Elisa Tenni
Let $f:S \fr B$ be a surface fibration with fibres of genus 5. We find a linear relation between the fundamental invariants of the surface. Namely $K_f^2=χ_f+N$ where $N$ is the number of trigonal fibres. Our proof is based on the analysis of the relative canonical algebra $\cal{R}(f)$.
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We consider relatively minimal fibrations of curves of genus two on rational surfaces whose Picard numbers are not maximal. By birational morphisms, such fibred surfaces are interpreted as pencils of plane curves. We show that only four are canonical, among a variety of possible models. For each canonical pencil, we give an example with trivial Mordell-Weil group.
We generalise a method of Xiao Gang to construct 'prototypes' of fibred surfaces with maximal irregularity without being a product. This enables us, in the case of fibre genus g=3 to describe the possible singular fibres and to calculate the invariants of these surfaces. We also prove structure theorems on the moduli space for fibred surfaces with fibre genus g=2 and g=3.
This note aims to improve known numerical bounds proved earlier by Chen \cite{PAMS} and Chen-Hacon \cite{Chen-Hacon} and to present some new examples of smooth minimal 3-folds canonically fibred by surfaces (resp. curves) of geometric genus as large as 19 (resp. 13). As an interesting by-product, we present a new class of general type surfaces which are canonically fibred by curves of genus 13.
We prove that the bicanonical map of a surfaces of general S type with p_g=q=0 is non birational if there exists a pencil on S whose general member is an hyperelliptic curve of genus 3.
We show that the strata $ \mathcal{M}_{11,6}(k) \subset \mathcal{M}_{11} $ of $ 6-$gonal curves of genus $ 11 $, equipped with $k$ mutually independent and type I pencils of degree six, have a unirational irreducible component for $5\leq k\leq 9$. The unirational families arise from degree $ 9 $ plane curves with $ 4 $ ordinary triple and $ 5 $ ordinary double points that dominate an irreducible component of expected dimension. We will further show that the family of degree $ 8 $ plane curves with $ 10 $ ordinary double points covers an irreducible component of excess dimension in $ \mathcal{M…
Let X be a smooth curve of genus g. When pi>=3g and d>=pi-2g+1 we show the existence of a double covering gamma:C-->X where C a smooth curve of genus pi with a base-point-free pencil of degree d which is not the pull-back of a pencil on X.
In this paper, the Euler characteristic formula for projective logarithmic minimal degenerations of surfaces with Kodaira dimension zero over a 1-dimensional complex disk is proved under a reasonable assumption and as its application, the singularity of logarithmic minimal degenerations are determined in the abelian or hyperelliptic case. By globalizing this local analysis of singular fibres via generalized canonical bundle formulae due to Fujino-Mori, we bound the number of singular fibres of abelian fibred Calabi-Yau threefolds from above,which was previously done by Oguiso in the potentiall…
We consider a rational surface with a relatively minimal fibration. Picard number of a such fibred surface is bounded in terms of the genus of a general fibre. When Picard number is the maximum for any given genus, we characterize a such fibred surface whose Mordell-Weil group is trivial by singular fibres. Furthermore, we describe the defining equation explicitly.
We prove that the genus $g$, the relative irregularity $q_f$ and the Clifford index $c_f$ of a non-isotrivial fibration $f$ satisfy the inequality $q_f \leq g - c_f$. This gives in particular a proof of Xiao's conjecture for fibrations whose general fibres have maximal Clifford index.
We construct two pencils of bielliptic curves of genus three and genus five. The first pencil is associated with a general abelian surface with a polarization of type $(1,2)$. The second pencil is related to the first by an unramified double cover, the Prym variety of which is canonically isomorphic to the Jacobian of a very general curve of genus two. Our results are obtained by analyzing suitable elliptic fibrations on the associated Kummer surfaces and rational double covers among them.
In this paper, we classify relatively minimal genus-$1$ holomorphic Lefschetz pencils up to smooth isomorphism. We first show that such a pencil is isomorphic to either the pencil on $\mathbb{P}^1\times \mathbb{P}^1$ of bi-degree $(2,2)$ or a blow-up of the pencil on $\mathbb{P}^2$ of degree $3$, provided that no fiber of a pencil contains an embedded sphere. (Note that one can easily classify genus-$1$ Lefschetz pencils with an embedded sphere in a fiber.) We further determine the monodromy factorizations of these pencils and show that the isomorphism class of a blow-up of the pencil on $\mat…
Let $f:\,S \to B$ be a locally non-trivial relatively minimal fibration of genus $g\geq 2$ with relative irregularity $q_f$. It was conjectured by Barja and Stoppino that the slope $λ_f\geq \frac{4(g-1)}{g-q_f}$. We prove the conjecture when $q_f$ is small with respect to $g$; we also construct counterexamples when $g$ is odd and $q_f=(g+1)/2$.
We explicitly construct Brill--Noether general $K3$ surfaces of genus $4,6$ and $8$ having the maximal number of elliptic pencils of degrees $3, 4$ and $5$, respectively, and study their moduli spaces and moduli maps to the moduli space of curves. As an application we prove the existence of Brill--Noether general $K3$ surfaces of genus $4$ and $6$ without stable Lazarsfeld--Mukai bundles of minimal $c_2$.
Monodromy of the Prym map and semicanonical pencils in genus 6
本文证明了亏格6中限制于奇半典范铅笔除子的Prym映射是双有理的，其单值群为Weyl群$WD_5$，并揭示了模空间中其他除子的次数及几何性质。
We prove that a complex surface S with irregularity q(S)=5 that has no irrational pencil of genus >1 has geometric genus p_g(S)>7. As a consequence, one is able to classify minimal surfaces S of general type with q(S)=5 and p_g(S)<8. This result is a negative answer, for q=5, to the question asked in arXiv:0811.0390 of the existence of surfaces of general type with irregularity q>3 that have no irrational pencil of genus >1 and with the lowest possible geometric genus p_g=2q-3. This gives some evidence for the conjecture that the only irregular surface with no irrational pencil of genus >1 and…
Slopes of trigonal fibred surfaces and of higher dimensional fibrations
本文推广Cornalba-Harris方法，在GIT半稳定条件下给出高维纤维化斜率下界，并证明偶数亏格三角纤维化斜率的锐利下界，从而证实Harris-Stankova猜想。
Locally non-trivial fibred surfaces with maximal unitary rank
该文研究具有最大酉秩$u_f=g-1$的局部非平凡纤维化曲面，证明强斜率不等式，结合三角曲线结果给出$g=6$、$u_f=5$情形的新约束，包括曲面指数严格为正及相对典范除子类的强限制。
The sextic plane curves that are invariant under the standard action of the icosahedral group on the projective plane make up a pencil of genus ten curves (spanned by a sum of six lines and a three times a conic). This pencil was first considered in a note by R.~M.~Winger in 1925 and is nowadays named after him. We gave this a modern treatment and proved among other things that it contains essentially every smooth genus ten curve with icosahedral symmetry. We here consider the monodromy group and the period map naturally defined by the icosahedral symmetry. We showed that this monodromy group …
We consider relationships between families of K3 surfaces, in the context of string theory. An important ingredient of string theory also of interest in algebraic geometry is T-duality. Donagi and Pantev have extended the original duality on genus one fibred K3 surfaces with a section to the case of any genus one fibration, via a Fourier-Mukai transform. We investigate possibilities of extending this result to the more general case of non fibred K3s, in the context of Caldararu's conjecture. We also provide an explicit construction of a quasi-universal sheaf associated to the classical Mukai c…
Bound of automorphisms of fibred surfaces in positive characteristic
本文证明了在正特征 $p\geq 5$ 下，一般型极小纤维曲面的保纤维自同构群阶不超过 $15658(K_S^2)^4$，并给出例子说明指数 $4$ 最优。
We investigate the moduli of genus 10 curves that are endowed with a faithful action of the icosahedral group $\mathcal{A}_5$. We show among other things that this has the structure of a Deligne-Mumford stack whose underlying coarse moduli space essentially consists of two copies of the pencil of plane sextics that was introduced by Winger in 1924, but with the unique unstable member (a triple conic) replaced by a smooth non-planar curve. The orbifold defined by any member has genus zero and comes with 4 orbifold points. We show that by numbering the points, we get a fine moduli space whose ba…
Let $C$ be a smooth curve with gonality $k\ge 6$ and genus $g\ge 2k^2+5k-6$. We prove that $W^1_d({C})$ has the expected dimension and that the general element of any irreducible component of $W^1_d({C})$ is primitive if either $g-k+4\le d\le g-2$ or $d=g-k+3$ and either $k$ is odd or $C$ is not a double covering of a curve of gonality $k/2$ and genus $k-3$. Even in the latter case we prove the existence of a complete and primitive $g^1_{g-k+3}$.
Let f: X->B be a fibred surface of genus g whose general fibre is a double cover of a smooth curve of genus gamma. We show that, for g > 4gamma+1, the number 4(g-1)/(g-gamma) is a sharp lower bound for the slope of f, proving a conjecture of Barja. Moreover, we give a characterisation of the fibred surfaces that reach the bound. In the case g = 4gamma+1 we obtain the same sharp bound under the assumption that the involutions on the general fibres glue to a global involution on X.
Geometry of Prym semicanonical pencils and an application to cubic threefolds
本研究探讨模空间中半典范线束对应除子上的Prym映射，揭示其差异与低亏格几何特性，并通过对特定除子的分析得到三次三维簇直线的计数结果。
Genus $2$ pencils on surfaces with $p_g=K^2=1$, envelopes, and conics tangent to plane cubic curves
本文研究了具有p_g=K^2=1的曲面中曲率2 pencil的存在性，证明其在模空间中形成不可约子流形，并探讨了与平面三次曲线相切的圆锥线几何关系。
We present an explicit parametrization of the families of lines of the Dwork pencil of quintic threefolds. This gives rise to isomorphic curves which parametrize the lines. These curves are 125:1 covers of certain genus six curves. These genus six curves are first presented as curves in P^1*P^1 that have three nodes. It is natural to blow up P^1*P^1 in the three points corresponding to the nodes in order to produce smooth curves. The result of blowing up P^1*P^1 in three points is the quintic del Pezzo surface dP_5, whose automorphism group is the permutation group S_5, which is also a symmetr…
K. Konno proved a slope equality for fibred surfaces with fibres of odd genus and general fibre of maximal gonality. More precisely he found a relation between the invariants of the fibration and certain weights of special fibres (called the Horikawa numbers). We give an alternative and more geometric proof in the case of a genus 5 fibration, under generality assumptions. In our setting we are able to prove that the fibre with positive Horikawa numbers are precisely the trigonal ones, we compute their weights explicitly and thus we exhibit explicit examples of regular surfaces with assigned in…
In 1981 W.L. Edge discovered and studied a pencil $\mathcal{C}$ of highly symmetric genus $6$ projective curves with remarkable properties. Edge's work was based on an 1895 paper of A. Wiman. Both papers were written in the satisfying style of 19th century algebraic geometry. In this paper and its sequel [FL], we consider $\mathcal{C}$ from a more modern, conceptual perspective, whereby explicit equations are reincarnated as geometric objects.
In this paper we discuss topological properties of holomorphic Lefschetz pencils on the four-torus. Relying on the theory of moduli spaces of polarized abelian surfaces, we first prove that, under some mild assumption, the (smooth) isomorphism class of a holomorphic Lefschetz pencil on the four-torus is uniquely determined by its genus and divisibility. We then explicitly give a system of vanishing cycles of the genus-3 holomorphic Lefschetz pencil on the four-torus due to Smith, and obtain those of holomorphic pencils with higher genera by taking finite unbranched coverings. One can also obta…
In the present note we show that any surface of general type with $p_g=2$,$q=1$ and non-birational bicanonical map has a pencil of curves of genus 2. Combining this result with previous ones, one obtains thatan irregular surface $S$ of general type with $χ(S)\geq 2$ and non-birational bicanonical map has a pencil of curves of genus 2.
We describe the minimal number of critical points and the minimal number $s$ of singular fibres for a non isotrivial fibration of a surface $S$ over a curve $B$ of genus $1$, constructing a fibration with $s=1$ and irreducible singular fibre with $4$ nodes. Then we consider the associated factorizations in the mapping class group and in the symplectic group. We describe explicitly which products of transvections on homologically independent and disjoint circles are a commutator in the Symplectic group $Sp (2g, \mathbb{Z})$.
Fibred surfaces and their unitary rank
本文研究亏格至少为2的复纤维化曲面的酉秩，证明多个新斜率不等式，并由此得到非等平凡纤维化酉秩的Xiao型上界及Coleman-Oort猜想方向的新约束。
Related to the classification of regular foliations in a complex algebraic surface, we address the problem of classifying the complex surfaces which admit a flat pencil of foliations. On this matter, a classification of flat pencils which admit foliations with a first integral of genus one and isolated singularities was done by Lins Neto. In this work, we complement Lins Neto's work, by obtaining the classification of compact complex surfaces which have a pencil with an invariant tangency set.
In this thesis we give a complete description of the syzygies of irreducible, nonsingular, canoncial curves $C$ of genus 9. This includes a collection of all possible Betti tables for $C$. Moreover a direct correspondence between these Betti tables and the number and types of special linear series on $C$ is given. Especially for $\operatorname{Cliff}(C)=3$ the curve $C$ is contained in determinantal surface $Y$ on a 4-dimensional rational normal scroll $X\subset \mathbb{P}^8$ constructed from a base point free pencil of divisors of degree 5.
The sextic plane curves that are invariant under the standard action of the icosahedral group on the projective plane make up a pencil of genus ten curves (spanned by a sum of six lines and a three times a conic). This pencil was first considered in a note by R.~M.~Winger in 1925 and is nowadays named after him. The second author recently gave this a modern treatment and proved among other things that it contains essentially every smooth genus ten curve with icosahedral symmetry. We here show that the Jacobian of such a curve contains the tensor product of an elliptic curve with a certain inte…
Let X be a complex projective n-dimensional manifold of general type, whose canonical system is composite with a pencil. If the Albanese map is generically finite, but not surjective, or if the irregularity is strictly larger than n and the image of X in its Albanese variety of Kodaira dimension one, then the geometric genus of a general fibre of the canonical map is one and the latter factors through the Albanese map. The last part of this result holds true for any threefold with irregularity 5 or larger.
Let $C$ be an algebraic curve defined by a sufficiently generic bivariate Laurent polynomial with given Newton polygon $Δ$. It is classical that the geometric genus of $C$ equals the number of lattice points in the interior of $Δ$. In this paper we give similar combinatorial interpretations for the gonality, the Clifford index and the Clifford dimension, by removing a technical assumption from a recent result of Kawaguchi. More generally, the method shows that apart from certain well-understood exceptions, every base-point free pencil whose degree equals or slightly exceeds the gonality is 'co…
Symplectic mapping class relations from pencil pairs
本文研究了Weinstein 4-流形中正Dehn扭动沿拉格朗日球体的乘积之间的辛映射类关系，通过分析圆柱对和Fano 3-折叠、极化K3表面的分类结果得出相关结论。
Let f:S ->B be a relatively minimal fibred surface. In this note we give a partial affirmative answer to a conjecture of Xiao, proving that the direct image of the relative dualizing sheaf of $f$ is ample when the slope of the fibration is less than 4, if the general fibre of $f$ is non-hyperelliptic or the genus of the fibre or of the base curve is low.
The {\em Wiman-Edge pencil} is the universal family $C_t, t\in\mathcal B$ of projective, genus $6$, complex-algebraic curves admitting a faithful action of the icosahedral group $\Af_5$. The curve $C_0$, discovered by Wiman in 1895 \cite{Wiman} and called the {\em Wiman curve}, is the unique smooth, genus $6$ curve admitting a faithful action of the symmetric group $\Sf_5$. In this paper we give an explicit uniformization of $\mathcal B$ as a non-congruence quotient $Γ\backslash \Hf$ of the hyperbolic plane $\Hf$, where $Γ<\PSL_2(\Z)$ is a subgroup of index $18$. We also give modular interpret…
Let $(S,D)$ be a minimal log pair of general type with $S$ a smooth projective surface and $D$ a simple normal corssing reduced divisor on $S$. We assume that its log canonial linear system $|K_S+D|$ is composed of a penciel, let $f\colon S\to B$ be the fiberation induced by the linear system $|K_S+D|$ and $F$ be a general fiber of $f$. Let $b$ (resp. $g$) be the genus of the base curve $B$ (resp. general fiber $F$) and $k=D\cdot F$ the intersection number. We show that 1. If $k>0$ and $b\geq 2$ then $2\leq g+k \leq 3$, when $g+k=3$ we have $b=2$ and $h^1(S,K_S+D)=0$. 2. Suppose $p_a(D)\leq 2(…
Let X be a smooth genus g curve equipped with a simple morphism f: X -> C, where C is either the projective line or more generally any smooth curve whose gonality is computed by finitely many pencils. Here we apply a method developed by Aprodu to prove that if g is big enough then X satisfies both Green and Green-Lazarsfeld conjectures. We also partially address the case in which the gonality of C is computed by infinitely many pencils.
We describe the Griffiths group of the product of a curve $C$ and a surface $S$ as a quotient of the Albanese kernel of $S$ over the function field of $C$. When $C$ is a hyperplane section of $S$ varying in a Lefschetz pencil, we prove the nonvanishing in $\text{Griff}(C\times S)$ of a modification of the graph of the embedding $C\hookrightarrow S$ for infinitely many members of the pencil, provided the ground field $k$ is of characteristic $0$, the geometric genus of $S$ is $>0$, and $k$ is large or $S$ is "of motivated abelian type".
We study the pencils of minimal degree on the smooth curves lying on a K3 surface X which carries a fixed-point free involution. Generically, the gonality of these curves is totally governed by the genus 1 fibrations of X
The moduli space $\mathscr{M}$ of surfaces of general type with $p_g=q=1, K^2=g=3$ (where $g$ is the genus of the Albanese fibration) was constructed by Catanese and Ciliberto in \cite{CaCi93}. In this paper we characterize the subvariety $\mathscr{M}_2 \subset \mathscr{M}$ corresponding to surfaces containing a genus 2 pencil, and moreover we show that there exists a non-empty, dense subset $\mathscr{M}^0 \subset \mathscr{M}$ which parametrizes isomorphism classes of surfaces with birational bicanonical map.
Polarized K3 surfaces of genus thirteen and curves of genus three
本研究将亏格3曲线上秩2稳定向量丛的6维模空间上的完全交簇，刻画为亏格13的本原极化K3曲面，超椭圆曲线情形下进一步对应格拉斯曼簇中两个Spin(7)切触齐性簇的交，丰富了K3曲面的几何构造。
Towards the classification of semistable fibrations having exactly five singular fibers
本文研究光滑有理曲线上恰有5个奇异纤维的半稳定非平凡纤维化，将一般非超椭圆纤维亏格上界由17改进至11，并刻画其纤维gonality及由极小有理曲面曲线束解消得到的结构。
Irrational pencils, and characterization of Varieties isogenous to a product, via the Profinite completion of the Fundamental group
本文通过基本群的 profinite 完成，刻画了与积同构的代数簇，并证明了无理 pencil 与紧 Kahler 流形的对应关系。
Let us consider the locus in the moduli space of curves of genus 2k defined by curves with a pencil of degree k. Since the Brill-Noether number is equal to -2, such a locus has codimension two. Using the method of test surfaces, we compute the class of its closure in the moduli space of stable curves.
Let $R$ be an excellent Henselian discrete valuation ring with algebraically closed residue field $k$ of any characteristic. Fix integers $r,d$ with $r\ge 2$. Let $X_R$ be a regular fibred surface over Spec($R$) with special fibre denoted $X_k$, a generalised tree-like curve of genus $g \ge 2$. Let ${L}_R$ be a line bundle on $X_R$ of degree $d$ such that the degree of the restriction of ${L}_R$ on the rational components of $X_k$ is a multiple of $r$. In this article we prove the existence of a rank $r$ locally free sheaf on $X_R$ of determinant ${L}_R$ such that it is semistable on the fibre…
For a smooth canonically embedded curve $C$ of genus $9$ together with a pencil $|L|$ of degree $6$, we study the relative canonical resolution of $C\subset X\subset \mathbb{P}^8$, where $X$ is the scroll swept out by the pencil $|L|$. We show that the second syzygy bundle in this resolution of $C\subset X$ is unbalanced. The proof reveals a new geometric connection between the universal Brill--Noether variety $\mathcal{W}^1_{9,6}$ and a moduli space $\mathcal{F}^\mathfrak{h}$ of lattice polarized $K3$ surfaces (for a certain rank $3$ lattice $\mathfrak{h}$). As a by-product we prove the unira…
The Konno invariant of a projective variety X is the minimum geometric genus of the fiber of a rational pencil on X. It was computed by Konno for surfaces in P^3, and in general can be viewed as a measure of the complexity of X. We estimate Konno(X) for some natural classes of varieties, including sharp asymptotics for polarized K3 surfaces. In an appendix, we give a quick proof of a classical formula due to Deligne and Hoskin for the colength of an integrally closed ideal on a surface.
Severi curves of rational elliptic surfaces
本文研究平面三次曲线束对应椭圆纤维化中参数化有理双截面的 Severi 曲线，证明其连通且约化，并用拟模形式界定几何亏格，同时提出可约性与次数猜想。
We exhibit new examples of double Kodaira fibrations by using finite Galois covers of a product $Σ_b \times Σ_b$, where $Σ_b$ is a smooth projective curve of genus $b \geq 2$. Each cover is obtained by providing an explicit group epimorphism from the pure braid group $\mathsf{P}_2(Σ_b)$ to some finite Heisenberg group. In this way, we are able to show that every curve of genus $b$ is the base of a double Kodaira fibration; moreover, the number of pairwise non-isomorphic Kodaira fibred surfaces fibering over a fixed curve $Σ_b$ is at least $\boldsymbolω(b+1)$, where $\boldsymbolω \colon \mathbb…
Let $S$ be a smooth minimal surface of general type with a (rational) pencil of hyperelliptic curves of minimal genus $g$. We prove that if $K_S^2<4χ(\mathcal O_S)-6,$ then $g$ is bounded. The surface $S$ is determined by the branch locus of the covering $S\rightarrow S/i,$ where $i$ is the hyperelliptic involution of $S.$ For $K_S^2<3χ(\mathcal O_S)-6,$ we show how to determine the possibilities for this branch curve. As an application, given $g>4$ and $K_S^2-3χ(\mathcal O_S)<-6,$ we compute the maximum value for $χ(\mathcal O_S).$ This list of possibilities is sharp.
This research monograph focuses on the arithmetic, over number fields, of surfaces fibred into curves of genus 1 over the projective line, and of intersections of two quadrics in projective space. The first half takes up and develops further the technique initiated by Swinnerton-Dyer in 1993, and later generalised by Colliot-Thélène, Skorobogatov and Swinnerton-Dyer, for studying rational points on pencils of curves of genus 1. The second half, which builds upon the first, is devoted to quartic del Pezzo surfaces and to higher-dimensional intersections of two quadrics. Conditionally on two wel…
Fujita's second theorem for Kähler fibre spaces over a curve asserts that the direct image $V$ of the relative dualizing sheaf splits as the direct sum $ V = A \oplus Q$, where $A$ is ample and $Q$ is unitary flat. We focus on our negative answer (\cite{cd}) to a question by Fujita: is $V$ semiample? We give here an infinite series of counterexamples using hypergeometric integrals and we give a simple argument to show that the monodromy representation is infinite. Our counterexamples are surfaces of general type with positive index, explicitly given as abelian coverings with group $(\mathbb Z/…
Let S be a minimal complex surface of general type with $q(S)=0$. We prove the following statements concerning the algebraic fundamental group: I) Assume that K^2_S\leq 3χ(S). Then S has an irregular etale cover if and only if S has a free pencil of hyperelliptic curves of genus 3 with at least 4 double fibres. II) If K^2_S=3 and χ(S)=1, then S has no irregular etale cover. III) If K^2_S<3χ(S) and S does not have any irregular etale cover, then the order of the algebraic fundamental group is lesser or equal to 9, and if equality occurs then K^2_S=2, χ(S)=1.
Construction and Applications of Trisections of Low Genus on Del Pezzo Surfaces of Degree One
本文研究一次德尔佩佐曲面上低亏格三分截面的构造，通过限定椭圆曲面系数并利用过给定点三次的三分截面得到三次曲线束，进而在若干情形下证明有理点的Zariski稠密性。
Let $(S,L)$ be a smooth primitively polarized K3 surface of genus $g$ and $f:X \rightarrow \mathbb{P}^1$ the fibration defined by a linear pencil in $|L|$. For $f$ general and $g \geq 7$, we work out the splitting type of the locally free sheaf $Ψ^{*}_f T_{\overline{M}_g}$, where $Ψ_f$ is the modular morphism associated to $f$. We show that this splitting type encodes the fundamental geometrical information attached to Mukai's projection map $\mathcal{P}_g \rightarrow \overline{\mathcal{M}}_g$, where $\mathcal{P}_g$ is the stack parameterizing pairs $(S,C)$ with $(S,L)$ as above and $C \in |L|…
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