Say I have a complex Fano threefold of some given codimension, some given genus, embedded into some (possibly weighted) projective space.
My question is: do the genus, codimension and weights of the projective space always uniquely determine a Fano threefold, or do I need more data?
For example, according to the Graded Ring Database, if we pick genus $0$ and codimension $1$, then there are $32$ such Fano threefolds. From a quick inspection, it seems that the weights of the projective spaces that they're embedded into seem different for each Fano threefold. So I'd hazard a guess that the genus, codimension and weights are the only data we need. Is this true?
Thank you.
No. For instance $$ X = \mathbb{P}^1 \times \mathbb{P}^1 \times \mathbb{P}^1 $$ and a smooth $(1,1)$-divisor $$ Y \subset \mathbb{P}^2 \times \mathbb{P}^2 $$ both a Fano threefolds of genus 25 in codimension 4 with all weights equal to 1.