This question is motivated by the ongoing discussion under my answer to this question. I wrote the following there:

A $(p, q, r)$ Steiner system is a collection of $q$-element subsets $A$ (called blocks) of an $r$-element set $S$ such that every $p$-element subset of $S$ is contained in a unique element of $A$. Good examples come from considering as blocks the set of hyperplanes in $\mathbb{A}^n$ or $\mathbb{P}^n$ over a finite field. For example, $\mathbb{A}^2$ over $\mathbb{F}_3$ gives a $(2, 3, 9)$ Steiner system: it contains $9$ ($\mathbb{F}_3$-rational) points, and let the blocks be the lines, each of which consists of $3$ points. Then any $2$ points are contained in a unique line. This is the unique $(2, 3, 9)$ Steiner system.

In general, considering lines in $\mathbb{A}^n$ or $\mathbb{P}^n$ gives an analogous Steiner system, and papers such as this one contain similar constructions.

Loosely, my question is: which Steiner systems come from similar constructions? I'll make this more precise in a bit.

An artificial construction allows us to realize any Steiner system as the points of a variety in $\mathbb{A}^n$ over $\mathbb{F}_2$, the blocks of which are given by the intersection of the variety with some specified hyperplanes, as follows. (This construction is due to Jeremy Booher.) Say we have a $(p,q, r)$ Steiner system with $k$ blocks; consider the subvariety of $\mathbb{A}^k$ containing the point $y_j=(a_i)_{1\leq i\leq k}$ with $a_i=0$ if the $j$-th element in our Steiner system is in block $i$ and $1$ otherwise. Then the intersections with the hyperplanes $x_i=0$ give our blocks. And this subset is a variety as any subset of $\mathbb{A}^k$ is a variety, as it is finite.

So let us try for something harder: Which Steiner systems $(p, q, r)$ come from a subvariety $X$ of $\mathbb{A}^n$ or $\mathbb{P}^n$ containing $r$ ($\mathbb{F}_s$-rational) points, with the blocks given as the intersections with all $p+1$-dimensional hyperplanes? A slightly weaker version: when is there a subvariety $X$ of $\mathbb{A}^n$ or $\mathbb{P}^n$ with $r$ $\mathbb{F}_s$-rational points such that every $p+1$-plane intersects $X$ at $q$ points? In particular, this requires that any $p$ points in $X$ be in general position, so things like rational normal curves are natural candidates.

This is probably too hard, so perhaps the simpler question is tractable:

Can you prove that some Steiner system does not come from this construction?

EDIT: One way of doing this might be to find a Steiner system with $b$ blocks, where $b$ is not a $q$-binomial coefficient with $q$ a prime power; such systems exist for $p=1$ but I am looking for a non-trivial example.

  • 1
    $\begingroup$ Perhaps the "design-theory" tag could be added. That would increase the number of design theory questions on this site by 50%. $\endgroup$ – Will Orrick Jun 24 '10 at 3:42

It seems that no Steiner System of the form $(2, 3, 25)$ can be represented in this fashion---many such systems do exist; see here. In particular, such a system would contain $100$ blocks; but no Grassmannian of lines in $\mathbb{A}^n$ or $\mathbb{P}^n$ over a finite field contains $100$ points.

However, this leaves the possibility of the following (less appealing) construction. Say a scheme $S$ is $k$-rigid for a scheme $Y$ if for any $k$ points of $Y$, $S$ embeds uniquely in $Y$ (up to automorphisms of $S$) so that it passes through all $k$ points. (This seems like a natural definition; does it already have a name?) Then we may ask for varieties $S; T\subset U$ with $S$ $p$-rigid for $U$ such that any embedding of $S$ in $U$ intersects $T$ at $q$ points, where $T$ has $r$ points.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.