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Vanishing of weight one syzygies of projective varieties

Debjit Basu

TL;DR

The paper studies vanishing of weight-one Koszul cohomology $K_{i,1}(X,L)$ for smooth projective varieties embedded by complete linear series, introducing and exploiting Property $(M_q)$ alongside $(N_p)$ to understand minimal free resolutions. A general vanishing theorem is established for line bundles $L=\ell B$ with $B$ ample and base point free, and the results are sharpened for various classes: varieties with nef $K_X$, rational and Fano varieties, and ruled varieties, with adjoint line bundles playing a central role. The work connects algebraic properties of syzygies to geometric invariants like gonality on rational surfaces and uses a hyperplane-section strategy to descend from high to low dimensions. It also develops an extensive framework for adjoint and multiple line bundles, yielding new instances where $(M_q)$ holds and linking these to k-very ampleness and projective normality, with conjectures and boundary examples illustrating the sharpness of the bounds.

Abstract

In this article we study conditions under which weight one Koszul cohomology vanishes on projective varieties. As corollary of more general results, we obtain statements on the so-called property (M_q) reflecting on the higher syzygies of minimal surfaces and higher dimensional projective varieties. By considering both properties (M_q) and (N_p), we gain a significantly deeper understanding of the minimal free resolution.

Vanishing of weight one syzygies of projective varieties

TL;DR

The paper studies vanishing of weight-one Koszul cohomology for smooth projective varieties embedded by complete linear series, introducing and exploiting Property alongside to understand minimal free resolutions. A general vanishing theorem is established for line bundles with ample and base point free, and the results are sharpened for various classes: varieties with nef , rational and Fano varieties, and ruled varieties, with adjoint line bundles playing a central role. The work connects algebraic properties of syzygies to geometric invariants like gonality on rational surfaces and uses a hyperplane-section strategy to descend from high to low dimensions. It also develops an extensive framework for adjoint and multiple line bundles, yielding new instances where holds and linking these to k-very ampleness and projective normality, with conjectures and boundary examples illustrating the sharpness of the bounds.

Abstract

In this article we study conditions under which weight one Koszul cohomology vanishes on projective varieties. As corollary of more general results, we obtain statements on the so-called property (M_q) reflecting on the higher syzygies of minimal surfaces and higher dimensional projective varieties. By considering both properties (M_q) and (N_p), we gain a significantly deeper understanding of the minimal free resolution.

Paper Structure

This paper contains 22 sections, 45 theorems, 181 equations, 2 tables.

Table of Contents

  1. Preliminaries
  2. Koszul cohomology and the Syzygy Bundle
  3. Castelnuovo Mumford regularity and surjectivity of global sections
  4. Method of hyperplane sections
  5. Some results on Vector bundles on curves
  6. Castelnuovo Mumford Regularity and the $(M_{q})$ property
  7. Property $(M_{q})$ for multiple line bundles
  8. Property-$(M_{2})$ of multiple line bundles on surfaces
  9. Syzygies of varieties with nef canonical bundle
  10. Surfaces with nef canonical bundle
  11. General results
  12. K3-surfaces
  13. Enriques surfaces
  14. Abelian and Bielliptic surfaces
  15. Varieties with nef canonical bundle
  16. ...and 7 more sections

Key Result

Theorem 2

(Multiple line bundles on smooth projective varieties) Let $X$ be a smooth projective variety and let $B$ be an ample and base point free line bundle on $X.$ Then, the line bundle $L = \ell B$ satisfies Property-$(m_{q})$ for all $\ell \geq \ell^{\mathrm{ceil}}_{q},$ where $\ell^{\mathrm{ceil}}_{q}

Theorems & Definitions (94)

  • Definition 1
  • Theorem 2
  • Theorem 3
  • Conjecture 4
  • Conjecture 5
  • Theorem 6
  • Theorem 7
  • Theorem 8
  • Definition 9
  • Theorem 10
  • ...and 84 more