Related papers: Singularities on toric fibrations

I state a conjecture asserting that for all generic klt Fano varieties X, there exists a generalised cluster variety U and a surjection from the set of torus charts on U to the set of toric specializations of X. I prove the conjecture in…

We study torus fibrations over the 2-sphere and Hurwitz equivalence of their monodromies. We show that, if two torus fibrations over $S^2$ have the same type of singularities, then their global monodromies are Hurwitz equivalent after…

We prove fibration theorems \`a la Milnor for differentiable real maps with non isolated critical values. We study the situation for maps with linear discriminant, and prove that the concept of d-regularity is the key point for the…

Let $f\colon S\to B$ a complex fibred surface with fibres of genus $g\geq 2$. Let $u_f$ be its unitary rank, i.e., the rank of the maximal unitary summand of the Hodge bundle $f_*\omega_f$. We prove many new slope inequalities involving…

We prove that a general Fano fibration $\pi\colon V\to {\mathbb P}^1$, the fiber of which is a double Fano hypersurface of index 1, is birationally superrigid provided it is sufficiently twisted over the base. In particular, on $V$ there…

We are going to use the Euler's vector fields in order to show that for real quasi-homogeneous singularities with isolated critical value, the Milnor's fibration in a "thin" hollowed tube involving the zero level and the fibration in the…

In order to understand the deformations of determinants and Pfaffians resulting from deformations of matrices, we study the deformation theory of composites $f\circ F$, with isolated singularities, where $f:Y\to\C$ has Cohen-Macaulay…

In this work we prove a bound for the torsion in Mordell-Weil groups of smooth elliptically fibered Calabi-Yau 3- and 4-folds. In particular, we show that the set which can occur on a smooth elliptic Calabi-Yau $n$-fold for ($n\geq 3$) is…

We find that for many Calabi-Yau threefolds with elliptic or genus one fibrations mirror symmetry factorizes between the fiber and the base of the fibration. In the simplest examples, the generic CY elliptic fibration over any toric base…

This is the third and last in a series of papers working towards the construction of non-trivial Cayley fibrations using gluing methods. In this paper we will show two stability results for Cayley fibrations with certains types of conical…

We consider F-theory compactifications on genus-one fibered Calabi-Yau manifolds with their fibers realized as hypersurfaces in the toric varieties associated to the 16 reflexive 2D polyhedra. We present a base-independent analysis of the…

We prove homological mirror symmetry for Milnor fibers of simple singularities in dimensions greater than one, which are among the log Fano cases of Conjecture 1.5 in arXiv:1806.04345. The proof is based on a relation between matrix…

A consistent exposition of the arguments and constructions of the method of maximal singularities, the aim of which is to describe birational iso/automorphisms of Fano varieties and Fano fibrations. The principal elements of the method are…

We classify fibrations by integral plane projective rational quartic curves whose generic fibre is regular but admits a non-smooth point that is a canonical divisor. These fibrations can only exist in characteristic two. The geometric…

We consider pairs (X,A), where X is a variety with klt singularities and A is a formal product of ideals on X with exponents in a fixed set that satisfies the Descending Chain Condition. We also assume that X has (formally) bounded…

This paper proposes the use of $F$-split and globally $F$-regular conditions in the pursuit of BAB type results in positive characteristic. The main technical work comes in the form of a detailed study of threefold Mori fibre spaces over…

The main result of this note is an effective uniform bound for the number of deformation types of certain nonisotrivial families of canonically polarized manifolds. It extends the author's earlier such bound for the classical Shafarevich…

We classify singular fibres over general points of the discriminant locus of projective complex Lagrangian fibrations on 4-dimensional holomorphic symplectic manifolds. The singular fibre F is the following either one: F is isomorphic to…

This paper classifies all toric Fano 3-folds with terminal singularities. This is achieved by solving the equivalent combinatoric problem; that of finding, up to the action of GL(3,Z), all convex polytopes in Z^3 which contain the origin as…

Suppose a sequence $M_j$ of Alexandrov spaces collapses to a space $X$ with only weak singularities. Yamaguchi constructed a map $f_j:M_j\to X$ called an almost Lipschitz submersion for large $j$. We prove that if $M_j$ has a uniform…