Core Concepts

This research paper explores the birational geometry of projective spaces blown up at general points, focusing on the characterization of special subvarieties called Weyl r-planes and their role in understanding the structure of effective cones and chamber decompositions.

Abstract

**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:**

- The paper establishes the equivalence of Weyl r-planes, Weyl cycles of dimension r, and stable base locus subvarieties in Mori dream spaces of the form Xn

s (projective space blown up at s general points). - For non-Mori dream spaces, the authors prove the existence of infinitely many Weyl r-planes and demonstrate that they appear as stable base locus subvarieties.
- The study introduces the concept of a Weyl chamber decomposition for the pseudoeffective cone of divisors, which serves as a coarser decomposition compared to the Mori chamber decomposition.
- The authors conjecture that the Weyl and Mori chamber decompositions coincide for the specific case of s = n + 4.

**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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Deeper Inquiries

Answer:
As the number of blown-up points (s) increases beyond n+4, the geometry of Weyl r-planes becomes significantly more intricate and less predictable for several reasons:
Infinite Weyl Orbits: For s ≥ n + 5, the Weyl group action on the curve classes generates infinite orbits. This implies an infinite number of Weyl r-planes, making their classification and individual study much more challenging compared to the finite cases (s ≤ n + 4).
Loss of Mori Dream Space Property: When s > n + 3, the spaces Xn
s are no longer Mori dream spaces. Consequently, the powerful tools and properties associated with Mori dream spaces, such as the duality between cones of divisors and curves, and the coincidence of stable base locus decomposition with the Mori chamber decomposition, are no longer available. This makes understanding the structure and behavior of Weyl r-planes more difficult.
Complexity of Stable Base Loci: The stable base loci of effective divisors become more complex to describe in non-Mori dream spaces. While Weyl r-planes remain stable base locus subvarieties, there might be other components and structures within the stable base loci that are not fully understood, potentially influencing the behavior of Weyl r-planes.
Unknown Relationship with Mori Chamber Decomposition: The conjecture about the coincidence of Weyl and Mori chamber decompositions for s = n + 4 suggests a close relationship between Weyl r-planes and the Mori chamber decomposition. However, for s ≥ n + 5, this relationship remains unclear. It is unknown whether Weyl r-planes continue to play a defining role in the Mori chamber decomposition or if other geometric structures become more relevant.
In summary, as s increases beyond n + 4, the infinite orbits, loss of Mori dream space properties, and increased complexity of stable base loci make the geometry of Weyl r-planes significantly more intricate and less understood. Further research is needed to unravel their properties and relationships with other geometric structures in these cases.

Answer:
Yes, it is highly plausible that geometric structures beyond Weyl r-planes play a crucial role in the Mori chamber decomposition of non-Mori dream spaces like Xn
s for s ≥ n + 5. Here's why:
Limitations of Weyl r-planes: While Weyl r-planes are instrumental in Mori dream spaces, their connection to the Mori chamber decomposition in non-Mori dream spaces is conjectural (for s = n + 4) and unexplored for larger s. The infinite orbits of Weyl r-planes for s ≥ n + 5 raise questions about their effectiveness in fully capturing the complexities of the Mori chamber decomposition.
Richer Birational Geometry: Non-Mori dream spaces are known for their richer and often less tractable birational geometry. This suggests that a broader range of geometric objects and phenomena might contribute to their Mori chamber decomposition.
Potential Candidates: Several potential candidates for such structures exist:
Negative curves: Curves with negative self-intersection could define extremal rays in the Mori cone and lead to contractions other than those associated with Weyl r-planes.
Higher-dimensional cycles: The paper focuses on Weyl r-planes, which are codimension r + 1 cycles. Higher-dimensional cycles, especially those with special intersection properties, could be relevant.
Birational modifications of Weyl r-planes: Images of Weyl r-planes under more complicated birational transformations (beyond the standard Cremona transformations) might define new chambers.
Open Questions: The paper highlights the lack of knowledge about chambers not of Weyl type in non-Mori dream spaces. This underscores the possibility of undiscovered structures governing the Mori chamber decomposition.
In conclusion, the limitations of Weyl r-planes in non-Mori dream spaces, combined with the expectation of richer birational geometry, strongly suggest the existence of other crucial geometric structures influencing the Mori chamber decomposition. Identifying and understanding these structures is a significant open challenge in birational geometry.

Answer:
The potential coincidence between the Weyl and Mori chamber decompositions for Xn
n+4, if proven true, would have profound implications for our understanding of birational geometry and offer new avenues for applications:
Deeper Connection between Classical Geometry and Birational Geometry: The Weyl group action, rooted in classical projective geometry, would directly dictate the structure of the Mori chamber decomposition, a fundamental concept in birational geometry. This would bridge a significant gap between these two areas, providing new insights and tools for studying birational modifications.
Simplified Study of Rational Contractions: The Mori chamber decomposition governs all possible ways to contract curves on a variety. If it coincides with the Weyl chamber decomposition, understanding the relatively simpler structure of Weyl r-planes would provide a direct route to classifying and studying all rational contractions of Xn
n+4.
Explicit Description of the Effective Cone: The conjecture about the anticanonical divisor having non-negative intersection with all effective divisors, combined with the chamber decomposition, would lead to a more explicit and geometrically meaningful description of the effective cone of divisors on Xn
n+4.
New Tools for Studying Linear Systems: The concept of Weyl expected dimension, introduced in the paper, could be rigorously established as a tool for computing the dimensions of linear systems on Xn
n+4. This would have implications for understanding maps defined by these linear systems and the geometry of their images.
Generalizations and Analogies: The case of Xn
n+4 could serve as a crucial testing ground for developing new techniques and conjectures about the interplay between Weyl group actions and birational geometry. These insights could potentially generalize to other classes of varieties or inspire analogous constructions in different contexts.
In summary, the potential coincidence between the Weyl and Mori chamber decompositions for Xn
n+4 holds significant promise for advancing our understanding of birational geometry. It would provide a powerful link between classical and birational techniques, simplify the study of rational contractions, and offer new tools for investigating linear systems and effective cones. This case has the potential to act as a springboard for broader generalizations and inspire new research directions in the field.

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