds have toric Landau- Ginzburg models. More precisely, we prove that their Landau-Ginzburg models, represented as Laurent polynomials, admit compactifications to…

### On the Calabi–Yau Compactifications of Toric Landau–Ginzburg Models for Fano Complete Intersections

- Mathematics
- 2017

It is well known that Givental’s toric Landau–Ginzburg models for Fano complete intersections admit Calabi–Yau compactifications. We give an alternative proof of this fact. As a consequence of this…

### On Calabi-Yau compactifications of Landau-Ginzburg models for coverings of projective spaces

- Mathematics
- 2021

We suggest the procedure that constructs a log Calabi–Yau compactification of weak Landau–Ginzburg model of a Fano variety. We apply the suggestion for del Pezzo surfaces and coverings of projective…

### Toric Degenerations and Laurent Polynomials Related to Givental's Landau–Ginzburg Models

- MathematicsCanadian Journal of Mathematics
- 2016

Abstract For an appropriate class of Fano complete intersections in toric varieties, we prove that there is a concrete relationship between degenerations to specific toric subvarieties and…

### Fibers over infinity of Landau-Ginzburg models

- Mathematics
- 2020

We conjecture that the number of components of the fiber over infinity of Landau--Ginzburg model for a smooth Fano variety $X$ equals the dimension of the anticanonical system of $X$. We verify this…

### Laurent phenomenon for Landau-Ginzburg models of complete intersections in Grassmannians

- Mathematics
- 2014

In 1997 Batyrev, Ciocan-Fontanine, Kim, and van Straten suggested a construction of Landau–Ginzburg models for Fano complete intersections in Grassmannians similar to Givental’s construction for…

### Birational geometry via moduli spaces.

- Mathematics
- 2013

In this paper we connect degenerations of Fano threefolds by projections. Using Mirror Symmetry we transfer these connections to the side of Landau-Ginzburg models. Based on that we suggest a…

### LAURENT PHENOMENON FOR LANDAU-GINZBURG MODELS OF HIGH INDEX COMPLETE INTERSECTIONS IN GRASSMANNIANS

- Mathematics
- 2015

In 1997 Batyrev, Ciocan-Fontanine, Kim, and van Straten suggested a construction of Landau--Ginzburg models for Fano complete intersections in Grassmannians similar to Givental's construction for…

### Projecting Fanos in the mirror

- Mathematics
- 2019

In the paper "Birational geometry via moduli spaces" by I. Cheltsov, L. Katzarkov, and V. Przyjalkowski a new structure connecting toric degenerations of smooth Fano threefolds by projections was…

## References

SHOWING 1-10 OF 50 REFERENCES

### Weak Landau–Ginzburg models for smooth Fano threefolds

- Mathematics
- 2009

We consider Landau–Ginzburg models for smooth Fano threefolds of the principal series and prove that they can be represented by Laurent polynomials. We check that these models can be compactified to…

### Hodge numbers of Fano threefolds via Landau--Ginzburg models

- Mathematics
- 2009

For each smooth Fano threefold $X$ with Picard number 1 we consider a weak Landau--Ginzburg model, that is a fibration over $\mathbb C^1$ given by a certain Laurent polynomial. In the spirit of L.…

### Birational geometry via moduli spaces.

- Mathematics
- 2013

In this paper we connect degenerations of Fano threefolds by projections. Using Mirror Symmetry we transfer these connections to the side of Landau-Ginzburg models. Based on that we suggest a…

### Toric Degenerations of Low Degree Fano Threefolds

- Mathematics
- 2012

For fixed degree $d\leq 12$, we study the Hilbert scheme of degree $d$ smooth Fano threefolds in their anticanonical embeddings. We use this to classify all possible degenerations of these varieties…

### Hilbert Schemes and Toric Degenerations for Low Degree Fano Threefolds

- Mathematics
- 2012

For fixed degree $d\leq 12$, we study the Hilbert scheme of degree $d$ smooth Fano threefolds in their anticanonical embeddings. We use this to classify all possible degenerations of these varieties…

### Degenerations to unobstructed Fano Stanley–Reisner schemes

- Mathematics
- 2011

We construct degenerations of Mukai varieties and linear sections thereof to special unobstructed Fano Stanley–Reisner schemes corresponding to convex deltahedra. This can be used to find toric…

### On -Fano threefolds

- Mathematics
- 2015

We study Fano threefolds with terminal Gorenstein singularities admitting a `minimal' action of a finite group. We prove that under certain additional assumptions such a variety contains no planes.…

### Toric degenerations of spherical varieties

- Mathematics
- 2004

Abstract.We prove that any affine, resp. polarized projective, spherical variety admits a flat degeneration to an affine, resp. polarized projective, toric variety. Motivated by mirror symmetry, we…

### Conifold degenerations of Fano 3-folds as hypersurfaces in toric varieties

- Mathematics
- 2012

There exist exactly 166 4-dimensional reflexive polytopes such that the corresponding 4-dimensional Gorenstein toric Fano varieties have at worst terminal singularities in codimension 3 and their…

### On Landau--Ginzburg models for Fano varieties

- Mathematics
- 2007

We observe a method for finding weak Landau-Ginzburg models for Fano varieties and find them for smooth Fano threefolds of genera 9, 10, and 12.