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Weak Fano threefolds arising as the blowup of a hyperquadric in $\mathbb{P}^4$ along a curve

Anne Schnattinger

TL;DR

The paper delivers a complete numerical and geometric description for when the blowup of a smooth hyperquadric in P^4 along a smooth curve yields a weak Fano threefold, tying the geometric condition of C lying on a smooth hypercubic section to inequalities in the curve's genus and degree. It establishes a tight equivalence: X is weak Fano if and only if C sits on a smooth sextic K3 (a hyperquadric–hypercubic intersection) and satisfies a precise no-secant constraint; the approach leverages anticanonical geometry, base-point-freeness, and genus bounds on sextic K3 surfaces. The work then classifies extremal contractions and Sarkisov links arising from these weak Fano threefolds, detailing when the anticanonical map is an isomorphism, yields divisorial contractions, or requires flops, and ties these to explicit (g,d) pairs drawn from a carefully described finite set P. The results extend prior numeric classifications by proving geometric realizability for the constructed families and by clarifying when Sarkisov links occur, thereby enriching birational geometry with explicit threefold constructions anchored in curves on quadrics and K3 surfaces.

Abstract

We characterize smooth irreducible curves $C$ on a smooth hyperquadric $Y$ of $\mathbb{P}^4$ such that the blowup of $Y$ along $C$ is a weak Fano threefold. These are precisely the smooth irreducible curves $C$ of degree $d$ and genus $g$ lying on a smooth hypercubic section of $Y$ such that (i) $C$ has no 4-secant line and no 7-secant conic; (ii) $d< 18$ and $(g,d)\not \in \{(4,7),\:(10, 11)\}$; (iii) either $3d-26<g\leq\frac{d^2-1}{12}$ or $(g,d)\in \{(4,6),\:(13,12)\}$. We prove the geometric realizability of each case, thereby proving the existence of weak Fano threefolds and Sarkisov links constructed from them, which were previously known only as numerical possibilities.

Weak Fano threefolds arising as the blowup of a hyperquadric in $\mathbb{P}^4$ along a curve

TL;DR

The paper delivers a complete numerical and geometric description for when the blowup of a smooth hyperquadric in P^4 along a smooth curve yields a weak Fano threefold, tying the geometric condition of C lying on a smooth hypercubic section to inequalities in the curve's genus and degree. It establishes a tight equivalence: X is weak Fano if and only if C sits on a smooth sextic K3 (a hyperquadric–hypercubic intersection) and satisfies a precise no-secant constraint; the approach leverages anticanonical geometry, base-point-freeness, and genus bounds on sextic K3 surfaces. The work then classifies extremal contractions and Sarkisov links arising from these weak Fano threefolds, detailing when the anticanonical map is an isomorphism, yields divisorial contractions, or requires flops, and ties these to explicit (g,d) pairs drawn from a carefully described finite set P. The results extend prior numeric classifications by proving geometric realizability for the constructed families and by clarifying when Sarkisov links occur, thereby enriching birational geometry with explicit threefold constructions anchored in curves on quadrics and K3 surfaces.

Abstract

We characterize smooth irreducible curves on a smooth hyperquadric of such that the blowup of along is a weak Fano threefold. These are precisely the smooth irreducible curves of degree and genus lying on a smooth hypercubic section of such that (i) has no 4-secant line and no 7-secant conic; (ii) and ; (iii) either or . We prove the geometric realizability of each case, thereby proving the existence of weak Fano threefolds and Sarkisov links constructed from them, which were previously known only as numerical possibilities.
Paper Structure (20 sections, 39 theorems, 97 equations, 12 tables)

This paper contains 20 sections, 39 theorems, 97 equations, 12 tables.

Table of Contents

  1. Introduction
  2. Arithmetic genus of curves on a smooth hyperquadric
  3. Arithmetic genus of (possibly singular) curves
  4. Genus of curves on hypersurfaces of a given degree
  5. Curves contained on a smooth hypercubic section
  6. Properties of the anticanonical linear system -KX
  7. Weak Fano implies that the curve lies on a smooth hypercubic section
  8. Bounding the genus of curves on a smooth hypercubic section
  9. Curves on a smooth hypercubic section yielding a weak Fano threefold
  10. Existence of curves yielding a weak Fano threefold
  11. Curves on a hyperplane or smooth hyperquadric section
  12. Existence of curves on a hyperplane section
  13. Curves on a hyperplane section yielding a weak Fano threefold
  14. Existence of curves on a smooth hyperquadric section
  15. Curves on a hyperquadric section yielding a weak Fano threefold
  16. ...and 5 more sections

Key Result

Theorem A

Let $C \subseteq Y$ be a smooth irreducible curve of genus $g$ and degree $d$ that lies on a smooth hyperquadric $Y \subseteq \mathbb{P}^4$ and denote the blowup of $Y$ along $C$ by $X$. Then $X$ is weak Fano if and only if

Theorems & Definitions (90)

  • Theorem A
  • Theorem B
  • Remark 2.1
  • Lemma 2.2
  • proof
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 80 more