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Background:

Let $\mathbb{F}$ be an algebraically closed field. Let $X \subset \mathbb{F}^n$ be an affine variety. Let $\pi(X)$ be the projection of $X$ to the first $m < n$ coordinates, $$ \pi(X) = \{(x_1,\ldots,x_m): x \in X\}, $$ and for a point $a \in \mathbb{F}^m$ let $\phi(a,X)$ be the fiber of $X$ over $a$, $$ \phi(a,X) = \{x \in X: x_1=a_1,\ldots,x_m=a_m\}. $$ It is known that $\dim(\pi(X)) + \dim(\phi(a,X)) \ge \dim(X)$ for all points $a$, and that equality holds for all $a \in U$ where $U \subset \mathbb{F}^m$ is a Zariski open set (so, dimension equality holds for "typical" fibers).

Question:

Are "a-typical" fibers, where the dimension equality doesn't hold, have lower degree than $X$?

That is, for all fibers we have that $\deg(\phi(a,X)) \le \deg(X)$ since they are the intersection of $X$ with the degree $1$ variety given by $x_1=a_1,\ldots,x_m=a_m$. Can it be that when $\dim(\pi(X))+\dim(\phi(a,X))>\dim(X)$ it implies that $\deg(\phi(a,X))<\deg(X)$?

Example:

Let $X$ be defined by $x_1 x_3 + x_2 x_4=0$. Then $\dim(X)=3,\deg(X)=2$. The projection of $X$ to the first $2$ coordinates $x_1,x_2$ has dimension $2$. Fibers over $(a_1,a_2)$ if $(a_1,a_2) \ne (0,0)$ have as expected dimension $3-2=1$. However, the fiber over $(0,0)$ has dimension $2$ (which is $>1$) but degree $1$ (which is $<2$).

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Sandor: By degree I mean what MP wrote - the typical (e.g. maximal) number of points in the intersection of X with a linear subspace. MP: first, if I am not mistake the correct degree of the embedding of F will be $deg(F) * n^{dim(F)} + O(n^{dim(F)-1})$, but for large $n$ this is not important. Second, and more importantly, it seems to me that the embedding of a fiber of X will not give a fiber of the image in the sense that I wrote (i.e. a fiber of a linear projection). –  Shachar Jan 19 '12 at 2:06
    
Thanks for the answers! So excess dimension can incur excess degree. Dan (or Sandor) - as this is not my area, this might be a trivial question. Why if $\deg(\pi(X))>1$ then all fibers have degree lower than $X$? what is the relation between the degrees of $X$, $\pi(X)$ and the fibers? –  Shachar Jan 19 '12 at 14:24
    
Shachar, it would be more effective if you made comments below the answer to which the comment belongs. For one thing, then the author of the answer gets notified that there is a comment, but it is also more readable to others. –  Sándor Kovács Jan 19 '12 at 19:09
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2 Answers 2

up vote 2 down vote accepted

This is an interesting idea, but I think there are some issues with it. First of all, it seems to me that if the image $\pi(X)$ has degree $>1$, then all fibers will have strictly smaller degree than $\mathrm{deg}X$. So one might ask, if it is true that the fibers with excess dimension have smaller degree than those fibers whose dimension is the expected one. However, this actually fails.

Example: Let $X\subseteq \mathbb A^5$ be defined by $x_1=x_3x_4,x_2=x_3x_5$, and $x_4x_5=1$ and project to the first three coordinates.

For points on $\pi(X)$ with $x_3\neq 0$ the fiber consists of the single point $\bigg(x_1,x_2,x_3,\dfrac{x_1}{x_3}, \dfrac{x_2}{x_3}\bigg)$. If $x_3=0$, then the first two equations of $X$ imply that $x_1=x_2=0$ as well, so the only point on $\pi(X)$ with $x_3=0$ is $(0,0,0)$. The fiber over that is the conic $\{(0,0,0,t,u) | tu=1 \}$. So, the general fiber has degree $1$, while the (only) higher dimensional fiber has degree $2$.

This degree is still smaller than the degree of $X$, but perhaps suggest that the defect is not a consequence of the excess dimension. In fact, notice that $\pi(X)\subseteq \mathbb A^3$ is contained in the quadric cone defined by $x_1x_2=x_3^2$, and my guess is that a proper degree estimate would have to take the degree of the image into account.

Addendum to answer Shachar's question in the comments to the original question.
Here is why I think that the degree of $\pi(X)$ matters:
1) Your definition of degree seems to agree with the following: Take the projective closure of $X$ and take the degree of that in $\mathbb P^n$.
2) To determine the degree of $X$ you need a linear space of appropriate dimension in general position. That may be described as the intersection of an appropriate number of hyperplanes in the target space of the projection which would give $\mathrm{deg} \pi(X)$ number of points.
3) The intersection of the hyperplanes that project onto these ones would give the same number of fibers. To get down to finitely many points, just take the appropriate remaining number of hyperplanes to intersect.
4) So now we have $\deg X$ number of points on $\deg\pi(X)$ number of fibers, therefore the expected number of points on a single fiber is $\deg X/\deg\pi(X)$.
5) The reason I'd prefer to work projectively is that then the intersections would always give the right number of points counted with multiplicities.
6) This is not a rigorous argument, but it seems convincing to me. Cheers!

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This is not true in general. Take your example and reembed everything using the $n$-th Veronese map. A subvariety $F$ of $X$ is reembedded as a variety of degree ${\rm deg}(F) n^{\dim(F)}$. In particular, bigger dimensional fibers will have larger degree than smaller dimensional fibers, provided $n$ is large enough.

EDIT: This is what I had in mind. Choose a positive integer $d$. Let $Z=(z_1,\ldots,z_N)$ be a list of all the monomials of degree at most $d$ in $x_1,\ldots,x_n$ such that $(z_1,\ldots,z_M)$ is a list of all the monomials of degree at most $d$ in $x_1,\ldots,x_m$. Using the monomials $Z$ we obtain an embedding $v_d \colon X \to \mathbb{A}^N$, called the $d$-th Veronese embedding. A property of $v_d$ is that for every subvariety $F$ of $\mathbb{A}^n$ the equality \[ deg(v_d(F)) = deg(F) \, d^{dim(F)} \] holds. Let $\pi_d \colon X \to \mathbb{A}^M$ be the projection on the first $M$ coordinates. The morphism $\pi_d$ is identical to the morphism $\pi$ (whatever that means), but the fibers of $\pi_d$ as subschemes of $\mathbb{A}^N$ are isomorphic to the image under the $d$-th Veronese embedding of the fibers of $\pi$. Thus, for $d$ large enough, the fibers of larger dimension will have degree larger than the fibers of smaller dimension.

I hope this clarifies your doubts!

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Given Sándor's comment, here is the definition that I am using of degree: the number of points in the intersection of $X$ with general linear subspace. –  M P Jan 19 '12 at 0:28
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