# 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 interests.

• Stable birational geometry of 3-folds and 4-folds fibrations
• Degeneration in equal and mixed characteristics
• Local structure of quadric fibrations in characteristic 2
• Brauer groups and ramification theory in $p$-primary contextes
• Log/crystalline differential forms in positive characteristic

#### Research statement.

My current interests are broadly focused on the generalised stable Lüroth problem; this consists in determining 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 varieties having a structure of conic, quadric or cubics fibration.

The Lüroth problem (and in particular the stable version) has been challenging mathematicians for almost a century, until the first example of unirational but stably irrational variety was given by Artin and Mumford in 1971. Several other examples have been found thereafter, but a systematic way of approaching this problem was not available.

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. Broadly speaking, the method is applied as following: let us assume 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. Under some reasonable hypotheses on the singularities of $Y$, one can conclude that 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 on it. These obstructions, in most cases, are not easy to produce 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. The interest in conic fibrations stems from their relationship with quaternion algebras and Brauer groups. Every conic fibration gives rise to a Brauer class of order 2 in $\mathrm{Br}(k(X))$ and, more interestingly, if the ground field has characteristic prime with 2, the discriminant locus of $\pi$ (namely, the set of points of $X$ over which the fibre is a singular conic) has to respect certain reciprocity conditions, which are well expressed by the so called reciprocity sequence, part of the Gersten complex. With this theory, conic bundles with wisely chosen discriminant locus can be constructed, and this machinery can be successfully used to induce some non-trivial class in $\mathrm{Br}(\widetilde{Y})[2]$, hence proving stable irrationality of $\widetilde{Y}$ and fitting this into the degeneration method.

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. From a heuristic point of view, reduction to small characteristic has proven to be an effective tool in dealing with birationality problems of varieties in high dimension (see the Kollar's method for irrationality via p-cyclic coverings).

The advantage of degenerating to characteristic 2 in the case of conic bundles can be explained via multiple instances:

• conics (and, more generally, quaternion algebras) have an exotic behaviour in characteristic 2 only
• geometry in positive characteristics becomes coarser, and some obstacles typical of characteristic 0 fall down (e.g. in positive characteristics there are coverings of the projective line ramified at a single point only)
• the structure of discriminant locus of a conic fibration appears to have richer local structure than the classical case
• the classical reciprocity sequence is truncated, which might release the conditions needed for a bunch of divisors to become the discriminant of a conic fibration
• the classical theory is insufficient to describe the behaviour of differentials in the dyadic context, which could be better understood by crystalline differential operators.

#### Contact informations.

Office: B0.15.

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

#### Publications.

1. (with A.Auel, Ch. Boehning, H.Ch. Graf von Bothmer) Universal triviality of the Chow group of 0-cycles and the Brauer group, preprint 2018 (arXiv)
2. (with A.Auel, Ch. Boehning, H.Ch Graf von Bothmer) Unramified Brauer group of conic bundle threefolds in characteristic two preprint, 2018 (arXiv)

#### Further 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.