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Caucher Birkar

Publications and source records attributed to Caucher Birkar.

At least 19 recordsLinked to original sources
arXiv2026

Singularities of rational maps: foundations and surfaces

We develop a theory of singularities of rational maps, focusing on maps $$ f\colon X\dashrightarrow \mathbb P^n, $$ and using their polarised graphs and normalised polarised graphs even when the source $X$ is very singular. To measure singularities of the map at a point $x\in X$, i.e. how far it is from being regular, we introduce invariants including the normalised graph fibre degree $\delta_x(f)$, a generalised lc threshold $\lambda_x(f)$, and invariants measuring the singularities of the normalised graph itself. Numerous examples show that the resulting invariants measure genuinely different aspects of map singularities. We investigate the surface case in detail. We relate the normalised graph fibre degree to multiplicity, prove a sharp threshold--degree inequality for klt surface germs, and develop a detailed theory of linear type maps on smooth and singular surfaces. In particular, we connect the existence of linear type maps to existence of smooth curves through the given point, and with local class groups and complement theory. We conclude with questions and future directions concerning higher dimensions, complements and boundedness, moduli, Cremona groups, commutative algebra, curve-counting theories, and positive characteristic.

math.AG↗
arXiv2026

Stein degree of proper morphisms

The notion of degree begins in field theory as the dimension of a field extension. In algebraic geometry, this idea reappears as the degree of a finite morphism, defined using the induced extension of function fields. For proper morphisms that are not necessarily finite, Stein factorization isolates the finite part of the map and leads to the notion of Stein degree. This invariant is especially useful in birational geometry, where it interacts naturally with singularities of pairs and the study of log Calabi-Yau fibrations. In this article we give an expository introduction to these ideas, discuss motivating examples, and explain a boundedness problem for Stein degree arising in recent work of the author and collaborators.

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arXiv2026

Sheaf stable pairs on projective surfaces and birational geometry

We study moduli space of higher rank marginally stable pairs (E,s:= (s_1,..., s_r)) consisting of torsion free coherent sheaf E of rank r and r sections (s_1,..., s_r) on a smooth projective surface. Having fixed the Chern character of E, the resulting moduli space is isomorphic to some subscheme of the Quot-scheme parametrising quotient sheaves of appropriate Chern character. We establish a connection between moduli space of higher rank stable pairs and stable minimal models induced by the sheaf E and sections s_i and the relative lc model of base surface, and use birational geometry of minimal models to analyse in detail the components of the fibre of the Hilbert-Chow morphism from the moduli space to the Hilbert scheme of effective Cartier divisors on the base surface.

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arXiv2025

Stein degree on log Calabi-Yau fibrations

We prove a conjecture proposed by the first author on boundedness of Stein degree of divisors on log Calabi-Yau fibrations. More precisely, for $d\in \mathbb{N}$ and $t\in (0,1]$, let $(X, B)\to Z$ be a log Calabi-Yau fibration of relative dimension $d$, and let $S$ be a horizontal$/Z$ irreducible component of $B$ whose coefficient in $B$ is $\ge t$. We show that the number of irreducible components of a general fibre of $S\to Z$ is bounded from above depending only on $d,t$.

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arXiv2025

Explicit Bounds on the Spectrum of 6d N=(1,0) Supergravity

We propose a novel strategy to derive explicit and uniform upper bounds on the particle spectrum of six-dimensional gravitational theories with minimal supersymmetry, focusing initially on the tensor sector. The strategy is motivated by considerations of F-theory compactifications on elliptic Calabi-Yau 3-folds. However, it admits a clear bottom-up interpretation and is thus applicable to general supergravity theories modulo certain physical conjectures. At the heart of the strategy are two key structures, most natural in birational geometry: one concerns the singularity of the natural pairs on the base manifolds, and the other, the fibration generically exhibited by the bases. Put physically, the former structure keeps the effective theories from decompactifying and the latter ensures the (generic) presence of a heterotic string. We sketch our bounding strategy and present, for an illustration, the explicit bounds thereby derived on the tensor spectrum, with the technical details relegated to a companion paper [1].

hep-th↗
arXiv2025

A Picard rank bound for base surfaces of elliptic Calabi-Yau 3-folds

We compute an explicit rank bound on the Picard group of the compact surfaces, which can serve as the base of an elliptic Calabi-Yau variety with canonical singularities. To bound the Picard rank from above, we develop a novel strategy in birational geometry, motivated in part by the physics of six-dimensional vacua of F-theory as discussed in a companion paper [1] to this one. The derivation of the concrete bound illustrates the strategy and clarifies the origin of the boundedness.

hep-th↗
arXiv2024

Singularities on vertical $\epsilon$-log canonical Fano fibrations

Given a Fano type log Calabi-Yau fibration $(X,B)\to Z$ with $(X,B)$ being $\epsilon$-lc, the first author in \cite{Bi23} proved that the generalised pair $(Z,B_Z+M_Z)$ given by the canonical bundle formula is generalised $\delta$-lc where $\delta>0$ depends only on $\epsilon$ and $\dim X-\dim Z$, which confirmed a conjecture of Shokurov. In this paper, we prove the above result under a weaker assumption. Instead of requiring $(X,B)$ to be $\epsilon$-lc, we assume that $(X,B)$ is $\epsilon$-lc vertically over $Z$, that is, the log discrepancy of $E$ with respect to $(X,B)$ is $\geq \epsilon$ for any prime divisor $E$ over $X$ whose center on $X$ is vertical over $Z$.

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arXiv2024

Sheaf stable pairs, Quot-schemes, and birational geometry

In this paper we build bridges between moduli theory of sheaf stable pairs on one hand and birational geometry on the other hand. We will in particular treat moduli of sheaf stable pairs on smooth projective curves in detail and present some calculations in low degrees. We will also outline problems in various directions.

math.AG↗
arXiv2024

Irrationality of degenerations of Fano varieties

In this paper we investigate the degrees of irrationality of degenerations of $\epsilon$-lc Fano varieties of arbitrary dimensions. We show that given a generically $\epsilon$-lc klt Fano fibration $X\to Z$ of dimension $d$ over a smooth curve $Z$ such that $(X, t F)$ is lc for a positive real number $t$ where $F$ is the reduction of an irreducible central fibre of $X$ over a closed point $z\in Z$, then $F$ admits a rational dominant map $\pi\colon F\dashrightarrow C$ to a smooth projective variety $C$ with bounded degree of irrationality depending only on $\epsilon, d, t$ such that the general fibres of $\pi$ are irreducible and rational. This proves the generically bounded case of a conjecture proposed by the first author and Loginov for log Fano fibrations of dimensions greater than three. One of the key ingredients in our proof is to modify the generically $\epsilon$-lc klt Fano fibration $X\to Z$ to a toroidal morphism of toroidal embeddings with bounded general fibres.

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arXiv2023

On explicit bounds of Fano threefolds

In this paper, we study the explicit geometry of threefolds, in particular, Fano varieties. We find an explicitly computable positive integer $N$, such that all but a bounded family of Fano threefolds have $N$-complements. This result has many applications on finding explicit bounds of algebraic invariants for threefolds. We provide explicit lower bounds for the first gap of the $\mathbb R$-complementary thresholds for threefolds, the first gap of the global lc thresholds, the smallest minimal log discrepancy of exceptional threefolds, and the volume of log threefolds with reduced boundary and ample log canonical divisor. We also provide an explicit upper bound of the anti-canonical volume of exceptional threefolds. While the bounds in this paper may not and are not expected to be optimal, they are the first explicit bounds of these invariants in dimension three.

math.AG↗
arXiv2023

Singularities on Fano fibrations and beyond

In this paper, we investigate singularities on fibrations and related topics. We prove conjectures of McKernan and Shokurov on singularities on Fano type fibrations and a conjecture of the author on singularities on log Calabi-Yau fibrations. From these we derive a variant of a conjecture of McKernan and Prokhorov on rationally connected varieties with nef anti-canonical divisor. We present further applications to other problems including boundedness of klt complements for Fano fibrations over curves, torsion index of rationally connected Calabi-Yau pairs, and gonality of fibres of del Pezzo fibrations. We prove a general result on controlling multiplicities of fibres of certain fibrations (not necessarily of Fano type) which is the key ingredient of the proofs of the above results.

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arXiv2022

Moduli of algebraic varieties

We develop a moduli theory of algebraic varieties and pairs of non-negative Kodaira dimension. We define stable minimal models and construct their projective coarse moduli spaces under certain natural conditions. This can be applied to a wide range of moduli problems in algebraic geometry.

math.AG↗
arXiv2022

Geometry of polarised varieties

In this paper we investigate the geometry of projective varieties polarised by ample and more generally nef and big Weil divisors. First we study birational boundedness of linear systems. We show that if $X$ is a projective variety of dimension $d$ with $ε$-lc singularities for $ε>0$, and if $N$ is a nef and big Weil divisor on $X$ such that $N-K_X$ is pseudo-effective, then the linear system $|mN|$ defines a birational map for some natural number $m$ depending only on $d,ε$. This is key to proving various other results. For example, it implies that if $N$ is a big Weil divisor (not necessarily nef) on a klt Calabi-Yau variety of dimension $d$, then the linear system $|mN|$ defines a birational map for some natural number $m$ depending only on $d$. It also gives new proofs of some known results, for example, if $X$ is an $ε$-lc Fano variety of dimension $d$ then taking $N=-K_X$ we recover birationality of $|-mK_X|$ for bounded $m$. We prove similar birational boundedness results for nef and big Weil divisors $N$ on projective klt varieties $X$ when both $K_X$ and $N-K_X$ are pseudo-effective (here $X$ is not assumed $ε$-lc). Using the above, we show boundedness of polarised varieties under some natural conditions. We extend these to boundedness of semi-log canonical Calabi-Yau pairs polarised by effective ample Weil divisors not containing lc centres. We will briefly discuss applications to existence of projective coarse moduli spaces of such polarised Calabi-Yau pairs.

math.AG↗
arXiv2022

Boundedness of Fano type fibrations

In this paper, we prove various results on boundedness and singularities of Fano fibrations and of Fano type fibrations. A Fano fibration is a projective morphism $X\to Z$ of algebraic varieties with connected fibres such that $X$ is Fano over $Z$, that is, $X$ has "good" singularities and $-K_X$ is ample over $Z$. A Fano type fibration is similarly defined where $X$ is assumed to be close to being Fano over $Z$. This class includes many central ingredients of birational geometry such as Fano varieties, Mori fibre spaces, flipping and divisorial contractions, crepant models, germs of singularities, etc. We develop the theory in the more general framework of log Calabi-Yau fibrations. Dans cet article, nous prouvons divers résultats sur les limites et les singularités de fibrations de Fano et les fibrations de type Fano. Une fibration de Fano est un morphisme projectif $X\to Z$ de variétés algébriques à fibres connexes tel que $X$ est Fano sur $Z$, c'est-à-dire que $X$ a de "bonnes" singularités et $-K_X$ est ample sur $Z$. Une fibration de type Fano est définie de façon similaire quand $X$ est supposé être proche d'être Fano sur $Z$. Cette classe comprend de nombreux ingrédients centraux de géométrie birationnelle tels que les variétés de fano, les espaces de fibres Mori, le flip et les contractions divisorielles, les modèles répétiteurs, les germes de singularités, etc. Nous développons la théorie dans le cadre plus général des log-fibrations de Calabi-Yau.

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