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Syzygies of Curves in Products of Projective Spaces

John Cobb

Abstract

Motivated by toric geometry, we lift machinery for understanding syzygies of curves in projective space to the setting of products of projective spaces. Using this machinery, we show an analogue of an influential result of Gruson, Peskine, and Lazarsfeld that gives a bound on the regularity of a possibly singular curve given its degree and the dimension of the ambient projective space. To do so, we show new results linking the shape of multigraded resolutions of a sheaf to its regularity region.

Syzygies of Curves in Products of Projective Spaces

Abstract

Motivated by toric geometry, we lift machinery for understanding syzygies of curves in projective space to the setting of products of projective spaces. Using this machinery, we show an analogue of an influential result of Gruson, Peskine, and Lazarsfeld that gives a bound on the regularity of a possibly singular curve given its degree and the dimension of the ambient projective space. To do so, we show new results linking the shape of multigraded resolutions of a sheaf to its regularity region.
Paper Structure (5 sections, 8 theorems, 61 equations, 3 figures)

This paper contains 5 sections, 8 theorems, 61 equations, 3 figures.

Table of Contents

  1. Background and Notation
  2. Multigraded Regularity
  3. Main Result and Proof
  4. Multiregularity of Curves in Products of Projective Spaces
  5. Examples

Key Result

Theorem A

Let $C\subseteq {\mathbb{P}}^{\bm{r}}$$(n>1)$ be a smooth nondegenerate curve of multidegree $\bm{d} = (d_1,\dots, d_n)\in {\mathbb N}^n$. Setting $a \coloneqq \max\Set*{d_i+d_j - r_i - r_j \nonscript\:\delimsize\vert\nonscript\:\mathopen{} i\neq j} + 2$, then the ideal sheaf ${\mathcal{I}}_C$ admit such that each ${\mathscr{E}}_i$ is a direct sums of line bundles ${\mathcal{O}}_{{\mathbb{P}}^{\bm

Figures (3)

  • Figure 1: Suppose that the shifts in ${\mathscr{E}}_0$ are $(0,2)$ and $(3,1)$, e.g. ${\mathscr{E}}_0 = {\mathcal{O}}_{{\mathbb{P}}^{\bm{r}}}(0,2) \oplus {\mathcal{O}}_{{\mathbb{P}}^{\bm{r}}}(3,1)$. The possible shifts for ${\mathscr{E}}_3$ are displayed in red unfilled circles.
  • Figure 2: The two regions inside $\mathop{\mathrm{Pic}}\nolimits({\mathbb{P}}^1\times {\mathbb{P}}^1) \cong {\mathbb{Z}}^2$ from Example \ref{['twopointsinP1xP1']}; $(2,1) + {\mathbb N}^2$ in red and $\mathop{\mathrm{reg}}\nolimits({\mathcal{I}}_X)$ in green.
  • Figure 3: Depicts $\mathop{\mathrm{reg}}\nolimits {\mathcal{I}}_C$ of the curve defined in Example \ref{['standardexample']}.

Theorems & Definitions (31)

  • Theorem A
  • Example
  • Proposition B
  • Definition 1.1
  • Proposition 1.2: Compare with Theorem 7.2 in MaclaganSmith04
  • Example 1.3
  • proof
  • Lemma 1.4: Compare with Lemma 7.1(2) in MaclaganSmith04
  • proof
  • Definition 1.5
  • ...and 21 more