[Paper Review] Multigraded regularity of complete intersections

Marc Chardin, Navid Nemati|arXiv (Cornell University)|Dec 29, 2020

Commutative Algebra and Its Applications6 references5 citations

TL;DR

This paper establishes explicit formulas for the bigraded Hilbert function of 0-dimensional complete intersection schemes in $\mathbb{P}^n \times \mathbb{P}^m$, showing that many values depend only on the bidegrees of the defining forms. It provides a sharp upper bound for multigraded regularity and proves that for generic complete intersections of bidegree $(d,e)$, the Hilbert function stabilizes to the degree of the scheme except on a finite set of bidegrees.

ABSTRACT

$V$ is a complete intersection scheme in a multiprojective space if it can be defined by an ideal $I$ with as many generators as $ extrm{codim}(V)$. We investigate the multigraded regularity of complete intersections scheme in $\mathbb{P}^n imes \mathbb{P}^m$. We explicitly compute many values of the Hilbert functions of $0$-dimensional complete intersections. We show that these values only depend upon $n,m$, and the bidegrees of the generators of $I$. As a result, we provide a sharp upper bound for the multigraded regularity of $0$-dimensional complete intersections.

Motivation & Objective

• To understand the multigraded regularity of complete intersection schemes in multiprojective spaces, particularly $\mathbb{P}^n \times \mathbb{P}^m$.
• To determine which homological invariants of 0-dimensional complete intersections are determined solely by the bidegrees of the defining forms.
• To compute explicit values of the bigraded Hilbert function for such schemes and identify regions where these values are independent of the specific choice of forms.
• To establish a sharp upper bound for multigraded regularity in the 0-dimensional case.
• To prove that for generic complete intersections of bidegree $(d,e)$, the Hilbert function stabilizes to the degree of the scheme except on a finite set of bidegrees.

Proposed method

• The authors define multigraded regularity in the bigraded setting using local cohomology modules with respect to the irrelevant ideal $B = B_1 \cdot B_2$.
• They use the minimal free resolution and graded Betti numbers to analyze the structure of the coordinate ring of the complete intersection.
• They exploit duality properties in the Hilbert function to compute values in specific regions of $\mathbb{N}^2$ independent of the choice of defining forms.
• They apply results from algebraic geometry and commutative algebra, including genericity arguments and dimension theory, to analyze the fiber structure of the universal family of complete intersections.
• They use localization and Jacobian-type arguments to show that the natural projections from the universal family to $\mathbb{P}^n$ and $\mathbb{P}^m$ are isomorphisms over a dense open set.
• They prove birationality of projection maps via the construction of an inverse map using partial derivatives and localization at non-vanishing Jacobian minors.

Experimental results

Research questions

• RQ1 Which values of the bigraded Hilbert function of a 0-dimensional complete intersection in $\mathbb{P}^n \times \mathbb{P}^m$ are independent of the choice of defining forms and depend only on their bidegrees?
• RQ2 Can a sharp upper bound for multigraded regularity be established for 0-dimensional complete intersections in $\mathbb{P}^n \times \mathbb{P}^m$?
• RQ3 What is the structure of the Hilbert function for generic complete intersections of bidegree $(d,e)$, and how does it stabilize?
• RQ4 How do duality properties manifest in the Hilbert function values of complete intersection points in multiprojective spaces?
• RQ5 Under what conditions is the natural projection from the universal family of complete intersections to $\mathbb{P}^n$ or $\mathbb{P}^m$ an isomorphism over a dense open subset?

Key findings

• For a 0-dimensional complete intersection in $\mathbb{P}^n \times \mathbb{P}^m$ defined by forms of given bidegrees, many values of the bigraded Hilbert function are independent of the specific choice of forms and depend only on $n$, $m$, and the bidegrees.
• The values of the Hilbert function in certain regions exhibit a duality, meaning they are determined by symmetric or complementary bidegrees.
• The paper provides a sharp upper bound for the multigraded regularity of 0-dimensional complete intersections in $\mathbb{P}^n \times \mathbb{P}^m$, depending only on the bidegrees of the defining forms.
• For generic complete intersections of bidegree $(d,e)$, the Hilbert function stabilizes to the degree of the scheme for all sufficiently large bidegrees, with only finitely many exceptions.
• The natural projections from the universal family of complete intersections to $\mathbb{P}^n$ and $\mathbb{P}^m$ are isomorphisms over a dense open subset, implying that the fibers are reduced and the scheme is generically smooth.
• The authors prove that the projection map from the universal family to $\mathbb{P}^n$ is birational over a dense open subset, and construct an explicit inverse via localization at non-vanishing partial derivatives.