On the birational geometry of ${\mathbb {Q}}$-Fano threefolds of large Fano index, I

Abstract

We investigate the rationality problem for ${\mathbb {Q}}$-Fano threefolds of Fano index $\ge 2$.

Appendix A

In this section we present a computer algorithm (see [28, § 3] or [5, § 3]) that alow to list all the numerical possibilities for ${\mathbb {Q}}$-Fano threefolds of index at least 3. Let X be a ${\mathbb {Q}}$-Fano threefold with $q:=\textrm{q}_{\mathbb {Q}}(X)\ge 3$ and let $T\in \text {Cl}(X)_{\textrm{t}}$ be an element of order N.

Step 1

By [12] we have the inequality

$$\begin{aligned} 0<-K_X\cdot c_2(X)= 24-\sum _{P\in {\textbf{B}}} \frac{r_P-1}{r_P}. \end{aligned}$$

(A.1.1)

This produces a finite (but huge) number of possibilities for the basket ${\textbf{B}}(X)$ and the number $-K_X\cdot c_2(X)$.

Step 2

Theorem 2.2.4 implies that $q\in \{3,\dots ,11, 13,17,19\}$. In each case we compute $A_X^3$ by the formula

$$\begin{aligned} A_X^3=\frac{12}{(q-1)(q-2)}\Bigl ( 1-\frac{A_X\cdot c_2(X)}{12}+\sum _{P\in B} c_P(-A_X) \Bigr ) \end{aligned}$$

(see [31]), where $c_P$ is the correction term in the orbifold Riemann-Roch formula [29]. The number $rA_X^3$ must be a positive integer by Theorem 2.2.4(iii).

Step 3

On this step we can use an improved version of Bogomolov–Miyaoka inequality [17] instead of the one used in [12] and [31]. Thus we have

$$\begin{aligned} (-K_X)^3\ {\left\{ \begin{array}{ll} < 3(-K_X)\cdot c_2(X)&{}\hbox { if}\ \textrm{q}_{\mathbb {Q}}(X)\ne 4,\, 5, \\ \le \frac{25}{8}(-K_X)\cdot c_2(X)&{}\text {otherwise}. \end{array}\right. } \end{aligned}$$

This removes a lot of possibilities.

Step 4

In a neighborhood of each point $P\in X$ we can write $A_X\sim l_PK_X$, where $0\le l_P<r_P$. There is a finite number of possibilities for the collection $\{(l_P)\}$. If ${\text {q}}_{{\text {W}}}(X)=\textrm{q}_{\mathbb {Q}}(X)$, then $\gcd (q,r)=1$ by Theorem 2.2.4. In this case the numbers $l_P$ are uniquely determined by $1+ql_P\equiv 0\mod r_P$ because $K_X+qA_X\sim 0$.

Step 5

Similarly, a neighborhood of each point $P\in X$ we can write $T\sim l_P'K_X$, where $0\le l_P'<r_P$. The collection $\{(l_P')\}$ and the number N satisfy the following properties:

$$\begin{aligned} \chi (X,\, {\mathscr {O}}_X(NT))=1\qquad \hbox { and}\qquad \chi (X,\, {\mathscr {O}}_X(jT))=0\quad \text {for}\ j=1,\dots ,N-1 \end{aligned}$$

(by the Kawamata–Viehweg vanishing). Thus we obtain a finite number of possibilities for $\{(l_P')\}$ and N.

Step 6

Finally, applying Kawamata–Viehweg vanishing we obtain

$$\begin{aligned} \chi (X,\, {\mathscr {O}}_X(mA_X+jT))={\text {h}}^0(X,\, {\mathscr {O}}_X(mA_X+jT))=0. \end{aligned}$$

for $-q<m<0$ and $0\le j<n$. Again, we check this condition using orbifold Riemann-Roch and remove a lot of possibilities.

Step 7

We obtain a list of collections $\left( q, {\textbf{B}}(X), A_X^3, \{(l_P)\}, n \{(l_P')\}\right)$. In each case we compute ${\text {g}}(X)$ and ${{\text {h}}}_X(t,\sigma )$ by using the orbifold Riemann-Roch theorem. For example,

$$\begin{aligned} {\text {g}}(X)= -\frac{1}{2} K_X^3+ 1-\sum _{P\in {\textbf{B}}} \frac{b_P(r_P-b_P)}{2r_P}. \end{aligned}$$

(A.1.2)