An official website of the United States government
Abstract This paper contains a method to prove the existence of smooth curves in positive characteristic whose Jacobians have unusual Newton polygons. Using this method, I give a new proof that there exist supersingular curves of genus$$4$$in every prime characteristic. More generally, the main result of the paper is that, for every$$g \geq 4$$and primep, every Newton polygon whosep-rank is at least$$g-4$$occurs for a smooth curve of genusgin characteristicp. In addition, this method resolves some cases of Oort’s conjecture about Newton polygons of curves.
more »
« less
Monodromy of Kodaira fibrations of genus 3
Abstract A Kodaira fibration is a non‐isotrivial fibration from a smooth algebraic surfaceSto a smooth algebraic curveBsuch that all fibers are smooth algebraic curves of genusg. Such fibrations arise as complete curves inside the moduli space of genusgalgebraic curves. We investigate here the possible connected monodromy groups of a Kodaira fibration in the case and classify which such groups can arise from a Kodaira fibration obtained as a general complete intersection curve inside a subvariety of parametrizing curves whose Jacobians have extra endomorphisms.
more »
« less
- Award ID(s):
- 1645877
- PAR ID:
- 10443964
- Publisher / Repository:
- Wiley Blackwell (John Wiley & Sons)
- Date Published:
- Journal Name:
- Mathematische Nachrichten
- Volume:
- 295
- Issue:
- 11
- ISSN:
- 0025-584X
- Page Range / eLocation ID:
- p. 2130-2146
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
NA (Ed.)Recall that a non-singular planar quartic is a canonically embedded non-hyperelliptic curve of genus three. We say such a curve is symmetric if it admits non-trivial automorphisms. The classification of (necessarily finite) groups appearing as automorphism groups of non-singular curves of genus three dates back to the last decade of the 19th century. As these groups act on the quartic via projective linear transformations, they induce symmetries on the 28 bitangents. Given such an automorphism group G=Aut(C), we prove the G-orbits of the bitangents are independent of the choice of C, and we compute them for all twelve types of smooth symmetric planar quartic curves. We further observe that techniques deriving from equivariant homotopy theory directly reveal patterns which are not obvious from a classical moduli perspective.more » « less
-
A question of Griffiths–Schmid asks when the monodromy group of an algebraic family of complex varieties is arithmetic. We resolve this in the affirmative for a class of algebraic surfaces known as Atiyah–Kodaira manifolds, which have base and fibers equal to complete algebraic curves. Our methods are topological in nature and involve an analysis of the ‘geometric’ monodromy, valued in the mapping class group of the fiber.more » « less
-
Abstract Let be a simple algebraic group over an algebraically closed field . Let be a finite group acting on . We classify and compute the local types of ‐bundles on a smooth projective ‐curve in terms of the first nonabelian group cohomology of the stabilizer groups at the tamely ramified points with coefficients in . When , we prove that any generically simply connected parahoric Bruhat–Tits group scheme can arise from a ‐bundle. We also prove a local version of this theorem, that is, parahoric group schemes over the formal disc arise from constant group schemes via tamely ramified coverings.more » « less
-
Abstract Consider the algebraic function $$\Phi _{g,n}$$ that assigns to a general $$g$$ -dimensional abelian variety an $$n$$ -torsion point. A question first posed by Klein asks: What is the minimal $$d$$ such that, after a rational change of variables, the function $$\Phi _{g,n}$$ can be written as an algebraic function of $$d$$ variables? Using techniques from the deformation theory of $$p$$ -divisible groups and finite flat group schemes, we answer this question by computing the essential dimension and $$p$$ -dimension of congruence covers of the moduli space of principally polarized abelian varieties. We apply this result to compute the essential $$p$$ -dimension of congruence covers of the moduli space of genus $$g$$ curves, as well as its hyperelliptic locus, and of certain locally symmetric varieties. These results include cases where the locally symmetric variety $$M$$ is proper . As far as we know, these are the first examples of nontrivial lower bounds on the essential dimension of an unramified, nonabelian covering of a proper algebraic variety.more » « less
