I'm interested in algebraic geometry.
More specifically, I like to apply non-archimedean analytic geometry (especially from the point of view of Berkovich) to problems in birational geometry (singularity theory, motivic integration), arithmetic geometry (models of curves and ramification), and combinatorics (tropical geometry).
"Links of sandwiched surface singularities and self-similarity" (with C. Favre and M. Ruggiero) – submitted, 35 pages (arXiv)
We characterize sandwiched singularities in terms of their link in two different settings.
We first prove that such singularities are precisely the normal surface singularities having self-similar non-archimedean links.
We describe this self-similarity both in terms of Berkovich analytic geometry and of the combinatorics of weighted dual graphs.
We then show that a complex surface singularity is sandwiched if and only if its complex link can be embedded in a Kato surface in such a way that its complement remains connected.
"Motivic and analytic nearby fibers at infinity and bifurcation sets" (with M. Raibaut) – submitted, 17 pages (Pdf)
In this paper we use motivic integration and non-archimedean analytic geometry to study the singularities at infinity of the fibers of a polynomial map f: 𝔸ℂd → 𝔸ℂ1.
We show that the motivic nearby cycles at infinity Sf,a∞ of f for a value a is a motivic generalization of the classical invariant λf(a), an integer that measures a lack of equisingularity at infinity in the fiber f-1(a).
We then introduce a non-archimedean analytic nearby fiber at infinity Ff,a∞ whose motivic volume recovers the motive Sf,a∞.
With Sf,a∞ and Ff,a∞ can be naturally associated a motivic and an analytic bifurcation sets respectively; we show that the first one always contains the second, and that both contain the classical topological bifurcation set of f if f has isolated singularities at infinity.
We study the Galois descent of semi-affinoid non-archimedean analytic spaces. These are the non-archimedean analytic spaces which admit an affine special formal scheme as model over a complete discrete valuation ring, such as for example open or closed polydiscs or polyannuli. Using Weil restrictions and Galois fixed loci for semi-affinoid spaces and their formal models, we describe a formal model of a K-analytic space X, provided that X⊗KL is semi-affinoid for some finite tamely ramified extension L of K. As an application, we study the forms of analytic annuli that are trivialized by a wide class of Galois extensions that includes totally tamely ramified extensions. In order to do so, we first establish a Weierstrass preparation result for analytic functions on annuli, and use it to linearize finite order automorphisms of annuli. Finally, we explain how from these results one can deduce a non-archimedean analytic proof of the existence of resolutions of singularities of surfaces in characteristic zero.
"Normalized Berkovich spaces and surface singularities" – to appear in Transactions of the American Mathematical Society, doi:10.1090/tran/7209, 51 pages (arXiv)
We define normalized versions of Berkovich spaces over a trivially valued field k, obtained as quotients by the action of ℝ>0 defined by rescaling semivaluations. We associate such a normalized space to any special formal k-scheme and prove an analogue of Raynaud's theorem, characterizing categorically the spaces obtained in this way. This construction yields a locally ringed G-topological space, which we prove to be G-locally isomorphic to a Berkovich space over the field k((t)) with a t-adic valuation. These spaces can be interpreted as non-archimedean models for the links of the singularities of k-varieties, and allow to study the birational geometry of k-varieties using techniques of non-archimedean geometry available only when working over a field with non-trivial valuation. In particular, we prove that the structure of the normalized non-archimedean links of surface singularities over an algebraically closed field k is analogous to the structure of non-archimedean analytic curves over k((t)), and deduce characterizations of the essential and of the log essential valuations, i.e. those valuations whose center on every resolution (respectively log resolution) of the given surface is a divisor.
"Faithful realizability of tropical curves" (with M. Cheung, J. Park, and M. Ulirsch) – International Mathematics Research Notices, 2016(15), 4706–4727, 2016 (arXiv)
We study whether a given tropical curve Γ in ℝn can be realized as the tropicalization of an algebraic curve whose non-archimedean skeleton is faithfully represented by Γ. We give an affirmative answer to this question for a large class of tropical curves that includes all trivalent tropical curves, but also many tropical curves of higher valence. We then deduce that for every metric graph G with rational edge lengths there exists a smooth algebraic curve in a toric variety whose analytification has skeleton G, and the corresponding tropicalization is faithful. Our approach is based on a combination of the theory of toric schemes over discrete valuation rings and logarithmically smooth deformation theory, expanding on a framework introduced by Nishinou and Siebert.