5
$\begingroup$

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 $\deg(X)-{\rm codim}(X,\mathbb{P}^N)+1$, where $\deg(X)$ stands for the degree of $X$. This conjecture was stated by Eisenbud and Goto over an algebraically closed field $k$.

Eisenbud-Goto conjecture is known to hold for arithmetically Cohen-Macaulay algebraic varieties in arbitrary dimension and arbitrary field $k$ by Eisenbud and Goto.

It is proved for complex smooth curves by Gruson-Lazarsfeld-Peskine, and later on this result was generalized for connected curves in an arbitrary algebraically close field.

It is also known for smooth complex surfaces by Lazarsfeld and Pinkham, for some singular complex surfaces by Niu and for most smooth $3$-folds by Ran and Kwak.

Roughly speaking, the main trick to tackle this conjecture was introduced by Greenberg and it consist in studying the length of the special fibers certain finite map. The same procedure seems not to work in positive characteristic.

The conjecture was proved to be false by Peeva and McCullough, in any characteristic and dimension sufficiently large. However, it seems that there is no smooth (or with "reasonable" singularities) counter-example to the conjecture yet.

I wanted to ask you the following about positive characteristic:

If EG conjecture holds for $X$ over the complex numbers, does it follows trivially that EG conjecture holds for the reduction mod $p$ of $X$? What about for $p$ large enough?

If yes, what is a good reference to read about this in the literature?

Improve this question
$\endgroup$
2
  • 1
    $\begingroup$ For an integral Noetherian ring $A$, for a closed subscheme $X_A=\text{Zero}(\mathcal{I})$ of projective space $\mathbb{P}^n_A$, for a morphism from a field $\text{Spec}\ k\to \text{Spec}\ A$, the Castelnuovo-Mumford regularity of $X_k$ is less than or equal to $r$ precisely if the cohomology group $H^q_k:=H^q(\mathbb{P}^n_k,\mathcal{I}_k(r-q))$ vanishes for $q=1,\dots,n$. By cohomology and base change, for each $q$ there exists a dense Zariski open subset $U_q$ of $\text{Spec}\ A$ such that for points of $U_q$, the rank of $H^q_k$ is independent of $k$ . . . $\endgroup$
    – Jason Starr
    Commented Sep 30, 2017 at 11:28
  • 1
    $\begingroup$ . . . Thus, for the dense open $U=U_1\cap \dots \cap U_k$, for every point $\text{Spec}\ k \to U$, the fiber $X_k$ has Castelnuovo-Mumford regularity $\leq r$ if and only if the generic fiber has Castelnuovo-Mumford regularity $\leq r.$ So the answer to your second question is positive by cohomology and base change. There are many examples where the Castelnuovo-Mumford regularity jumps up in families, e.g., a quartic rational normal curve has regularity $1,$, but it specializes to a linearly degenerate smooth quartic rational curve with regularity $2.$ $\endgroup$
    – Jason Starr
    Commented Sep 30, 2017 at 11:33

0

Reset to default

You must log in to answer this question.