# When is an algebraic geometer skilful enough?

I ran into the following lovely quote by the late Andrei Tyurin:

An algebraic geometer is skillful enough if he or she can recognize the geometric person under many guises of different dimensions.

The reason for this quote (and the quote itself) can be read in his 1995 lecture notes for the summer school on algebraic geometry in Ankara. One of the examples that a skilful algebraic geometer must know are the objects:

- in dimension 0: 6 distinct points on the projective line up to projective equivalence
- in dimension 1: a genus 2 curve
- in dimension 2: a cubic surface with a unique ordinary double point
- in dimension 3: a nonsingular intersection of 2 quadric hypersurfaces in $\mathbb{P}^5$

and more importantly their relation to each other:

- the genus 2 curve is a double cover of the projective line ramified in 6 distinct points
- blowing up 6 points on a conic (so the points are not in general position!) in $\mathbb{P}^2$ gives a singular cubic surface
- the pencil spanned by the two quadric hypersurfaces has 6 singular members

Recently, many developments in the study of derived categories of smooth projective varieties have been of this nature too, e.g. Kuznetsov's homological projective duality, or the Segal--Thomas example of a Calabi–Yau threefold embedding in a Fano elevenfold. In noncommutative algebraic geometry there is also the intricate connection between noncommutative planes and planar cubic curves which comes to mind.

He uses the word *person* at another point in the text where I would say *object*. I don't know whether he has his own idiosyncratic vocabulary just like Erdős did.