The matroid stratification of the Hilbert scheme of k points in P^1
The matroid stratification of the Hilbert scheme of k points in P^1
复制标题
P^1 中 k 点的希尔伯特格式的拟阵分层
DOI：
--
复制
发表时间：
2019
期刊：
影响因子：
--
通讯作者：
Robert Silversmith
中科院分区：
文献类型：
--
作者：
Robert Silversmith
Given a homogeneous ideal $I$ in a polynomial ring over a field, one may record, for each degree $d$ and for each polynomial $fin I_d$, the set of monomials in $f$ with nonzero coefficients. These sets form a sequence of matroids indexed by $d$, collectively called the tropicalization of $I$. Tropicalizing ideals induces a "matroid stratification" on any (multi-)graded Hilbert scheme. Very little is known about the structure of these stratifications; our motivation in studying them is to understand torus orbits in the Hilbert scheme of points $(mathbb{A}^2)^{[n]}.$
In this paper, we explore many examples of matroid strata, and give a convenient way of visualizing them. We focus on the special case of principal ideals in $K[x,y],$ i.e. collections of points in $mathbb{P}^1$. In this case, we find that the matroid strata in $(mathbb{P}^1)^{[k]}$ (of which there are infinitely many, so they are not Zariski-locally closed) are exactly cut out by the Schur polynomials in $k$ variables. We find certain minimal strata (rational curves) in $(mathbb{P}^1)^{[k]}$ that are in natural bijection with binary necklaces with $k$ black beads and any number of white beads, if we are over a field containing appropriate roots of unity. We then give an application: by intersecting these special principal ideals with monomial ideals, we find corresponding strata in $(mathbb{A}^2)^{[n]}$. This proves the existence of certain new edges in the $T$-graph of $(mathbb{A}^2)^{[n]};$ classifying this graph is a longstanding open problem studied by Altmann-Sturmfels, Hering-Maclagan, and others.
Given a homogeneous ideal $I$ in a polynomial ring over a field, one may record, for each degree $d$ and for each polynomial $fin I_d$, the set of monomials in $f$ with nonzero coefficients. These sets form a sequence of matroids indexed by $d$, collectively called the tropicalization of $I$. Tropicalizing ideals induces a "matroid stratification" on any (multi-)graded Hilbert scheme. Very little is known about the structure of these stratifications; our motivation in studying them is to understand torus orbits in the Hilbert scheme of points $(mathbb{A}^2)^{[n]}.$
In this paper, we explore many examples of matroid strata, and give a convenient way of visualizing them. We focus on the special case of principal ideals in $K[x,y],$ i.e. collections of points in $mathbb{P}^1$. In this case, we find that the matroid strata in $(mathbb{P}^1)^{[k]}$ (of which there are infinitely many, so they are not Zariski-locally closed) are exactly cut out by the Schur polynomials in $k$ variables. We find certain minimal strata (rational curves) in $(mathbb{P}^1)^{[k]}$ that are in natural bijection with binary necklaces with $k$ black beads and any number of white beads, if we are over a field containing appropriate roots of unity. We then give an application: by intersecting these special principal ideals with monomial ideals, we find corresponding strata in $(mathbb{A}^2)^{[n]}$. This proves the existence of certain new edges in the $T$-graph of $(mathbb{A}^2)^{[n]};$ classifying this graph is a longstanding open problem studied by Altmann-Sturmfels, Hering-Maclagan, and others.