Claus Diem, Gerhard Frey
For every odd prime number p, we give examples of non-constant smooth families of genus 2 curves over fields of characteristic p which have pro-Galois (pro-étale) covers of infinite degree with geometrically connected fibers. The Jacobians of the curves are isomorphic to products of elliptic curves.
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Galois lines for a canonical curve of genus 4, III: non-cyclic Galois lines
研究四次典范曲线的Galois线，证明S3线和K4线数量上限分别为10和15。
We show all the possible structures of finite subgroups of the automorphism groups of elliptic curves. Here the automorphism means the biholomorphic transformation. Using the result, we determine every Galois group G at outer Galois point for genus-one curve with singular points. In particular, if G is abelian, then its order is at most nine. However if not so, the order is unbounded. Furthermore, we give every defining equation of the curve with the Galois point in the case where G is abelian.
Galois subspaces for compact Riemann surfaces of genus 2
研究二阶紧凑黎曼面的伽罗瓦子空间结构，分析当非常量除子由自动群子群诱导时，伽罗瓦子空间分解中投影空间的维度。
In this article, we develop the geometry of canonical stratifications of the spaces $\mathcal{M}_{0,n}$ and prepare ground for studying the action of the Galois group $Gal (\overline{\mathbb{Q}} /\mathbb{Q})$ upon strata. We define and introduce a version of a gravity operad constructed for a class of moduli spaces $\mathcal{M}_{0,n}$, equipped with a hidden holomorphic involution. This additional symmetry is associated to a split quaternionic structure. We introduce a categorical framework to present this object. Interaction between the geometry, physics and the arithmetics are discussed. An …
Let $ϕ:\,X\rightarrow Y$ be a (possibly ramified) cover between two algebraic curves of positive genus. We develop tools that may identify the Prym variety of $ϕ$, up to isogeny, as the Jacobian of a quotient curve $C$ in the Galois closure of the composition of $ϕ$ with a well-chosen map $Y\rightarrow \mathbb{P}^1$. This method allows us to recover all previously obtained descriptions of a Prym variety in terms of a Jacobian that are known to us, besides yielding new applications. We also find algebraic equations for some of these new cases, including one where $X$ has genus $3$, $Y$ has genu…
Given a Prym-Teichmüller curve in $\mathcal{M}_3$, this note provides an invariant that sorts the cusp prototypes of Lanneau and Nguyen by component. This can be seen as an analogue of McMullen's genus $2$ spin invariant, although the source of this invariant is different. Moreover, we describe the Galois action on the cusps of these Teichmüller curves, extending the results of Bouw and Möller in genus $2$. We use this to show that the components of the genus $3$ Prym-Teichmüller curves are homeomorphic.
Given a family of Galois coverings of the projective line we give a simple sufficient condition ensuring that the closure of the image of the family via the period mapping is a special (or Shimura) subvariety in A_g. By a computer program we get the list of all families in genus up to 8 satisfying our condition. There is no family in genus 8, all of them are in genus at most 7. These examples are related to a conjecture of Oort. Among them we get the cyclic examples constructed by various authors (Shimura, Mostow, De Jong-Noot, Rohde, Moonen and others) and the abelian non-cyclic examples foun…
The minimum genus of Galois covers of curves
在正特征代数闭域上，针对有限循环p-群G和素数l≠p，研究G-稳定子模并给出构造支配Y且分支在S_X外的Γ-覆盖的最小亏格方法，以及ℙ¹上在𝔸¹外平展的Γ-覆盖的最小亏格。
On a Galois cover of the Hermitian curve of genus $\mathfrak{g}=\frac{1}{8}(q-1)^2$
本文研究亏格为 $\frac{1}{8}(q-1)^2$、$q\equiv1\pmod4$ 的 Hermitian 曲线的 4 阶循环 Galois 覆盖，分析其 Frobenius 嵌入、Weierstrass 半群与自同构群。
Given an embedding of a smooth projective curve $X$ of genus $g\geq1$ into $\mathbb{P}^N$, we study the locus of linear subspaces of $\mathbb{P}^N$ of codimension 2 such that projection from said subspace, composed with the embedding, gives a Galois morphism $X\to\mathbb{P}^1$. For genus $g\geq2$ we prove that this locus is a smooth projective variety with components isomorphic to projective spaces. If $g=1$ and the embedding is given by a complete linear system, we prove that this locus is also a smooth projective variety whose positive-dimensional components are isomorphic to projective bund…
Rationality of Kontsevich moduli of stable maps, and their Galois descents
本文研究域k上几何有理簇X的低次数零亏格稳定映射的Kontsevich模空间的k-合理性，定义其扭曲伽罗瓦下降，针对不同类型的X展开分析，复数域齐次簇情形对应Kim-Pandharipande的已知结论。
In this article we classify quadruple Galois canonical covers $ϕ$ of singular surfaces of minimal degree. This complements the work done in math.AG/0302045, so the main output of both papers is the complete classification of quadruple Galois canonical covers of surfaces of minimal degree, both singular and smooth. Our results show that the covers $X$ studied in this article are all regular surfaces and form a bounded family in terms of geometric genus $p_g$. In fact, the geometric genus of $X$ is bounded by 4. Together with the results of Horikawa and Konno for double and triple covers, a stri…
We study wildly ramified G-Galois covers $ϕ:Y \to X$ branched at B (defined over an algebraically closed field of characteristic p). We show that curves Y of arbitrarily high genus occur for such covers even when G, X, B and the inertia groups are fixed. The proof relies on a Galois action on covers of germs of curves and formal patching. As a corollary, we prove that for any nontrivial quasi-p group G and for any sufficiently large integer $σ$ with $p \nmid σ$, there exists a G-Galois étale cover of the affine line with conductor $σ$ above the point $\infty$.
For an abelian surface A over a number field k, we study the limiting distribution of the normalized Euler factors of the L-function of A. This distribution is expected to correspond to taking characteristic polynomials of a uniform random matrix in some closed subgroup of USp(4); this Sato-Tate group may be obtained from the Galois action on any Tate module of A. We show that the Sato-Tate group is limited to a particular list of 55 groups up to conjugacy. We then classify A according to the Galois module structure on the R-algebra generated by endomorphisms of A_Qbar (the Galois type), and e…
A point on a plane curve is said to be Galois (for the curve) if the projection from the point as a map from the curve to a line induces a Galois extension of function fields. It is known that the number of Galois points is finite except for a certain explicit example. We establish upper bounds for the number of Galois points for all plane curves other than the example in terms of the genus, degree and the generic order of contact, and settle curves attaining the bounds.
Let $\mathbb{F}$ be the finite field of order $q^2$, $q=p^h$ with $p$ prime. It is commonly atribute to J.P. Serre the fact that any curve $\mathbb{F}$-covered by the Hermitian curve $\mathcal{H}_{q+1}:\, y^{q+1}=x^q+x$ is also $\mathbb{F}$-maximal. Nevertheless, the converse is not true as the Giulietti-Korchmáros example shows provided that $q>8$ and $h\equiv 0\pmod{3}$. In this paper, we show that if an $\mathbb{F}$-maximal curve $\mathcal{X}$ of genus $g\geq 2$ where $q=p$ is such that $|Aut(\mathcal{X})|>84(g-1)$ then $\mathcal{X}$ is Galois-covered by $\mathcal{H}_{p+1}$. Also, we show t…
We give a new example of a curve C algebraically, but not rationally, uniformized by radicals. This means that C has no map onto the projective line P^1 with solvable Galois group, while there exists a curve C' that maps onto C and has a finite morphism to P^1 with solvable Galois group. We construct such a curve C of genus 9 in the second symmetric product of a general curve of genus 2. It is also an example of a genus 9 curve that does not satisfy condition S(4,2,9) of Abramovich and Harris.
Automorphy of mod 2 Galois representations associated to certain genus 2 curves over totally real fields
本文研究全实域上亏格2超椭圆曲线的模2 Galois表示，在其像为$S_5$且在$\mathrm{GSp}_4(\mathbb{F}_2)\cong S_6$下传递时，证明它对应平行权2的Hilbert–Siegel Hecke特征尖点形式。
For a plane curve, a point on the projective plane is said to be Galois if the projection from the point as a map from the curve to a line induces a Galois extension of function fields. We present upper bounds for the number of Galois points, if the genus is greater than zero. If the curve is not an immersed curve, then we have at most two Galois points. If the degree is not divisible by two nor three, then the number of outer Galois points is at most three. As a consequence, a conjecture of Yoshihara is true in these cases.
Rational functions over finite fields with Galois closure of genus zero
本文分类了有限域上伽罗瓦闭包亏格零的可分有理函数，并证明此类函数置换有限域当且仅当例外，给出置换扩张次数与多项式代表元。
Exceptional covers of the projective line with genus-one Galois closure: arithmetic forms over finite fields
本文对几何伽罗瓦闭包亏格1的可分不可约驯服例外射影直线映射分类，给出有限域上算术固定域形式的编码准则，获秩2支撑公式等结果，明确特征与障碍的相关性质。
The metabelian Grothendieck conjecture for genus zero curves over finitely generated fields
研究有限生成域上零亏格曲线的亚交换格罗滕迪克猜想，通过证明两条双曲零亏格曲线在基域上同构（正特征下经弗罗贝尼乌斯扭转）的充要条件是其几何极大亚交换驯基本群在基域绝对伽罗瓦群上同构来得出结论。
Klein coverings over hyperelliptic genus 3 curves
本文研究了超椭圆亏格为3的曲线的克莱因覆盖的模空间，证明了普里姆映射的单射性，并展示了其普里姆映射的有限性。
In this paper we construct Galois towers with good asymptotic properties over any non-prime finite field $\mathbb F_{\ell}$; i.e., we construct sequences of function fields $\mathcal{N}=(N_1 \subset N_2 \subset \cdots)$ over $\mathbb F_{\ell}$ of increasing genus, such that all the extensions $N_i/N_1$ are Galois extensions and the number of rational places of these function fields grows linearly with the genus. The limits of the towers satisfy the same lower bounds as the best currently known lower bounds for the Ihara constant for non-prime finite fields. Towers with these properties are imp…
In this paper, we explicitly determine the automorphism group of every nonhyperelliptic superspecial curve of genus $4$ over $\mathbb{F}_{11}$. Our algorithm determining automorphism groups works for any nonhyperelliptic curves of genus $4$ over finite fields. With this computation, we show the compatibility between the enumeration of superspecial curves of genus $4$ over $\mathbb{F}_{11}$ obtained computationally by the first and second authors in 2017 and an enumeration by Galois cohomology theory.
Indecomposable rational functions over finite fields with Galois closure of genus one: Reconstruction and exceptionality
该研究针对有限域上带群作用的椭圆曲线间的等变同态，建立了其诱导有理函数的置换性与例外性的等价判据，明确了相关周期与密度，给出低阶循环商群的显式判据。
The Frey--Mazur conjecture states that an elliptic curve over $\mathbb{Q}$ is determined up to isogeny by its $p$-torsion Galois representation for $p\geq 17$. We study a geometric analog of this conjecture, and show that the map from isogeny classes of "fake elliptic curves"---abelian surfaces with quaternionic multiplication---to their $p$-torsion Galois representations is one-to-one over function fields of small genus complex curves for sufficiently large $p$ relative to the genus.
Let $f:X \to S$ be a Galois cover of Riemann surfaces, with Galois group $G$. In this paper we analyze the $G$-invariant divisors on $X$, and their associated spaces of meromorphic functions, differentials, and $q$-differentials. We generalize the trace formula for non-trivial elements of $G$ on $q$-differentials, as well as the Chevalley--Weil Formula. When $G$ is Abelian or when the genus of $S$ is 0 we prove additional results, and we also determine the non-special $G$-invariant divisors when both conditions are satisfied.
Let $k$ be an algebraically closed field of characteristic $p>0$. Let $G$ be $Z/\ell Z$ semi-direct product $Z/pZ$ where $\ell$ is a prime distinct from $p$. In this paper, we study Galois covers $ψ:Z \to P^1_k$ ramified only over $\infty$ with Galois group $G$. We find the minimal genus of a curve $Z$ that admits such a cover and show that it depends only on $\ell$, $p$, and the order $a$ of $\ell$ modulo $p$. We also prove that the number of curves $Z$ of this minimal genus which admit such a cover is at most $(p-1)/a$.
For each linearly normal elliptic curve $C$ in $\mathbb P^3$, we determine Galois lines and their arrangement. The results are as follows: the curve $C$ has just six $V_4$-lines and in case $j(C)=1$, it has eight $Z_4$-lines in addition. The $V_4$-lines form the edges of a tetrahedron, in case $j(C)=1$, for each vertex of the tetrahedron, there exist just two $Z_4$-lines passing through it. We obtain as a corollary that each plane quartic curve of genus one does not have more than one Galois point.
For each nonsingular hyperelliptic curve of arbitrary genus, we construct a natural injection from the Galois cohomology of 2-torsion subgroups of Jacobian varieties of the curve to the set of isomorphism classes of nonsingular complete intersections of two quadrics. This gives a generalization of the result of Flynn and Skorobogatov.
In a talk at the Banff International Research Station in 2015 Asher Auel asked questions about genus one curves in Severi-Brauer varieties $SB(A)$. More specifically he asked about the smooth cubic curves in Severi-Brauer surfaces, that is in $SB(D)$ where $D/F$ is a degree three division algebra. Even more specifically, he asked about the Jacobian, $E$, of these curves. In this paper we give a version of an answer to both these questions, describing the surprising connection between these curves and properties of the algebra $A$. Let $F$ contain $ρ$, a primitive third root of one. Since $D/F$…
In this article we consider the moduli space of smooth $n$-pointed non-hyperelliptic curves of genus 3. In the pursuit of cohomological information about this space, we make $\mathbb{S}_n$-equivariant counts of its numbers of points defined over finite fields for $n \leq 7$. Combining this with results on the moduli spaces of smooth pointed curves of genus 0, 1 and 2, and the moduli space of smooth hyperelliptic curves of genus 3, we can determine the $\mathbb{S}_n$-equivariant Galois and Hodge structure of the ($\ell$-adic respectively Betti) cohomology of the moduli space of stable curves of…
Let $X$ be a smooth projective connected curve of genus $g\ge 2$ defined over an algebraically closed field $k$ of characteristic $p>0$. Let $G$ be a finite group, $P$ a Sylow $p$-subgroup of $G$ and $N_G(P)$ its normalizer in $G$. We show that if there exists an étale Galois cover $Y\to X$ with group $N_G(P)$, then $G$ is the Galois group wan étale Galois cover $\mathcal{Y}\to\mathcal{X}$, where the genus of $\mathcal{X}$ depends on the order of $G$, the number of Sylow $p$-subgroups of $G$ and $g$. Suppose that $G$ is an extension of a group $H$ of order prime to $p$ by a $p$-group $P$ and $…
The V_4-lines for each linearly normal space elliptic curve form the edges of a tetrahedron, however in case the elliptic curve has j=12^3, there exist Z_4-lines in addition. We show the arrangement of V_4 and Z_4-lines explicitly for the curve. As a corollary we obtain that each irreducible quartic curve with genus one has at most two Galois points.
We give a complete list of rational functions $A$ such that the genus $g$ of the Galois closure of $\mathbb C(z)/\mathbb C(A)$ equals zero. We also provide a geometric description of $A$ for which $g=1.$
Pryms of $\mathbb{Z}_3\times\mathbb{Z}_3$ coverings of genus 2 curves
本文研究了Z_3×Z_3覆盖的二阶曲线的Prym变种和映射，证明了Prym-Teorelli定理成立，并探讨了任意交换群G的覆盖情况。
We study the relationship between rational points and Galois points for a plane curve over a finite field. It is known that the set of Galois points coincides with that of rational points of the projective plane if the curve is the Hermitian, Klein quartic or Ballico-Hefez curves. We propose a problem: Does the converse hold true? When the curve of genus at most one has a rational point, we will have an affirmative answer.
To figure properties of a curve of form $C_{f,g} = {(x,y)| f(x) - g(y)= 0}$ you must address the genus 0 and 1 components of its projective normalization $\tilde C_{f,g}$. For $f$ and $g$ polynomials with $f$ indecomposable, [Fr73a] distinguished $\tilde C_{f,g}$ with $u=1$ versus $u > 1$ components (Schinzel's problem). For $u = 1$, [Prop. 1, Fr73b] gave a direct genus formula. To complete $u > 1$ required an adhoc genus computation. [Pak22] dropped the indecomposable and polynomial restrictions but added $\tilde C_{f,g}$ is irreducible ($u = 1$). He showed - for fixed $f$ - unless the Galois…
We classify, up to isomorphism, maximal curves covered by the Hermitian curve \mathcal H by a prime degree Galois covering. We also compute the genus of maximal curves obtained by the quotient of \mathcal H by several automorphisms groups. Finally we discuss the value for the third largest genus that a maximal curve can have.
Given a smooth projective curve $X$ of genus at least 2 over a number field $k$, Grothendieck's Section Conjecture predicts that the canonical projection from the étale fundamental group of $X$ onto the absolute Galois group of $k$ has a section if and only if the curve has a rational point. We show that there exist curves where the above map has a section over each completion of $k$ but not over $k$. In the appendix Victor Flynn gives explicit examples in genus 2. Our result is a consequence of a more general investigation of the existence of sections for the projection of the étale fundament…
In this article we study the bicanonical map $ϕ_2$ of quadruple Galois canonical covers X of surfaces of minimal degree. We show that $ϕ_2$ has diverse behavior and exhibit most of the complexities that are possible for a bicanonical map of surfaces of general type, depending on the type of X. There are cases in which $ϕ_2$ is an embedding, and if so happens, $ϕ_2$ embeds $X$ as a projectively normal variety, and cases in which $ϕ_2$ is not an embedding. If the latter, $ϕ_2$ is finite of degree 1, 2 or 4. We also study the canonical ring of X, proving that it is generated in degree less than o…
Families of cyclic curve coverings with maximal monodromy
本文研究了具有最大单值性的循环曲线覆盖族，通过Hodge结构变分的G分解条件，证明了单值性最大性的判据，并探讨了双伽罗华覆盖和三伽罗华覆盖的测地性问题。
Invariant vector bundles and Hitchin systems
本文研究亏格至少为2的光滑复射影曲线伽罗瓦覆盖上半稳定不变量向量丛的模空间，分类其连通分支，并确定相应希钦系统在光滑情形下的纤维结构。
We consider the moduli space $\Hh_{g,n}$ of $n$-pointed smooth hyperelliptic curves of genus $g$. In order to get cohomological information we wish to make $\s_n$-equivariant counts of the numbers of points defined over finite fields of this moduli space. We find recurrence relations in the genus that these numbers fulfill. Thus, if we can make $\s_n$-equivariant counts of $\Hh_{g,n}$ for low genus, then we can do this for every genus. Information about curves of genus 0 and 1 is then found to be sufficient to compute the answers for $\Hh_{g,n}$ for all $g$ and for $n \leq 7$. These results ar…
This survey is about Galois theory of curves in characteristic p, a topic which has inspired major research in algebraic geometry and number theory and which contains many open questions. We illustrate important phenomena which occur for covers of curves in characteristic p. We explain key results on the structure of fundamental groups. We end by describing areas of active research and giving two new results about the genus and p-rank of certain covers of the affine line.
In this article we classify quadruple Galois canonical covers of smooth surfaces of minimal degree. The classification shows that they are either non-simple cyclic covers or bi-double covers. If they are bi-double then they are all fiber products of double covers. We construct examples to show that all the possibilities in the classification do exist. There are implications of this classification that include the existence of families with unbounded geometric genus and families with unbounded irregularity, in sharp contrast with the case of double and triple canonical covers. Together with the…
Let $K$ be a number field not containing a CM subfield. For any smooth projective curve $Y/K$ of genus $\geq2$, we prove that the image of the "Selmer" part of Grothendieck's section set inside the $K_v$-rational points $Y(K_v)$ is finite for every finite place $v$. This gives an unconditional verification of a prediction of Grothendieck's section conjecture. In the process of proving our main result, we also refine and extend the method of Lawrence and Venkatesh, with potential consequences for explicit computations.
We study Shimura subvarieties of $\mathsf{A}_g$ obtained from families of Galois coverings $f: C \rightarrow C'$ where $C'$ is a smooth complex projective curve of genus $g' \geq 1$ and $g= g(C)$. We give the complete list of all such families that satisfy a simple sufficient condition that ensures that the closure of the image of the family via the Torelli map yields a Shimura subvariety of $\mathsf{A}_g$ for $g' =1,2$ and for all $g \geq 2,4$ and for $g' > 2$ and $g \leq 9$. In a previous work of the first and second author together with A. Ghigi [FGP] similar computations were done in the c…
On 7-adic Galois representations for elliptic curves over $\mathbb{Q}$
本文研究7进制伽罗瓦表示的分类，证明模曲线X_{ns}^+(49)无非CM有理点，并将其归结为求解一般费马方程和确定单个平面四次曲线的有理点。
This paper is devoted to the explicit description of the Galois descent obstruction for hyperelliptic curves of arbitrary genus whose reduced automorphism group is cyclic of order coprime to the characteristic of their ground field. Along the way, we obtain an arithmetic criterion for the existence of a hyperelliptic descent. The obstruction is described by the so-called arithmetic dihedral invariants of the curves in question. If it vanishes, then the use of these invariants also allows the explicit determination of a model over the field of moduli; if not, then one obtains a hyperelliptic mo…
We study group actions on regular models of curves. If $X$ is a smooth curve defined over the fraction field $K$ of a complete d.v.r. $R$, every tamely ramified extension $K'/K$ with Galois group $G$ induces a $G$-action on $X_{K'}$. In this paper we study the extension of this $G$-action to certain regular models of $X_{K'}$. In particular, we obtain a formula for the Brauer trace of the endomorphism induced by a group element on the alternating sum of the cohomology groups of the structure sheaf of the special fiber of such a regular model. Inspired by this global study, we also consider sim…
On Shimura curves generated by families of Galois $G$-covers of curves
本文利用 Higgs 丛理论和 Viehweg-Zuo 刻画，证明了在特定素因子条件下，椭圆曲线及任意亏格曲线的 $\Z_n$-覆盖族无法生成位于 Torelli 轨迹中的非紧 Shimura 曲线。
The Absolute Anabelian Geometry of Virtual Curves of Arbitrary Genus
本文研究虚曲线的绝对单形几何，扩展了之前关于亏格零情况的基底群理论拉回结果，引入了CAVC型包含和虚拟双螺旋范畴概念，建立虚拟曲线几何性的判据。
In this article we construct for any prime power $q$ and odd $n \ge 5$, a new $\mathbb{F}_{q^{2n}}$-maximal curve $\mathcal X_n$. Like the Garcia--G\" uneri--Stichtenoth maximal curves, our curves generalize the Giulietti--Korchmáros maximal curve, though in a different way. We compute the full automorphism group of $\mathcal X_n$, yielding that it has precisely $q(q^2-1)(q^n+1)$ automorphisms. Further, we show that unless $q=2$, the curve $\mathcal{X}_n$ is not a Galois subcover of the Hermitian curve. Finally, we find new values of the genus spectrum of $\mathbb{F}_{q^{2n}}$-maximal curves, …
In this article, we show that in each of four standard families of hyperelliptic curves, there is a density-$1$ subset of members with the property that their Jacobians have adelic Galois representation with image as large as possible. This result constitutes an explicit application of a general theorem on arbitrary rational families of abelian varieties to the case of families of Jacobians of hyperelliptic curves. Furthermore, we provide explicit examples of hyperelliptic curves of genus $2$ and $3$ over $\mathbb Q$ whose Jacobians have such maximal adelic Galois representations.
In this paper we study genus 2 curves whose Jacobians admit a polarized (4,4)-isogeny to a product of elliptic curves. We consider base fields of characteristic different from 2 and 3, which we do not assume to be algebraically closed. We obtain a full classification of all principally polarized abelian surfaces that can arise from gluing two elliptic curves along their 4-torsion and we derive the relation their absolute invariants satisfy. As an intermediate step, we give a general description of Richelot isogenies between Jacobians of genus 2 curves, where previously only Richelot isogenies …
Let $X$ be a smooth curve defined over the fraction field $K$ of a complete d.v.r. $R$, and let $K'/K$ be a tame extension. We study extensions of the $G = \Gal(K'/K)$-action on $ X_{K'} $ to certain regular models of $X_{K'}$ over $R'$, the integral closure of $R$ in $K'$. In particular, we consider the induced action on the cohomology groups of the structure sheaf of the special fiber of such a regular model, and obtain a formula for the Brauer trace of the endomorphism induced by a group element on the alternating sum of the cohomology groups. We apply these results to study a natural filtr…
We study triple covers of K3 surfaces, following Miranda's theory of triple covers. We relate the geometry of the covering surfaces with the properties of both the branch locus and the Tschirnhausen vector bundle. In particular, we classify Galois triple covers computing numerical invariants of the covering surface and of its minimal model. We provide examples of non Galois triple covers, both in the case in which the Tschirnhausen bundle splits into the sum of two line bundles and in the case in which it is an indecomposable rank 2 vector bundle. We provide a criterion to construct rank 2 vec…
Complete Families of Curves in the Moduli Space of Genus g Curves
通过推广González Díez和Harvey的构造，在亏格g≥3的模空间M_g中构造新的完全曲线，并利用函数域的伽罗瓦理论给出每条曲线T的亏格公式。
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