# Grothendieck--Lefschetz type theorems for the local Picard group

@article{Kollar2012GrothendieckLefschetzTT, title={Grothendieck--Lefschetz type theorems for the local Picard group}, author={J'anos Koll'ar}, journal={arXiv: Algebraic Geometry}, year={2012} }

We propose a strengthening of the Grothendieck--Lefschetz hyperplane theorem for the local Picard group, prove some special cases and derive several consequences to the deformation theory of log canonical singularities. Version 2: Main conjecture changed, following recent results of Bhatt and de Jong. Small changes to the proofs.

## 5 Citations

GAGA theorems.

- Mathematics
- 2018

We prove a new and unified GAGA theorem for proper schemes and algebraic spaces. This recovers all analytic and formal GAGA results in the literature, and is also valid in the non-noetherian setting.

On the smoothings of non-normal isolated surface singularities

- Mathematics
- 2015

We show that isolated surface singularities which are non-normal may have Milnor fibers which are non-diffeomorphic to those of their normalizations. Therefore, non-normal isolated singularities…

Rationality of an $$S_6$$S6-Invariant Quartic 3-Fold

- MathematicsBulletin of the Brazilian Mathematical Society, New Series
- 2019

We complete the study of rationality problem for hypersurfaces $$X_t\subset \mathbb {P}^4$$Xt⊂P4 of degree 4 invariant under the action of the symmetric group $$S_6$$S6.

On the relationship between depth and cohomological dimension

- MathematicsCompositio Mathematica
- 2015

Let $(S,\mathfrak{m})$ be an $n$-dimensional regular local ring essentially of finite type over a field and let $\mathfrak{a}$ be an ideal of $S$. We prove that if…

How much of the Hilbert function do we really need to know

- Mathematics
- 2015

We describe several examples where the leading coefficient of a Hilbert function tells us everything we need. Based on my lectures at Oberwolfach and Stony Brook.

## References

SHOWING 1-10 OF 30 REFERENCES

The Grothendieck-Lefschetz theorem for normal projective varieties

- Mathematics
- 2005

We prove that for a normal projective variety $X$ in characteristic 0, and a base-point free ample line bundle $L$ on it, the restriction map of divisor class groups $\Cl(X)\to \Cl(Y)$ is an…

Log canonical singularities are Du Bois

- Mathematics
- 2010

A recurring difficulty in the Minimal Model Program is that while log terminal singularities are quite well behaved (for instance, they are rational), log canonical singularities are much more…

A CONSTRUCTION OF Q-GORENSTEIN SMOOTHINGS OF INDEX TWO

- Mathematics
- 1992

The notion of Q-Gorenstein smoothings has been introduced by Kollar. ([KoJ], 6.2.3). This notion is essential for formulating Kollar's conjectures on smoothing components for rational surface…

Existence of log canonical closures

- Mathematics
- 2011

Let f:X→U be a projective morphism of normal varieties and (X,Δ) a dlt pair. We prove that if there is an open set U0⊂U, such that (X,Δ)×UU0 has a good minimal model over U0 and the images of all the…

Singularities of the minimal model program

- Mathematics
- 2013

Preface Introduction 1. Preliminaries 2. Canonical and log canonical singularities 3. Examples 4. Adjunction and residues 5. Semi-log-canonical pairs 6. Du Bois property 7. Log centers and depth 8.…

Moduli of varieties of general type

- Mathematics
- 2010

(1.1) Identify a class of objects whose moduli theory is nice. In some cases the answer is obvious, for instance, we should study the moduli theory of smooth projective curves. In other cases, it…

Algebraic Topology

- Mathematics

The focus of this paper is a proof of the Nielsen-Schreier Theorem, stating that every subgroup of a free group is free, using tools from algebraic topology.

Existence of log canonical flips and a special LMMP

- Mathematics
- 2011

Let (X/Z,B+A) be a Q-factorial dlt pair where B,A≥0 are Q-divisors and KX+B+A∼Q0/Z. We prove that any LMMP/Z on KX+B with scaling of an ample/Z divisor terminates with a good log minimal model or a…

Deformations of singularities

- Mathematics
- 2003

Introduction.- Deformations of singularities.- Standard bases.- Infinitesimal deformations.- Example: the fat point of multiplicity four.- Deformations of algebras.- Formal deformation theory.-…