Cornalba-Harris equality for semistable hyperelliptic curves in positive characteristic

Trans. Amer. Math. Soc.

2011

Abstract. Using the discriminant modular form and the Noether formula it is possible to write the admissible self-intersection of the relative dualising sheaf of a semistable hyperelliptic curve over a number field or function field as a sum, over all places, of a certain adelic invariant χ. We provide a simple geometric interpretation for this invariant χ, based on the arithmetic of symmetric roots. We propose the conjecture that the invariant χ coincides with the invariant ϕ introduced in a recent paper by S.-W. Zhang. This conjecture is true in the genus 2 case, and we obtain a new proof of the Bogomolov conjecture for curves of genus 2 over number fields.

Section: ωη and D(x)mentioning

confidence: 99%

Symmetric roots and admissible pairing

Trans. Amer. Math. Soc.

2011

“…In [7] a complete study is made of ϕ(G, q) for all stable polarized graphs of genus three. In [30] [31] the invariant ϕ(G, q) is studied for so-called hyperelliptic polarized graphs. In our proof of Theorem 1.3 below we will make essential use of a result from [31] that relates ε(G, q) and ϕ(G, q) for hyperelliptic polarized graphs.…”

Section: Polarized Metric Graphs and Zhang's Graph Invariantmentioning

confidence: 99%

Point-like limit of the hyperelliptic Zhang–Kawazumi invariant

Pure and Applied Mathematics Quarterly

2015

The behavior near the boundary in the Deligne-Mumford compactification of many functions on M h,n can be conveniently expressed using the notion of "point-like limit" that we adopt from the string theory literature. In this note we study a function on M h that has been introduced by N. Kawazumi and S. Zhang, independently. We show that the point-like limit of the Zhang-Kawazumi invariant in a family of hyperelliptic Riemann surfaces in the direction of any hyperelliptic stable curve exists, and is given by evaluating a combinatorial analogue of the Zhang-Kawazumi invariant, also introduced by Zhang, on the dual graph of that stable curve.

“…3.1). In this case the inequalities follow from a well-known identity due, in increasing order of generality, to CornalbaHarris [5], Kausz [14], Maugeais [21] and Yamaki [35].…”

Section: ⊗−1mentioning

confidence: 99%

Local invariants attached to Weierstrass points

2009

manuscripta math.

Abstract. Let X/S be a hyperelliptic curve of genus g over the spectrum of a discrete valuation ring. Two fundamental numerical invariants are attached to X/S: the valuation d of the hyperelliptic discriminant of X/S, and the valuation δ of the Mumford discriminant of X/S (equivalently, the Artin conductor). For a residue field of characteristic 0 as well as for X/S semistable the invariants d and δ are known to satisfy certain inequalities. We prove an exact formula relating d and δ with intersection theoretic data determined by the distribution of Weierstrass points over the special fiber, in the semistable case. We also prove an exact formula for the stable Faltings height of an arbitrary curve over a number field, involving local contributions associated to its Weierstrass points.