Moduli Space Of Hyperelliptic Curves Research Articles

Explicit moduli of superelliptic curves with level structure

In this article we give an explicit construction of the moduli space of trigonal superelliptic curves with level 3 structure. The construction is given in terms of point sets on the projective line and leads to a closed formula for the number of connected (and irreducible) components of the moduli space. The results of the article generalise the description of the moduli space of hyperelliptic curves with level 2 structure, due to Dolgachev and Ortland, Runge and Tsuyumine.

Geometric aspects on Humbert-Edge curves of type 5, Kummer surfaces and hyperelliptic curves of genus 2

AbstractIn this work, we study the Humbert-Edge curves of type 5, defined as a complete intersection of four diagonal quadrics in ${\mathbb{P}}^5$ . We characterize them using Kummer surfaces, and using the geometry of these surfaces, we construct some vanishing thetanulls on such curves. In addition, we describe an argument to give an isomorphism between the moduli space of Humbert-Edge curves of type 5 and the moduli space of hyperelliptic curves of genus 2, and we show how this argument can be generalized to state an isomorphism between the moduli space of hyperelliptic curves of genus $g=\frac{n-1}{2}$ and the moduli space of Humbert-Edge curves of type $n\geq 5$ where $n$ is an odd number.

The rational Chow rings of moduli spaces of hyperelliptic curves with marked points

The rational Chow rings of moduli spaces of hyperelliptic curves with marked points

Borel subgroups of the plane Cremona group

Abstract It is well known that all Borel subgroups of a linear algebraic group are conjugate. Berest, Eshmatov, and Eshmatov have shown that this result also holds for the automorphism group Aut ⁢ ( 𝔸 2 ) {{\mathrm{Aut}}({\mathbb{A}}^{2})} of the affine plane. In this paper, we describe all Borel subgroups of the complex Cremona group Bir ⁢ ( ℙ 2 ) {{\rm Bir}({\mathbb{P}}^{2})} up to conjugation, proving in particular that they are not necessarily conjugate. In principle, this fact answers a question of Popov. More precisely, we prove that Bir ⁢ ( ℙ 2 ) {{\rm Bir}({\mathbb{P}}^{2})} admits Borel subgroups of any rank r ∈ { 0 , 1 , 2 } {r\in\{0,1,2\}} and that all Borel subgroups of rank r ∈ { 1 , 2 } {r\in\{1,2\}} are conjugate. In rank 0, there is a one-to-one correspondence between conjugacy classes of Borel subgroups of rank 0 and hyperelliptic curves of genus ℊ ≥ 1 {\mathcal{g}\geq 1} . Hence, the conjugacy class of a rank 0 Borel subgroup admits two invariants: a discrete one, the genus ℊ {\mathcal{g}} , and a continuous one, corresponding to the coarse moduli space of hyperelliptic curves of genus ℊ {\mathcal{g}} . This moduli space is of dimension 2 ⁢ ℊ - 1 {2\mathcal{g}-1} .

Ekedahl–Oort strata on the moduli space of curves of genus four

We study the induced Ekedahl–Oort stratification on the moduli of curves of genus 4 in positive characteristic by computing the de Rham cohomology of curves. For moduli space of hyperelliptic curves of genus 4 in characteristic 3, we determine the dimension and reducibility of Ekedahl–Oort strata. For moduli space of curves of genus 4 in odd characteristic, we show the existence of certain Ekedahl–Oort strata and discuss the existence of superspecial curves.

Picard Groups of Moduli Spaces of Curves with Symmetry

Abstract We study the Picard groups of moduli spaces of smooth complex projective curves that have a group of automorphisms with a prescribed topological action. One of our main tools is the theory of symmetric mapping class groups. In the 1st part of the paper, we show that, under mild restrictions, the moduli spaces of smooth curves with an abelian group of automorphisms of a fixed topological type have finitely generated Picard groups. In certain special cases, we are able to compute them exactly. In the 2nd part of the paper, we show that finite abelian level covers of the hyperelliptic locus in the moduli space of smooth curves have finitely generated Picard groups. We also compute the Picard groups of the moduli spaces of hyperelliptic curves of compact type.

Tautological classes on the moduli space of hyperelliptic curves with rational tails

We study tautological classes on the moduli space of stable n-pointed hyperelliptic curves of genus g with rational tails. The method is based on the approach of Yin in comparing tautological classes on the moduli of curves and the universal Jacobian. Our result gives a complete description of tautological relations. It is proven that all relations come from the Jacobian side. The intersection pairings are shown to be perfect in all degrees. We show that the tautological algebra coincides with its image in cohomology via the cycle class map. The latter is identified with monodromy invariant classes in cohomology.

An Outline of the Log Minimal Model Program for the Moduli Space of Curves

ABSTRACTWe consider the log minimal model program for the moduli space of stable curves. It is widely believed now that there is a descending sequence of critical $α$ values where the log canonical model for the moduli space of stable curves with respect to $αδ$ changes, where $δ$ denotes the divisor of singular curves. We derive a conjectural formula for the critical values in two different ways: By working out the intersection theory of the moduli space of hyperelliptic curves and by computing the GIT stability of certain curves with tails and bridges. The results give a rough outline of how the log minimal model program would proceed, predicting when the log canonical model changes and which curves are to be discarded and acquired at the critical steps.

Traces of high powers of the Frobenius class in the moduli space of hyperelliptic curves

The Zeta function of a curve C over a finite field may be expressed in terms of the characteristic polynomial of a unitary matrix \(\Theta _C\). Following the work of Rudnick (Acta Arith. 143(1), 81–99, 2010), we compute the expected value of \({{\mathrm{tr}}}(\Theta _C^n)\) over the moduli space of hyperelliptic curves of genus g, over a fixed finite field \({\mathbb F}_q\), in the limit of large genus. As an application, we compute the expected value of the number of points on C in \({\mathbb F}_{q^n}\) as the genus tends to infinity. We also look at biases in both expected values for small values of n.

Moduli spaces of hyperelliptic curves with A and D singularities

We introduce moduli spaces of quasi-admissible hyperelliptic covers with at worst A and D singularities. The stability conditions for these moduli spaces depend on two rational parameters describing allowable singularities. For the extreme values of the parameters, we obtain the stacks of stable limits of \(A_n\) and \(D_n\) singularities, and the quotients of the miniversal deformation spaces of these singularities by natural \(\mathbb G _m\)-actions. We interpret the intermediate spaces as log canonical models of the stacks of stable limits of \(A_n\) and \(D_n\) singularities.

Square-tiled surfaces and rigid curves on moduli spaces

We study the algebro-geometric aspects of Teichmüller curves parameterizing square-tiled surfaces with two applications. (a) There exist infinitely many rigid curves on the moduli space of hyperelliptic curves. They span the same extremal ray of the cone of moving curves. Their union is a Zariski dense subset. Hence they yield infinitely many rigid curves with the same properties on the moduli space of stable n-pointed rational curves for even n. (b) The limit of slopes of Teichmüller curves and the sum of Lyapunov exponents for the Teichmüller geodesic flow determine each other, which yields information about the cone of effective divisors on the moduli space of curves.

The p-rank strata of the moduli space of hyperelliptic curves

We prove results about the intersection of the p-rank strata and the boundary of the moduli space of hyperelliptic curves in characteristic p ⩾ 3 . This yields a strong technique that allows us to analyze the stratum H g f of hyperelliptic curves of genus g and p-rank f. Using this, we prove that the endomorphism ring of the Jacobian of a generic hyperelliptic curve of genus g and p-rank f is isomorphic to Z if g ⩾ 4 . Furthermore, we prove that the Z / ℓ -monodromy of every irreducible component of H g f is the symplectic group Sp 2 g ( Z / ℓ ) if g ⩾ 3 , and ℓ ≠ p is an odd prime (with mild hypotheses on ℓ when f = 0 ). These results yield numerous applications about the generic behavior of hyperelliptic curves of given genus and p-rank over finite fields, including applications about Newton polygons, absolutely simple Jacobians, class groups and zeta functions.

The integral monodromy of hyperelliptic and trielliptic curves

We compute the $$\mathbb{Z}/\ell$$ and $$\mathbb{Z}_{\ell}$$ monodromy of every irreducible component of the moduli spaces of hyperelliptic and trielliptic curves. In particular, we provide a proof that the $$\mathbb{Z}/\ell$$ monodromy of the moduli space of hyperelliptic curves of genus g is the symplectic group $${\rm S}_{P2g}(\mathbb{Z}/\ell)$$ . We prove that the $$\mathbb{Z}/\ell$$ monodromy of the moduli space of trielliptic curves with signature (r,s) is the special unitary group $${\rm SU}_{(r,s)}(\mathbb{Z}/\ell\otimes\mathbb{Z}[\zeta_{3}])$$ .

Explicit Mumford isomorphism for hyperelliptic curves

Using an explicit version of the Mumford isomorphism on the moduli space of hyperelliptic curves we derive a closed formula for the Arakelov–Green function of a hyperelliptic Riemann surface evaluated at its Weierstrass points.

Orbits of Galois Invariant n-Sets of [formula omitted]1 under the Action of PGL2

For any finite field k we count the number of orbits of galois invariant n-sets of P1(k̄) under the action of PGL2(k). For k of odd characteristic, this counts the number of k-points of the moduli space of hyperelliptic curves of genus g over k. We get in this way an explicit formula for the number of hyperelliptic curves over k of genus g, up to k-isomorphism and quadratic twist.

On Moduli Spaces, Equidistribution, Estimates, and Rational Points of Algebraic Curves

We consider the moduli spaces of hyperelliptic curves, Artin–Schreier coverings, and some other families of curves of this type over fields of characteristic p. By using the Postnikov method, we obtain expressions for the Kloosterman sums. The distribution of angles of the Kloosterman sums was investigated on a computer. For small prime p, we study rational points on curves y2 = f(x). We consider the problem of the accuracy of estimates of the number of rational points of hyperelliptic curves and the existence of rational points of curves of the indicated type on the moduli spaces of these curves over a prime finite field.

Manifolds of support sets of chebyshev polynomials

We consider a parameterization of the set of polynomialsT n(E, x) whose deviation from zero is the least on a systemE consisting of several intervals on the real axis. We point out a new way for obtaining the equations which describe the boundary of the maximum set of the least deviationE + ⊃E. We describe the geometry of the variety of all possible setsE +; this manifold is embedded in the moduli space of hyperelliptic curves with real branch points.

Chebyshev polynomials and navigation in the moduli space of hyperelliptic curves

Article Chebyshev polynomials and navigation in the moduli space of hyperelliptic curves was published on January 1, 1999 in the journal Russian Journal of Numerical Analysis and Mathematical Modelling (volume 14, issue 3).

THE RATIONALITY OF MODULI SPACES OF HYPERELLIPTIC CURVES

The rationality of the moduli spaces of hyperelliptic curves of genus is proved. Bibliography: 5 titles.

RATIONALITY OF THE ORBIT SPACES OF IRREDUCIBLE REPRESENTATIONS OF THE GROUP SL2

In this paper the rationality of the moduli spaces of hyperelliptic curves of genus g≥15 is proved.Bibliography: 6 titles.