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Motivated by algebraic geometry, I've come up with a purely combinatorial definition within the theory of matroids. The question is: is this concept known?

If you like matroids but not algebraic geometry, skip to the definition below.

Let $n\choose k$ denote the collection of all $k$-element subsets of $[1,n]$ (rather than the number thereof). We can and will identify this collection both with the set of $T$-fixed points on the $k$-Grassmannian $Gr(k,n)$, where $T$ is the $n$-torus that acts (unfaithfully), and also with the set of Plücker coordinates.

Let $C \subseteq {n\choose k}$ be a subcollection. Then, following Neil White, we can define a subscheme $\Pi_C$ of $Gr(k,n)$ by killing all Plücker coordinates $p_S, S \notin C$. This subscheme is $T$-invariant, and its $T$-fixed points are exactly $C$.

Easy fact: if $\Pi_C$ is irreducible, then $C$ is a matroid. The non-Pappus matroid shows the converse is false. (This is my own motivation for matroids -- they serve as combinatorial stand-ins for subvarieties of Grassmannians.)

I'm interested in the smooth points of $\Pi_M$, where $M$ is a matroid. Perhaps the most efficient way to describe $M$ is by listing its connected flats $F$, and for each, giving the rank. (Saying that $rank(F) \leq r$ means that for each $S$ that intersects $F$ too much, $p_F = 0$. I'm pretty sure that the connected flats gives the shortest list of $F$s to give all the $S$.)

If $M \subseteq {n\choose k}$ is a matroid, call $\lambda \in M$ a smooth base if for any connected flat $F$, $rank(F) \leq = |\lambda \cap F|$.

Note that $\geq$ is required for $\lambda$ to be a base at all. It's pretty easy to prove that $\lambda$ is a smooth point of $\Pi_M$ iff $\lambda$ is a smooth base of $M$ in the sense above.

Is this concept known to matroid theorists? Is this characterization of smooth points known to anybody?

Example: let $M$ be the Schubert matroid for a $\lambda \in {n\choose k}$, i.e. for each $i \notin \lambda, i+1 \in \lambda$, we have a connected flat $[1,i]$ with rank $|[1,i] \cap \lambda|$. Then $\lambda$ is a smooth base of $M$. And indeed, $\lambda$ is the point in the Bruhat cell whose closure is the Schubert variety $\Pi_M$.

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Motivated by algebraic geometry, I've come up with a purely combinatorial definition within the theory of matroids. The question is: is this concept known?

If you like matroids but not algebraic geometry, skip to the definition below.

Let $n\choose k$ denote the collection of all $k$-element subsets of $[1,n]$ (rather than the number thereof). We can and will identify this collection both with the set of $T$-fixed points on the $k$-Grassmannian $Gr(k,n)$, where $T$ is the $n$-torus that acts (unfaithfully), and also with the set of Plücker coordinates.

Let $C \subseteq {n\choose k}$ be a subcollection. Then, following Neil White, we can define a subscheme $\Pi_C$ of $Gr(k,n)$ by killing all Plücker coordinate coordinates $p_S, S \notin C$. This subscheme is $T$-invariant, and its $T$-fixed points are exactly $C$.

Easy fact: if $\Pi_C$ is irreducible, then $C$ is a matroid. The non-Pappus matroid shows the converse is false. (This is my own motivation for matroids -- they serve as combinatorial stand-ins for subvarieties of Grassmannians.)

I'm interested in the smooth points of $\Pi_M$, where $M$ is a matroid. Perhaps the most efficient way to describe $M$ is by listing its connected flats $F$, and for each, giving the rank. (Saying that $rank(F) \leq r$ means that for each $S$ that intersects $F$ too much, $p_F = 0$. I'm pretty sure that the connected flats gives the shortest list of $F$s to give all the $S$.)

If $M \subseteq {n\choose k}$ is a matroid, call $\lambda \in M$ a smooth base if for any connected flat $F$, $rank(F) \leq |\lambda \cap F|$.

It's pretty easy to prove that $\lambda$ is a smooth point of $\Pi_M$ iff $\lambda$ is a smooth base of $M$ in the sense above.

Is this concept known to matroid theorists? Is this characterization of smooth points known to anybody?

Example: let $M$ be the Schubert matroid for a $\lambda \in {n\choose k}$, i.e. for each $i \notin \lambda, i+1 \in \lambda$, we have a connected flat $[1,i]$ with rank $|[1,i] \cap \lambda|$. Then $\lambda$ is a smooth base of $M$. And indeed, $\lambda$ is the point in the Bruhat cell whose closure is the Schubert variety $\Pi_M$.

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# Smooth bases of matroids

Motivated by algebraic geometry, I've come up with a purely combinatorial definition within the theory of matroids. The question is: is this concept known?

If you like matroids but not algebraic geometry, skip to the definition below.

Let $n\choose k$ denote the collection of all $k$-element subsets of $[1,n]$ (rather than the number thereof). We can and will identify this collection both with the set of $T$-fixed points on the $k$-Grassmannian $Gr(k,n)$, where $T$ is the $n$-torus that acts (unfaithfully), and also with the set of Plücker coordinates.

Let $C \subseteq {n\choose k}$ be a subcollection. Then, following Neil White, we can define a subscheme $\Pi_C$ of $Gr(k,n)$ by killing all Plücker coordinate $p_S, S \notin C$. This subscheme is $T$-invariant, and its $T$-fixed points are exactly $C$.

Easy fact: if $\Pi_C$ is irreducible, then $C$ is a matroid. The non-Pappus matroid shows the converse is false. (This is my own motivation for matroids -- they serve as combinatorial stand-ins for subvarieties of Grassmannians.)

I'm interested in the smooth points of $\Pi_M$, where $M$ is a matroid. Perhaps the most efficient way to describe $M$ is by listing its connected flats $F$, and for each, giving the rank. (Saying that $rank(F) \leq r$ means that for each $S$ that intersects $F$ too much, $p_F = 0$. I'm pretty sure that the connected flats gives the shortest list of $F$s to give all the $S$.)

If $M \subseteq {n\choose k}$ is a matroid, call $\lambda \in M$ a smooth base if for any connected flat $F$, $rank(F) \leq |\lambda \cap F|$.

It's pretty easy to prove that $\lambda$ is a smooth point of $\Pi_M$ iff $\lambda$ is a smooth base of $M$ in the sense above.

Is this concept known to matroid theorists? Is this characterization of smooth points known to anybody?

Example: let $M$ be the Schubert matroid for a $\lambda \in {n\choose k}$, i.e. for each $i \notin \lambda, i+1 \in \lambda$, we have a connected flat $[1,i]$ with rank $|[1,i] \cap \lambda|$. Then $\lambda$ is a smooth base of $M$. And indeed, $\lambda$ is the point in the Bruhat cell whose closure is the Schubert variety $\Pi_M$.