# Alessandro Bigazzi

I'm a Mathematics PhD student in Algebraic Geometry, working under the supervision of Christian Böhning.

I've earned a Bachelor of Sciences in pure mathematics at Università di Firenze (Italy) and a Master of Science in pure mathematics at Università Roma Tre (Italy), with the final dissertation Castelnuovo-Mumford regularity for projective curves, advised by Professor Edoardo Sernesi.

Research.

My current interests are broadly focused on the generalised stable Lüroth problem, namely studying the existence of unirational varieties (of dimension at least 3) which are not stably rational. In particular, I'm interested in the stable irrationality of conic fibration defined over a field of characteristc 2, a setting that arises when trying to degenerate cubic 3-folds.

The Lüroth problem (like the stable version) has been challenging mathematicians for almost a century, until the first example was given by Artin and Mumford in 1971. Since then, similar examples have been provided but the process of finding them has remained cumbersome and somehow confined to reworking Artin-Mumford's original example.

The degeneration method by Claire Voisin in 2013 has brought new life to the whole topic, and a plethora of new examples has been computed with the aid of the new tools. Roughly speaking, it links the classical algebraic obstructions for stable irrationality (Brauer group, differential forms...) to a deformation-theoretic perspective, via an abstract intersection-theoretic invariant (universal triviality of $\mathrm{CH}_0$ group). The method is applied as following: suppose we have a flat family $\mathfrak{X}\longrightarrow B$ of algebraic varieties over an algebraically closed field $k$, and let $Y:=\mathfrak{X}_0$, the spcial fibre, be stably irrational. If $Y$ admits a resolution of singularities which is universally $\mathrm{CH}_0$ trivial, then the geometric generic fibre $\overline{\mathfrak{X}_\eta}$ is stably irrational as well.

However, the advantage of allowing flat limits in the theory, while being remarkable as stable rationality is not deformation invariant, can not be fully exploited if the special fibre does not have good properties. In particular, we requirea good control over the obstructions of stable irrationality in order to produce new examples. For instance, the stable rationality of $Y$ could be obstructed by checking non-triviality of $\mathrm{Br}(\widetilde{Y})[2]$, where $\widetilde{Y}$ is an appropriate desingularisation, or by the existence of some non-trivial differential form. These, in general, are not easy conditions to check for a general variety.

Amongst others, conic fibrations have proved to be an interesting specimen of varieties in this theory. By conic fibration it is meant a flat, proper morphism $\pi : Y\longrightarrow X$, whose fibres are isomorphic to conics. Very recently, stable irrationality of conic fibrations has been related to the geometry of its discriminant locus, namely the locus of points $p\in X$ above which the fibre is a singular conic. This was done using arithmetic-geometry tools like unramified cohomology and has been established when the base $X$ is a rational surface or 3-fold; the whole theory relies heavily on residue maps theory coming from Galois cohomology and by some cohomological purity argument. Additionally, it is required that the fields have characteristic prime with 2.

It turns out that many interesting examples of degenerations appear in the so called "unequal characteristic" case: namely, the special fibre is defined over a field of positive characteristic, while the generic one is defined over a field of characteristic 0. A relevant example of that is the case of cubic hypersurfaces of dimension 3, which have a conic bundle structure and could be degenerated to characteristic 2 (even if the actual degeneration has to be found).

This of course raises the problem of re-interpreting the above machinery in a positive characteristic context. In particular, one needs to care about interpreting Brauer group's torsion by means of unramified cohomology, whose description by means of residue maps fails in characteristic 2.

Contact informations.

Office: B0.15.

E-mail: A.Bigazzi <at> warwick.ac.uk

Material.

Some notes on Van Kampen's theorem, written during an undergraduate course in algebraic topology.

An Essay on the stable Lüroth problem, written in my first year of doctoral study.