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Birational geometry of weighted complete intersections of type $(12, 14)$ in $\mathbb{P} (1, 2, 3, 4, 7, 11)$

Takuzo Okada

TL;DR

The paper proves that the codimension‑2 Fano 3‑fold $X = X_{12,14} \subset \mathbb{P} (1,2,3,4,7,11)$ is birationally solid (hence irrational) by constructing a birational link to a degree‑7 hypersurface $\hat{X} \subset \mathbb{P} (1,1,1,2,3)$ with a terminal $cE_6$ singularity and analyzing all potential Mori fiber space centers via the Sarkisov program. The analysis shows that $X$ has a unique elementary link to $\hat{X}$ (and in the $\lambda\neq0$ case, a second link $\sigma'$ exists on the hatX side, yielding a birational involution). On the hatX side, the singular structure includes points $\mathsf{p}_t$, $\mathsf{p}_v$, and $\hat{\mathsf{q}}$, with a finite, explicitly bounded set of divisors of discrepancy $1$; no additional maximal centers occur. The results provide the first explicit birationally solid example for a Fano 3‑fold WCI of codimension $\ge2$ and index $\ge2$, supporting a broader conjecture on solidity in this family.

Abstract

We show that any quasismooth Fano threefold weighted complete intersections of type $(12, 14)$ in $\mathbb{P} (1, 2, 3, 4, 7, 11)$ is birationally solid.

Birational geometry of weighted complete intersections of type $(12, 14)$ in $\mathbb{P} (1, 2, 3, 4, 7, 11)$

TL;DR

The paper proves that the codimension‑2 Fano 3‑fold is birationally solid (hence irrational) by constructing a birational link to a degree‑7 hypersurface with a terminal singularity and analyzing all potential Mori fiber space centers via the Sarkisov program. The analysis shows that has a unique elementary link to (and in the case, a second link exists on the hatX side, yielding a birational involution). On the hatX side, the singular structure includes points , , and , with a finite, explicitly bounded set of divisors of discrepancy ; no additional maximal centers occur. The results provide the first explicit birationally solid example for a Fano 3‑fold WCI of codimension and index , supporting a broader conjecture on solidity in this family.

Abstract

We show that any quasismooth Fano threefold weighted complete intersections of type in is birationally solid.

Paper Structure

This paper contains 10 sections, 14 theorems, 90 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Maximal singularities
  4. Weighted projective spaces and rank $2$ toric varieties
  5. Special divisors over a $3$-fold germ
  6. Exclusion for $X_7 \subset \mathbb{P} (1, 1, 1, 2, 3)$
  7. Proof of Theorem \ref{['mainthm']}
  8. Analysis of $\hat{X}$
  9. Construction of a link in the case $\lambda \ne 0$
  10. Proof of Theorem \ref{['mainthm']}

Key Result

Theorem 1.1

Let $X = X_{12, 14} \subset \mathbb{P} (1, 2, 3, 4, 7, 11)$ be a quasismooth Fano $3$-fold weighted complete intersection of type $(12, 14)$. Then $X$ is birational to a Fano $3$-fold weighted hypersurface $\hat{X} \subset \mathbb{P} (1, 1, 1, 2, 3)$ of degree $7$ admitting a terminal singularity of

Theorems & Definitions (29)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.2: Ok3
  • Definition 2.3
  • Remark 2.4: OkSolid
  • Lemma 2.5
  • Lemma 2.6
  • proof
  • Lemma 3.1
  • proof
  • ...and 19 more