Rank-two reflexive sheaves on the projective space with second Chern class equal to four

TL;DR

The paper classifies spectra for rank-2 reflexive sheaves on ${\mathbb{P}}^3$ with $c_2=4$ and analyzes the moduli spaces ${\mathcal{R}}(c_1,c_2,c_3)$ for $c_1\in\{-1,0\}$ and $c_3\ge8$. Using Serre’s correspondence and Liaison Theory, it realizes all admissible spectra via explicit curve data, and provides a detailed stratification of the moduli spaces, including cases with $c_3=8,10,12$. The work gives both obstructed and unobstructed loci, proves many components are smooth and unirational, and supplies explicit constructions and cohomology data (with $\chi(F(l))$ and $h^i(F(l))$) for representative sheaves. Overall, it advances a complete picture of the rank-2 reflexive sheaf landscape on ${\mathbb{P}}^3$ at $c_2=4$, linking spectra, curves, and deformation theory. The results have implications for understanding the geometry of moduli spaces of stable sheaves on projective 3-folds and for the explicit construction of examples with controlled Chern data.

Abstract

We study rank-two reflexive sheaves on $\mathbb{P}^3$ with $c_2 =4$, expanding on previous results for $c_2\le3$. We show that every spectrum not previously ruled out is realized. Moreover, moduli spaces are studied and described in detail for $c_1=-1$ or $0$ and $c_3\ge8$.

Theorems & Definitions (52)

• Theorem 1
• Theorem 2
• Lemma 3: HRR
• Theorem 4: Hartshorne
• Remark 5
• Lemma 6
• proof
• Lemma 7
• proof
• Proposition 8