Constructing Q-Fano 3-folds \`a la Prokhorov & Reid

TL;DR

This paper extends the construction of Q-Fano 3-folds of indices 2 and 3, increasing known families and analyzing cases with identical Hilbert series to deepen understanding of their structure.

Contribution

It generalizes Prokhorov & Reid's method to produce five new families of Q-Fano 3-folds, expanding the classification of these algebraic varieties.

Findings

Five new families of Q-Fano 3-folds constructed

Two families share the same Hilbert series and are studied in detail

Enhanced understanding of the structure and classification of Q-Fano 3-folds

Abstract

We generalise a construction by Prokhorov & Reid of two families of Q-Fano 3-folds of index 2 to obtain five more families of Q-Fano 3-folds; four of index 2 and one of index 3. Two of the families constructed have the same Hilbert series and we study these cases in more detail.