[Paper Review] Decomposability of local determinantal representations of hypersurfaces

Dmitry Kerner, Victor Vinnikov|arXiv (Cornell University)|Sep 13, 2010

Commutative Algebra and Its Applications31 references3 citations

TL;DR

This paper establishes necessary and sufficient criteria for the decomposability of local determinantal representations of hypersurface singularities, particularly when the determinant is reducible. It shows that maximally generated and saturated determinantal matrices—common in algebraic geometry and commutative algebra—admit strong decomposability conditions, with explicit block-triangular or block-diagonal normal forms under equivalence.

ABSTRACT

Let M be a matrix whose entries are power series in several variables and determinant det(M) does not vanish identically. The equation det(M)=0 defines a hypersurface singularity and the (co)-kernel of M is a maximally Cohen-Macaulay module over the local ring of this singularity. Suppose the determinant det(M) is reducible, i.e. the hypersurface is locally reducible. A natural question is whether the matrix is equivalent to a block-diagonal or at least to an upper-block-triangular. (Or whether the corresponding module is decomposable or at least is an extension.) We give various necessary and sufficient criteria. Two classes of such matrices of functions appear naturally in the study of decomposability: those with many generators (e.g. maximally generated or Ulrich maximal) and those that descend from birational modifications of the hypersurface by pushforwards (i.e. correspond to modules over bigger rings). Their properties are studied.

Motivation & Objective

• To determine when a local determinantal representation of a hypersurface singularity is decomposable or an extension, especially when the determinant is reducible.
• To analyze the structure of determinantal matrices that are maximally generated or arise from birational modifications of the singularity.
• To provide explicit normal forms for such matrices under local equivalence, particularly in the case of plane curves.
• To establish conditions under which a matrix is equivalent to a block-diagonal or upper-block-triangular form based on algebraic and geometric invariants.
• To connect decomposability to properties of the adjoint matrix and relative ideal conditions in the context of modifications of the singularity.

Proposed method

• Use of local equivalence relations $\mathcal{M} \sim A\mathcal{M}B$ with $A,B \in GL(d, \mathcal{O}_{(k^n,0)})$ to classify matrices up to invertible transformations.
• Application of the adjoint matrix $\mathcal{M}^\vee$ satisfying $\mathcal{M} \mathcal{M}^\vee = \det(\mathcal{M}) \cdot \mathbf{1}$ to analyze structure and decomposability.
• Introduction of the concept of maximally generated determinantal representations, where $\operatorname{corank}(\mathcal{M}|_0) = \operatorname{mult}(X,0)$, to identify special classes amenable to strong criteria.
• Study of saturated determinantal representations that arise from pushforwards along birational modifications $\nu: (X',0) \to (X,0)$, linking decomposability to the structure of the pullback module.
• Use of column and row operations (e.g., conjugation by upper-triangular matrices) to reduce entries to normal forms with controlled $x_2$-order and degree constraints.
• Application of the relative adjoint ideal condition $\operatorname{Adj}_{(X',0)/(X,0)}$ to characterize decomposability, particularly in the context of tangential decompositions.

Experimental results

Research questions

• RQ1 When is a local determinantal representation of a reducible hypersurface singularity equivalent to a block-diagonal matrix?
• RQ2 Under what conditions is a determinantal matrix equivalent to an upper-block-triangular matrix, corresponding to an extension of modules?
• RQ3 How do maximally generated or saturated determinantal representations behave under decomposability criteria?
• RQ4 What is the role of the relative adjoint ideal in determining whether a matrix descends from a birational modification and is decomposable?
• RQ5 What explicit normal forms can be achieved for maximally generated determinantal representations of plane curves with multiple tangent lines?

Key findings

• A determinantal representation is completely decomposable if and only if the relative adjoint ideal $\operatorname{Adj}_{(X',0)/(X,0)}$ is generated by the $f_i$'s, where $f_i$ are the irreducible components of $\det(\mathcal{M})$.
• For maximally generated determinantal representations of plane curves, the matrix is equivalent to an upper-triangular form with diagonal entries $f_i$, and off-diagonal entries satisfying order and degree constraints.
• If the $x_2$-order of an off-diagonal entry $\mathcal{M}_{i,i+1}$ is at least $\min(l_i, l_{i+1})$, it can be eliminated via elementary row and column operations.
• The remaining non-zero off-diagonal entries can be normalized to $x^{n_i}$ with $n_i < \min(l_i, l_{i+1})$ via conjugation by a suitable diagonal matrix $U$.
• For the curve $\{y(y^2 - x^{2l+1}) = 0\}$, all maximally generated determinantal representations are equivalent to a matrix with entries parameterized by $H^0(\mathcal{O}_{\mathbb{P}^1}(m-2)) \times H^0(\mathcal{O}_{\mathbb{P}^1}(2l-m))/\sim$, showing a moduli space of representations.
• The space of such representations is finite-dimensional and parameterized by homogeneous polynomials of specified degrees, modulo scaling, with explicit degree bounds on the entries.