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Alessandro Bigazzi


portrait

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 rationality problems for 3-folds and 4-folds quadric fibrations
  • Degeneration-like theorems in equal and mixed characteristics
  • Quadric fibrations in characteristic 2
  • Algebraic and (log) geometric obstructions to (stable) rationality
  • Behviour of p-fibrations in characteristic p

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 fibration, and my work aims to determine invariants of geometric or algebraic nature that can detect stable irrationality via this extra structure.

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 spirit of the method is 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. One would like to conclude that, upon some reasonable hypotheses on Y, 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.

There are several instances in which research can be carried over:

  • First of all, it is possible to keep the required mildness of singularities sufficiently reasonable to make possible the introduction of classical invariants like \mathrm{Br}(\widetilde{Y})[2], where \widetilde{Y} is an appropriate desingularisation, or global differential forms.
    • In this viewpoint, conic fibrations have proved to be quite well fitting to this method. 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). Intuitively speaking, one could expect this as a reflection of positive characteristic "breaking up" some geometric constraints typical of complex varieties, which are tightly related to their analytic counterparts (and so have a "rigid" geometry, in some sense).
    • However, the reciprocity sequence recalled above seems to have no immediate analogue in characteristic 2, which complicates the general theory. It is still matter of speculation whether a cohomological criterion could replace this sequence in the dyadic case.
  • On a second instance, it could be interesting to replace the condition on singularities with weaker conditions, which would lead to an improvement of the original degeneration method, since this would allow degenerations with more singular fibres.
    • In the original method by Voisin's, the leading idea is to transfer an intersection-theoretic property (decomposition of the diagonal) from the generic fibre of a flat family to its special fibre. This actually happens without any restriction on the singularities of the special fibre; the requirement of "mild singularities" is needed only to ensure that this same property is preserved in a smooth birational model of the special fibre, which in turn is needed as all known invariants make sense for smooth varieties only.
    • In this scenario, it would make sense to look for a property which obstruct existence of decomposition of diagonal in the singular special fibre, which however should "lift" to the general member of the family like the standard property does. This property might be likely found implementing some log geometry techniques in this setting.
    • On the other hand, once one has a sufficient control on this new log-geometric property (provided one succeeds in defining such thing), it is essential to determine an algebraic criterion that detects this property (similarly to what one wants to do with the Brauer group controlling existence of decomposition of the diagonal). This criterion might bear some relationship with log differential forms or may reside in some new object that one could define (e.g. a log Brauer group?).

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)

Teaching.

A.Y. 2018-19: TA for Geometry (2nd year).
A.Y. 2018-19: TA for Introduction to Topology (3rd year).
A.Y. 2018-19: Supervision of 2nd year Mathematics students.

A.Y. 2017-18: TA for Metric Spaces (2nd year, with exam marking)
A.Y. 2017-18:
TA for Commutative Algebra (3rd year)
A.Y. 2017-18:
TA for Introduction to Topology (3rd year)
A.Y. 2017-18:
Supervision of 2nd year Mathematics students.

A.Y. 2016-17: Supervision of 1st and 2nd year Mathematics students.

A.Y. 2015-16: Exercise sessions for Algebraic Geometry 2 - Schemes and Cohomology (Univ. Roma 3)

A.Y. 2012-13: Exercise sessions for Geometry III - Algebraic topology (Univ. di Firenze)
A.Y. 2012-13: Exercise sessions for Differential Topology (Univ. di Firenze)

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.