ASYMPTOTIC INVARIANTS OF LINE BUNDLES

@article{Ein2005ASYMPTOTICIO,
  title={ASYMPTOTIC INVARIANTS OF LINE BUNDLES},
  author={Lawrence Ein and Robert Lazarsfeld and Mircea Mustaţǎ and Michael Nakamaye and Mihnea Cristian Popa},
  journal={Pure and Applied Mathematics Quarterly},
  year={2005},
  volume={1},
  pages={379-403}
}
The purpose of the present expository note is to give an invitation to this circle of ideas. Our hope is that this informal overview might serve as a jumping off point for the more technical literature in the area. Accordingly, we sketch many examples but include no proofs. In an attempt to make the story as appealing as possible to non-specialists, we focus on one particular invariant — the “volume” — that measures the rate of growth of sections of powers of a line bundle. Unfortunately, we… 

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References

SHOWING 1-10 OF 50 REFERENCES

LECTURES ON LINEAR SERIES 1

The past decade has witnessed two important new developments in the study of linear series on algebraic varieties. First, vector bundles have emerged as powerful tools for analyzing linear series on

A numerical criterion for very ample line bundles

— Let X be a projective algebraic manifold of dimension n and let L be an ample line bundle over X . We give a numerical criterion ensuring that the adjoint bundle KX + L is very ample. The

Asymptotic cohomological functions on projective varieties

Our purpose here is to consider certain cohomological invariants associated to complete linear systems on projective varieties. These invariants — called asymptotic cohomological functions — are

Approximating Zariski decomposition of big line bundles

If H\ίπ*L)=H°(ίH) for every t>0 such that tE is a Z-divisor, then π*L = E+H is a Zariski decomposition of L. In such a case we have h°(V, lL) = dt/n !+(lower order terms) for d = H. However,

ON THE VOLUME OF A LINE BUNDLE

Using the Calabi–Yau technique to solve Monge-Ampere equations, we translate a result of T. Fujita on approximate Zariski decompositions into an analytic setting and combine this to the holomorphic

A subadditivity property of multiplier ideals.

Let X be a smooth complex quasi-projective variety, and let D be an effective Q-divisor on X. One can associate to D its multiplier ideal sheaf J (D) = J (X,D) ⊆ OX , whose zeroes are supported on

Mori dream spaces and GIT.

The main goal of this paper is to study varieties with the best possible Mori theoretic properties (measured by the existence of a certain decomposition of the cone of effective divisors). We call

Periodicity of the fixed locus of multiples of a divisor on a surface

In his paper "The theorem of Riemann-Roch for high multiples of an effective divisor on an algebraic surface" I-Z], Zariski studies the Riemann-Roch problem on an algebraic surface. His main results

Zero-estimates, intersection theory, and a theorem of Demailly

Let X be a smooth complex projective variety of dimension n, and let A be an ample line bundle on X . Fujita has conjectured that the adjoint bundle OX(KX+mA) is basepoint-free if m ≥ n + 1 and very

The pseudo-effective cone of a compact K\

We prove that a holomorphic line bundle on a projective manifold is pseudo-effective if and only if its degree on any member of a covering family of curves is non-negative. This is a consequence of a