Questions tagged [determinantal-ideals]

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2 votes
0 answers
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Dimension of the determinantal variety

Let $C$ be a smooth projective curve of genus $g$ over the complex field, embedded into $\mathbb{P}^r$ via a line bundle $L$ of degree $n>>0$. Let $D$ be a divisor of degree $d$ with $h^0(C,D)=s$...
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1 vote
0 answers
73 views

Determine whether a set generates a residue field of an invariant ring

Fix two positive integers $m>n$. Let $(A|Y)$ be an $m\times (n+1)$ augmented matrix consisted of $m\times (n+1)$ indeterminates, where $Y$ is a column symbolic vector of length $m$. Denote $R=\...
  • 301
1 vote
0 answers
42 views

For any initial ideal $I$ of the ideal of maximal minors, is it true that $I^n = I^{(n)}$?

Let $X$ denote a generic $n \times m$ (with $n \leq m$) matrix and $R = k[X]$, where $k$ is any field. Let $J := I_n (X)$. It is well-known that $J^t = J^{(t)}$ for all $t$ (where $-^{(t)}$ denotes ...
  • 381
3 votes
1 answer
106 views

"Classical" proof that maximal minors form a Grobner basis under diagonal term order

Let $R= k[x_{11} , x_{12} \dotsc , x_{nm}]$ denote the coordinate ring of a generic $n \times m$ matrix, $M$. It is well known that under the standard diagonal term order, the ideal of maximal minors ...
  • 381
3 votes
0 answers
102 views

Is the determinantal ideal of the span of a linearly independent set of rank-one matrices radical?

Let $k$ be an algebraically closed field, and let $X_1,\dots, X_n \in M_m(k)$ be rank-one matrices that are linearly independent over $k$. For a fixed integer $1 \leq r \leq m$, consider the ideal $I \...
  • 888
9 votes
2 answers
331 views

Can we recover all $k$-minors of a square matrix from some of them?

This is a cross-post. Let $k,n$ be natural numbers, $1<k<n$. Suppose we have an "unknown" invertible $n \times n$ matrix $A$ over a field of characteristic zero. (we do not know the entries of ...
  • 6,489
2 votes
1 answer
185 views

what are the possible approximations for ideals

(Fix some local ring $(R,\mathfrak{m})$ over a field of zero characteristic.) Suppose an ideal $J$ is defined by some complicated formula/procedure. And there is no hope of computing it/or writing ...
4 votes
3 answers
709 views

Equations for points to lie on a rational normal curve

$\def\PP{\mathbb{P}}$Let $z_1$, $z_2$, ..., $z_n$ be points in $\PP^{k-1}$. I am interested in equations for when the $z_i$ lie on a rational normal curve (or degeneration thereof.) Specifically, ...
6 votes
1 answer
1k views

Vanishing patterns of minors of matrix

Let $M$ be a $m\times n$ matrix with entries in, say $\mathbb{C}$; assume $n\leq m$. Denote by $I\subseteq\{1,2,\ldots, n\}$ a subset of the columns of M. I am interested in positive results to the ...
2 votes
1 answer
179 views

minimality/universality of the Springer resolution of a determinantal variety

Let $X\subset P^n$ be a singular determinantal variety and $S\to X$ its Springer resolution. Let $X'\to X$ another resolution of singularities (say, a blow-up). Does $S$ have some minimality/...
  • 3,727
1 vote
3 answers
668 views

Syzygies of determinantal varieties: Looking for English text

I would like to understand the syzygies of the determinantal ideal $I_r$, generated by the $r\times r$ minors of a matrix $(X_{ij})$ of indeterminantes in the polynomial ring over an algebraically ...
8 votes
1 answer
295 views

When two determinantal ideals together generate a power of the maximal ideal?

(A somewhat technical question, but maybe it is well known.) Consider matrices over the ring $k[[x_1,\dots,x_n]]$, whose entries vanish at the origin (i.e. belong to the maximal ideal $\mathfrak{m}$...