# Questions tagged [castelnuovo-mumford-regularity]

The castelnuovo-mumford-regularity tag has no usage guidance.

5
questions

4
votes

1
answer

140
views

### Castelnuovo-Mumford regularity for tensor products of vector bundles

I believe this should be a well known result, but I wasn’t able to prove or find a good reference for it.
Let $E$ and $F$ be $n$-regular, respectively $m$-regular vector bundles in the sense of ...

2
votes

0
answers

117
views

### $m$-regularity of sheaves

This is Lemma 1.4 on Green and Lazarsfeld's Some results on the syzygies of finite sets and algebraic curves. Let $X$ be a closed subscheme of $\mathbb{P}^r$. Suppose the ideal sheaf $\mathcal{I}$ of $...

5
votes

0
answers

548
views

### Eisenbud-Goto conjecture in Positive Characteristic

Eisenbud-Goto conjecture predicted that the Castelnuovo-Mumford regularity
${\rm reg}(X)$ of a non-degenerate projective variety $X\subset \mathbb{P}^N$
is bounded by the $\deg(X)-{\rm codim}(X,\...

3
votes

1
answer

439
views

### Castelnuovo-Mumford regularity in multigraded case

Let $R=\oplus_{n\geq 0}R_n$ be a standard Noetherian commuative graded ring over a local ring $(A,m)$ where $R_0=A.$ Put $R_+=\oplus_{n\geq 1}R_n.$ Let $M$ be a finitely generated $\mathbb Z$-graded $...

20
votes

4
answers

4k
views

### What is interesting/useful about Castelnuovo-Mumford regularity?

What is interesting/useful about Castelnuovo-Mumford regularity?