Research

The refined local lifting problem for cyclic covers asks: Given a cyclic cover Z → X of a curve over a positive-characteristic field k and a lift 𝒴 → 𝒳 of its subcover Y → X to a finite extension of the ring of Witt vectors W(k) (hence in characteristic 0), can one extend that lift to the entire cover 𝒵 → 𝒳 (potentially after a finite extension of W(k))?

   Spec(k)        ←──   X   ←──   Y   ←──   Z

      ↓                              ↓                 ↓                 ↓

   Spec(W(k))    ←──  𝒳   ←──  𝒴   ←──   𝒵

We develop a Hurwitz tree technique to address the problem. Furthermore, in characteristic 2, we demonstrate that the refined local lifting holds most of the time, in particular, for ℤ/4 × ℤ/m covers where m is odd.

We reprove a main result from an unpublished work of Sekiguchi and Suwa on Kummer-Artin-Schreier-Witt theory. In particular, we construct a short exact sequence of group schemes over (the algebraic closure of) ℤ₍ₚ₎[ζₚˢ]

0 → ℤ/𝑝ˢ → 𝒲ₛ → 𝒱ₛ → 0,

whose generic fiber is the Kummer short exact sequence of group schemes over ℚ[ζₚˢ]

0 → μₚˢ → 𝔾ₘ → 𝔾ₘ → 0,

and whose special fiber is the Artin-Schreier-Witt short exact sequence of group schemes over 𝔽ₚ

0 → ℤ/𝑝ˢ → 𝑊ₛ → 𝑊ₛ → 0.

To do so, we apply an algebrization process to Matsuda's morphism, derived from a deformed Artin-Hasse exponential function.

We demonstrate that cyclic covers of curves in characteristic p > 0 can be equicharacteristically deformed "in towers." Specifically, given a cyclic cover Z → X of a curve over a characteristic-p-field k and a deformation 𝒴 → 𝒳 of its subcover Y → X over an equal-characteristic complete discrete valuation ring R with residue k, we show that it is possible to extend that deformation to the entire cover 𝒵 → 𝒳 (potentially after a finite extension of R).

   Spec(k)    ←──   X   ←──   Y   ←──   Z

       ↓                         ↓                 ↓                 ↓

   Spec(R)    ←──  𝒳   ←──  𝒴   ←──   𝒵

To prove this result, we generalized the concept of Hurwitz tree and adapted the construction from the proof of the lifting conjecture for cyclic covers of curves.

Our research focusses on the classification of the deformations of Artin-Schreier covers of curves, a.k.a, ℤ/p-covers in characteristic p. We achieve this by analyzing the ramification jumps of the branch points on each fiber. To accomplish this, we employ a generalized version of the Hurwitz tree, a combinatorial-differential object that encodes the degeneration of a cover, as measured by the refined Swan conductors. One application of our main result is a more detailed description of the connectedness of the moduli space of the projective line's Artin-Schreier covers.

Additionally, we developed a program under my guidance to assist in checking the existence of a deformation with predetermined "type." William Winston, an undergraduate at UVA, completed this program as part of a directed reading project. 

We investigate the a-number of cyclic coverings of the projective line. In particular, we identify a family of ℤ/9-coverings whose a-numbers are determined by the covers' ramification data.

We study the moduli space of cyclic covers in positive characteristic. In particular, we identify the irreducible components and determine their dimensions. To achieve this, we analyze the ramification data of the represented curves and use it to classify all the irreducible components. Additionally, we investigate the geometry of the moduli space by studying the deformations of cyclic covers which vary the p-rank and the number of branch points.

We introduce the concept of Hurwitz tree obstructions to addressing the refined local lifting problem. We specifically explore the circumstances under which these obstructions vanish for cyclic groups. 

We show that ℤ/4-covers (of curves) in characteristic 2 lift in towers when the first level lift is "nice." 

The main result of this manuscript is that the Galois covers of curves whose inertia groups are dihedral of order 25 x 2 (resp. 27 x 2) in characteristic 5 (resp. characteristic 3) always lift to characeristic 0. To obtain it, we prove the existence of certain meromorphic differential forms over the projective line using the Gröbner bases technique. Their connection is established here. 

We show that the moduli space of Artin-Schreier covers (ℤ/p-covers in characteristic p)  of the projective line of genus g is connected when g is large.