# Questions tagged [determinantal-ideals]

The determinantal-ideals tag has no usage guidance.

15
questions

2
votes

0
answers

100
views

### Invariant factors and commuting matrices over a discrete valuation ring

$\DeclareMathOperator\Im{Im}\DeclareMathOperator\Ker{Ker}$Let $A$ be a discrete valuation ring with uniformizer $p$. Let $X, Y\in M_n(A)$ be square matrices such that $XY=YX$, and let $X^T$, $Y^T$ be ...

1
vote

0
answers

96
views

### On Serre's condition and singular locus of determinantal rings

Let $R$ be a Commutative Noetherian ring. Let $\mathbf X:=[X_{ij}]_{1\le i \le r, 1 \le s \le t}$ be a matrix of indeterminates. Let $t>1$ be an integer, and $I_t(\mathbf X)$ denote the ideal in $...

2
votes

0
answers

112
views

### Resolution of singularities of the resultant locus

We consider projective space of dimension $n$ as the parameter space of degree $n$ polynomials in one variable. Then, I am interested in resolving the singularities of the "resultant locus" $...

2
votes

0
answers

129
views

### 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 $h^0(C,D)=s$...

1
vote

0
answers

75
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=\...

1
vote

0
answers

48
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 ...

3
votes

1
answer

120
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 ...

3
votes

0
answers

111
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 \...

9
votes

2
answers

413
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 ...

2
votes

1
answer

189
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

729
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

2k
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

183
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/...

1
vote

3
answers

733
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

306
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 $\mathfrak{m}$...