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On Mori dreamness of blowups along space curves

Tiago Duarte Guerreiro, Sokratis Zikas

TL;DR

The paper analyzes when the blowup $X$ of $\mathbb{P}^3$ along a smooth space curve $C$ is a Mori Dream Space (MDS). It develops two complementary directions: constructive positive results showing $X$ is an MDS for curves that are complete/almost complete intersections or lie on low-degree surfaces, and obstructions from external geometry (super-rigid/linkage and extremal surfaces) that yield non-MDS on infinitely many Hilbert schemes components. Employing Hilbert-flag schemes and liaison theory, it demonstrates that Mori dreamness is not an open property in flat families and constructs degenerations illustrating multiple Mori chambers or non-semiample nef divisors. A key technical theme is the use of Pell-type arithmetic on quartic surfaces and the analysis of nef/movable/effective cones via extremal rays, enabling explicit criteria for (non-)MDS behavior and revealing intricate birational dynamics governed by linkage. Overall, the work bridges explicit curve geometry with birational MDS theory to produce both broad families of MDS blows-ups and systematic obstructions, including concrete counterexamples to openness in families.

Abstract

We study the problem of determining when the blowup $X \to \mathbb{P}^3$ along a smooth space curve $C$ is a Mori Dream Space. We obtain sufficient conditions, as well obstructions to the Mori dreamness of $X$ based on the external geometry of $C$. We furthermore find infinitely many pairs $(g,d)$ such that the corresponding Hilbert schemes $H_{g,d}^S$ admit components whose general element has these obstructions. As a consequence we show that Mori dreamness is not an open property in flat families and exhibit various degenerational pathologies.

On Mori dreamness of blowups along space curves

TL;DR

The paper analyzes when the blowup of along a smooth space curve is a Mori Dream Space (MDS). It develops two complementary directions: constructive positive results showing is an MDS for curves that are complete/almost complete intersections or lie on low-degree surfaces, and obstructions from external geometry (super-rigid/linkage and extremal surfaces) that yield non-MDS on infinitely many Hilbert schemes components. Employing Hilbert-flag schemes and liaison theory, it demonstrates that Mori dreamness is not an open property in flat families and constructs degenerations illustrating multiple Mori chambers or non-semiample nef divisors. A key technical theme is the use of Pell-type arithmetic on quartic surfaces and the analysis of nef/movable/effective cones via extremal rays, enabling explicit criteria for (non-)MDS behavior and revealing intricate birational dynamics governed by linkage. Overall, the work bridges explicit curve geometry with birational MDS theory to produce both broad families of MDS blows-ups and systematic obstructions, including concrete counterexamples to openness in families.

Abstract

We study the problem of determining when the blowup along a smooth space curve is a Mori Dream Space. We obtain sufficient conditions, as well obstructions to the Mori dreamness of based on the external geometry of . We furthermore find infinitely many pairs such that the corresponding Hilbert schemes admit components whose general element has these obstructions. As a consequence we show that Mori dreamness is not an open property in flat families and exhibit various degenerational pathologies.

Paper Structure

This paper contains 18 sections, 36 theorems, 107 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Mori Dream Spaces
  4. Linkage and the Hilbert-flag scheme
  5. $\mathbb{Q}$-canonical curves
  6. Curves on quartic surfaces
  7. Sufficient Conditions for Mori dreamness
  8. Complete and almost complete intersections
  9. Curves in low degree surfaces
  10. Components of $H_{g,d}^S$ with obstructions to Mori dreamness
  11. Super-rigid linkage
  12. Curves with extremal surfaces
  13. Rigid and super-rigid skew linkage
  14. Definition and basic properties
  15. Two contraction statements
  16. ...and 3 more sections

Key Result

Theorem 1

Let $(g,d)$ be integers satisfying $8g<d^2$ and let $C$ be a general element in $H_{g,d}^S$ contained in a quartic surface. Define $r = d^2-8(g-1)$ and suppose that the generalized Pell equations do not admit any integer solutions, and that either $d \geq 16$ or $64 - 8d + 2g -2 \leq 0$. Denote by $X \to \mathbb{P}^3$ the blowup along $C$. Then $\overline{\operatorname{Mov}}(X)$ has an irrational

Figures (5)

  • Figure 1: The nef cone of $X$ as a slice of that of $S$.
  • Figure 2: $\operatorname{NE}(X)$ with the rays spanned by the $\gamma_i$, the shaded region being the $D_a$-negative half space.
  • Figure 3: $\operatorname{Eff}(X)$ with the shaded and striped regions signifying the movable and nef cones respectively.
  • Figure 4: The inductive step in the proof of Proposition \ref{['prop:SQMci']}
  • Figure : $\operatorname{NE}(X)$ with the rays spanned by the $\gamma_i$, the shaded region being the $D_a$-negative half space.

Theorems & Definitions (90)

  • Theorem 1: Theorem \ref{['thm:nonMDSquartics']}
  • Theorem 2: Theorem \ref{['thm:mainThmRigid']} & Corollary \ref{['cor:degenToSemiample']}
  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Proposition 2.5: HarDef
  • Definition 2.6
  • Proposition 2.7: KleppeFlag
  • Proposition 2.8: KleppeFlag
  • ...and 80 more