Fano $3$-folds of Fano Index $\geq 3$, and Constructing GRDB Number 41058

We study Fano $3$-folds of Fano index $\geq 3$. In particular, we settle the question of which of the 24 codimension 4 candidates listed in GRDB exist. Our main new construction is GRDB No. 41058, that was previously unknown. We construct it starting from a codimension 2 complete intersection $Y$ containing an unprojection divisor $D$ that is a weighted projective plane in a nonnormal projective embedding. The ideal of $D$ is generated by the minors of a $2 \times 6$ matrix, as in the classical Type II$_{1}$ method of Reid [14] and Papadakis [9, 10].