We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
Published online by Cambridge University Press: 27 June 2025
This note describes some of the recent methods and effective results in the study of pluricanonical and adjoint linear systems on higherdimensional varieties. We describe an algebraic construction for the multiplier ideals and we use it to give a simple proof for the Reider theorem.
§1. The purpose of this note is to survey some of the recent results on pluricanonical and adjoint linear systems on algebraic varieties. Let A be a nef and big divisor on a smooth projective variety X. We would like to study the linear system |Kx + A|. For X a curve, it is well known that if deg A ≥ 2, the linear system |Kx + A| is free, and if deg A ≥ 3, then Kx + A is very ample. The canonical linear system |Kx| is very ample if and only if X is not a hyperelliptic curve. From the theory of curves, we know that the properties of these linear systems are closely related to the geometry of the curve. It is natural that one would like to obtain similar numerical criteria for freeness and very ampleness for adjoint linear systems on a higher-dimensional variety, and study their geometric properties.
Many results and ideas in this note are based on my joint work with R. Lazarsfeld. I would like to thank him for sharing with me many of his ideas. For surfaces of general type, Kodaira [Kod], Bombieri [Bmb] and many others have studied the behavior of the pluricanonical maps.
To save this book to your Kindle, first ensure no-reply@cambridge-org.demo.remotlog.com is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Find out more about the Kindle Personal Document Service.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.
