Related papers: On the Noether--Fano inequalities

We prove a more general and precise version of the Noether-Fano inequalities for birational maps between Mori fiber spaces. This is applied to give descriptions of global canonical thresholds on Fano varieties of Picard number one.

This is a report on some of the main developments in birational geometry in the last few years focusing on the minimal model program, Fano varieties, singularities and related topics, in characteristic zero.

We prove effective upper bounds on the global sections of nef line bundles of small generic degree over a fibered surface over a field of any characteristic. It can be viewed as a relative version of the classical Noether inequality for…

This is a resume of the talk delivered at the Symposium on Hodge theory, Degeneration and Complex surfaces, Tagajo, Miyagi, March 2004.

For a birational log Fano contraction, it is conjectured an inequality between the dimension of its exceptional locus and the minimal log discrepancy over the locus. The conjecture follows from the existence of the flip for the contraction…

In this technical note, we give two extensions of the classical Fano inequality in information theory. The first extends Fano's inequality to the setting of estimation, providing lower bounds on the probability that an estimator of a…

In this paper, we study the explicit geography problem of irregular Gorenstein minimal 3-folds of general type. We generalize the classical Noether-Castelnuovo inequalities for irregular surfaces to irregular 3-folds according to the…

We study the geography and birational geometry of 3-fold conic bundles over P^2 and cubic del Pezzo fibrations over P^1. We discuss many explicit examples and raise several open questions. This paper was submitted to the proceedings of the…

We study the geography of Gorenstein stable log surfaces and prove two inequalities for their invariants: the stable Noether inequality and the $P_2$-inequality. By constructing examples we show that all invariants are realised except…

This extended abstract is based on a talk given at the workshop and summer school ``Direct and Inverse Problems with Applications" in Ghent Analysis and PDE Centre in August 2024. It focuses on nonlinear diffusion equations of slow and fast…

Noether symmetries from geodetic Lagrangian for time-conformal plane symmetric spacetime are presented. Here, time conformal factor is used to find the approximate Noether symmetries. This is a generalization of the idea discussed by I.…

Noether gauge symmetry for F(R) theory of gravity has been explored recently. The fallacy is that, even after setting gauge to vanish, the form of F(R) \propto R^n (where n \neq 1, is arbitrary) obtained in the process, has been claimed to…

This paper presents a new Lie theoretic approach to fractal calculus, which in turn yields such new results as a Fractal Noether's Theorem, a setting for fractal differential forms, for vector fields, and Lie derivatives, as well as…

We construct exceptional Fano varieties with the smallest known minimal log discrepancies in all dimensions. These varieties are well-formed hypersurfaces in weighted projective space. Their minimal log discrepancies decay doubly…

We show that the minimal log discrepancy of any isolated Fano cone singularity is at most the dimension of the variety. This is based on its relation with dimensions of moduli spaces of orbifold rational curves. We also propose a…

We discuss the relation between Noether (point) symmetries and discrete symmetries for a class of minisuperspace cosmological models. We show that when a Noether symmetry exists for the gravitational Lagrangian then there exists a…

Given a rational dominant map $\phi: Y \dashrightarrow X$ between two generic hypersurfaces $Y,X \subset \mathbb{P}^n$ of dimension $\ge 3$, we prove (under an addition assumption on $\phi$) a "Noether-Fano type" inequality $m_Y \ge m_X$…

In this Lecture Notes we present, in a sufficiently self contained way, our contributions and interests in the field of Minimal Model Theory. We study Fano-Mori spaces, both from the biregular and the birational point of view. For the…

We present an addendum/erratum to the paper "Weyl Groups and Birational Transformations among Minimal Models" written by the author and published in 1995, adding the analysis of the "88-th" deformation type of a smooth Fano 3-fold with $B_2…

Consideration of the Noether variational problem for any theory whose action is invariant under global and/or local gauge transformations leads to three distinct theorems. These include the familiar Noether theorem, but also two equally…