Bibliographic Information: Brambilla, M. C., Dumitrescu, O., Postinghel, E., & Santana Sánchez, L. J. (2024). Birational geometry of blowups via Weyl chamber decompositions and actions on curves. arXiv preprint arXiv:2410.18008v1.
Research Objective: This paper aims to characterize a special class of subvarieties called Weyl r-planes in blown-up projective spaces and investigate their connection to the Mori chamber decomposition and stable base locus decomposition of effective cones.
Methodology: The authors utilize tools from birational geometry, including Cremona transformations, Weyl group actions, and the study of cones of curves and divisors. They analyze the effective Weyl orbits of curves and divisors to understand the structure of these cones and their decompositions.
Key Findings:
Main Conclusions: This work provides a comprehensive characterization of Weyl r-planes in blown-up projective spaces and highlights their significance in understanding the birational geometry of these varieties. The introduction of the Weyl chamber decomposition offers a new perspective on the structure of effective cones, particularly in non-Mori dream spaces.
Significance: This research contributes significantly to the field of birational geometry by providing new insights into the structure of blown-up projective spaces. The findings have implications for understanding the geometry of these varieties and their associated cones of curves and divisors.
Limitations and Future Research: The authors acknowledge that the birational geometry of non-Mori dream spaces is less understood and suggest further research into cases with s ≥ n + 5. The conjecture regarding the coincidence of Weyl and Mori chamber decompositions for s = n + 4 presents an intriguing avenue for future investigation.
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