---
abstract: 'In [@HMo theorem 2.5] Harris and Morrison construct semistable families $f\colon F\to Y$ of $k$-gonal curves of genus $g$ such that for every $k$ the corresponding modular curves give a sweeping family in the $k$-gonal locus $\overline{{\mathcal{M}}^{k}_{g}}$. Their construction depends on the choice of a smooth curve $X$. We show that if the genus $g(X)$ is sufficiently high with respect to $g$ then the ratio $\frac{K^{2}_{F}}{\chi({\mathcal{O}}_{F})}$ is $8$ asymptotically with respect to $g(X)$. Moreover, if the conjectured estimates given in [@HMo p. 351-352] hold, we show that if $g$ is big enough, then $F$ is a surface of positive index.'
address:
- |
    Dipartimento di Matematica e Geoscienze, Università degli studi di Trieste,\
    via Valerio 12/b, 34127 Trieste, Italy\
    `beorchia@units.it`
- |
    Dipartimento di Matematica e Informatica\
    via delle Scienze 206, Università degli studi di Udine\
    Udine, 33100 Italy\
    `Francesco.Zucconi@uniud.it`
author:
- Beorchia Valentina and Zucconi Francesco
title:  A note on Harris Morrison sweeping families of maximal gonality
---

Introduction.
=============

Let $\overline{\mathcal{M}}_{g}$ be the moduli space of stable curves of genus $g$. A family ${\mathcal{B}}$ of curves $Z\subset {\overline {\mathcal M}_g}$ is said to be a sweeping family if $\cup{\mathcal{B}}$ is a Zariski dense subset of $\overline{\mathcal{M}}_{g}$. The invariants of sweeping families can furnish important information on the geometry of $\overline{\mathcal{M}}_{g}$; see: [@CFM].

In this paper we focus on the geometry of the families of curves constructed in [@HMo Theorem 2.5]. We revise this construction in Section 2; here we point out only that it depends on choosing a smooth curve $X$ of genus $g(X)$. For the aim of [@HMo] $g(X)$ can be taken to be zero, but Harris Morrison construction applies then to any curve $X$. We compute the invariants of the smooth surface $F$ supporting the Harris Morrison family. We denote by $e(F)$, $\chi({\mathcal{O}}_{F})$, $K^{2}_{F}$ respectively the topological characteristic of $F$, its cohomological characteristic and the self-intersection of the canonical divisor. We also denote by $q(F)$ the irregularity of $F$. We have:

[**[Theorem]{}**]{}[*[ Let $g,k\in\mathbb N$ such that $k$ can occur as the gonality of a smooth curve of genus $g$. There exist Harris Morrison sweeping $k$-gonal semistable genus-$g$ fibrations $f \colon F\to Y$ such that the ratio $\frac{K^{2}_{F}}{\chi({\mathcal{O}}_{F})}$ is $8$ asymptotically with respect to $g(X)>0$.]{}*]{}

See Theorem \[secondaadifferenza\] for a refined statement. We stress that our Theorem is not obtained by a base change over a family as in [@HMo Theorem 2.5] starting from ${\mathbb{P}}^{1}$. We remark that the above result applies to any gonality $k$.

If we consider only those families with maximal gonality, then we show that (see Proposition \[secondo\]):

[**Proposition**]{} [*[Let $f\colon F\to Y$ be an Harris Morrison genus-$g$ fibration starting from any plane curve $X$ of genus $g(X)>>0$. Assume that the gonality $k$ is maximal and that the conjectured estimates in [@HMo p. 351-352] are true. If $g$ is big enough, then $F$ is a surface of positive index i.e. $K^{2}_{F}> 8\chi({\mathcal{O}}_{F})$. Moreover if $f\colon F\to Y$ is general in its class then the irregularity of $F$ is $g(Y)$. In particular, $f\colon F\to Y$ is the Albanese morphism of $F$.]{}*]{}

We recall that any surface of general type satisfies the Miyaoka-Yau inequality $K^{2}_{F}\leq 9\chi({\mathcal{O}}_{F})$, and equality holds if and only if $F$ is a ball quotient. Surfaces of positive index satisfy the inequalities $8\chi({\mathcal{O}}_{F})< K^{2}_{F}{\bf < }\
 9\chi({\mathcal{O}}_{F})$. These surfaces are still quite mysterious objects [@Re], [@MT], [@My].

The statement of the Proposition above is close to the recent results of Urz[ú]{}a. He has found in [@Ur1] simply connected projective surfaces of general type with $\frac{K_{F}^{2}}{e(F)}$ arbitrarily close to $\frac{71}{26}$; this improves previous estimates by Persson-Peters-Xiao [@PPX]. By his method, based on a generalisation of the method of line arrangements on ${\mathbb{P}}^{2}$, he finds interesting surfaces of positive index; see: [@Ur2]. Thanks to the Kapranov construction of the moduli space ${\overline{M}}_{0,d+1}$ of rational curves with $d+1$ marked points, see [@Ka1], [@Ka2], line arrangements correspond to curves inside ${\overline{M}}_{0,d+1}$. In this sense Urz[ú]{}a construction is related to the Harris Morrison one. Hence the fact that our results on Harris Morrison construction of $k$-gonal sweeping families match those obtained in [@Ur2] is an evidence for the conjectural results stated in [@HM].

The authors would like to thank D. Chen for his comments on the first version of the paper.

This research is supported by MIUR funds, PRIN project [*Geometria delle varietà algebriche*]{} (2010), coordinator A. Verra. The first author is also supported by funds of the Università degli Studi di Trieste - Finanziamento di Ateneo per progetti di ricerca scientifica - FRA 2011.

 The basic construction.
========================

Harris-Morrison families.
-------------------------

We review and we explain some features of the basic construction of [@HMo]. We could construct a slight refinement of the basic construction of [@HMo Section 2] using not a product surface $X\times{\mathbb{P}}^{1}$ but a ruled surface ${\mathbb{P}}({\mathcal{E}})\to X$ over $X$, but we have decided to follow [@HMo Section 2] for simplicity.

We denote by $[N]$ and $N$ a finite set and its order respectively. If $[M]$ is a subset of $[N]$ and is invariant under the action of $\mathfrak{S}_k$ we will denote the quotient $[M]/\mathfrak{S}_k$ by $[\widetilde M]$.

Let ${\overline{M_{0, b}}}$ be the compactification of the moduli space of $b$-pointed stable curves of genus $0$ and let ${\overline\beta}\colon{\overline{H_{k, 
b}}}\to {\overline{M_{0, b}}}$ be the natural morphism on the Hurwitz scheme of admissible $k$-covers of stable $b$-pointed curves of genus $0$ constructed in [@HM]. Then the morphism $\beta\colon H_{k,b}\to{\mathcal{M}}_{0,b}$ is a $\widetilde N(k,b)$ sheeted unramified covering, where $\widetilde N(k,b)$ counts the $k$-sheeted [*[connected]{}*]{} covers of ${\mathbb{P}}^{1}$ simply branched at $b$ fixed points. From now on we set $N:=N(k,b)$, where $N(k,b)$ is defined in [@HMo page 334]. Concerning the subset $[N]$ we recall that $\mathfrak{S}_k$ acts on $[N]$ and by [@HMo Lemma 1.24] this action is free if $k\geq 3$ and trivial for $k=2$, so $N=k!\ \widetilde N$ if $k\geq 3$ and $N=\widetilde N$ if $k=2$.

Let $X$ be any smooth complete curve and let $\pi_{X}\colon X\times{\mathbb{P}}^{1}\to X$ be the projection. By [@Ha] ${\rm{NS}}(X\times{\mathbb{P}}^{1})=[s]\mathbb Z\oplus [f]\mathbb 
Z$ where $[s]$ is the numerical class of a $\pi_{X}$-section $s$ and $[f]$ is the numerical class of a $\pi_{X}$-fiber $f$. In the sequel we will not distinguish between $s$ and its class $[s]$ if no danger of confusion can arise.

Let $s$ be a $\pi_{X}$-section with $s^{2}=0$. Let $b\in \mathbb N$ and let $C_{1},\ldots, C_{b}$ be effective divisors on $X$ of degrees $c_i >0$, such that ${\mathcal{O}}_{X\times{\mathbb{P}}^{1}}(s+\pi_X ^\star C_i)$ is very ample, $i=1,\ldots 
,b$. Let $\sigma_{i}\in |s+\pi_X ^\star C_i|$ be a smooth curve for every $i=1,\ldots, b$, such that the sections $\sigma_{i}$ meet transversely everywhere and such that each $\pi_{X}$-fiber contains at most one of these intersections.

We set $$\sigma:=\sum \sigma_{i}$$ and we point out that $[\sigma]= b[s]+[\pi_X ^\star C]=b[s]+c[f]$ where $C=\sum C_{i}$ and $c=\sum c_{i}$. The reader is warned that if $X={\mathbb{P}}^{1}$ the above costruction easily works. If $X$ is a curve of positive genus then it is sufficient to assume that for every $i=1,\ldots, b$ the number $c_{i}$ is at least equal to the degree of a very ample divisor on $X$.

Let $[I_{X}]$ be the set of nodes of $\sigma$ or equivalently the set of intersections of the $\sigma_{i}$’s. Then $$I_{X}=(b-1)c;$$ see: [@HMo Formula 2.1 on page 338].

Let $B_{X}$ be the blow-up of $X\times{\mathbb{P}}^{1}$ at $[I_{X}]$ and denote by $B_{x}$ its fiber over $x\in X$ and $\widetilde \sigma_{i}$ the strict transform of $\sigma_{i}$ on $B_{X}$. Let $\alpha\colon X\to{\overline{{\mathcal{M}}_{0,b}}}$ be the map which sends a point $x\in X$ to the class of the fiber $B_{x}$ marked by its $b$ points of intersections with the $\widetilde\sigma_{i}$ where $i=1,\ldots, b$. Let $Y:=X\times_{{\overline{{\mathcal{M}}_{0,b}}}}{\overline{H_{k, 
b}}}$. By the same argument of [@HMo pages 338-339] the induced map $\mu\colon Y\to X$ is a covering of degree $\widetilde N$. Let $\pi_{Y}\colon Y\times{\mathbb{P}}^{1}\to Y$ be natural projection. If we set $\nu:=\mu\times{\rm{id}}\colon Y\times{\mathbb{P}}^{1}\to 
X\times{\mathbb{P}}^{1}$ we see that $[I_{Y}]:=\nu^{\star}([I_{X}])$ is the scheme of singularities of the divisor $$\tau:=\nu^{\star}\sigma$$ on $Y\times{\mathbb{P}}^{1}$. Let $[J_{Y}]\subset
Y$ be the $\pi_{Y}$-projection of $[I_{Y}]$ and let $[J_{X}]\subset 
X$ be the $\pi_{X}$-projection of $[I_{X}]$. By construction we have a morphism from the complement of $[J_{Y}]$ to ${\overline{{\mathcal{M}}_{g}}}$, and we extend it to a morphism $\rho\colon Y\to {\overline{{\mathcal{M}}_{g}}}$ following the argument of [@HMo Theorem 2.15].

Here we recall first that $\mu\colon Y\setminus [J_{Y}]\to X\setminus [J_{X}]$ is unramified and that $[J_{Y}]$ is partitioned into three classes: $$[J_{Y}]=[J_{Y, (1)}]\cup [J_{Y, (2,2)}]\cup 
   [J_{Y, (3)}].$$ The topological description given in [@HMo Page 340] yields that $\mu$ is not ramified on $[J_{Y, (1)}]\cup [J_{Y, (2,2)}]$ and it is triply branched at points of $[J_{Y, (3)}]$. In particular $$I_{Y}=\left({\widetilde{N_{1}}}+{\widetilde{N_{2,2}}}+
   \frac{{\widetilde{N_{3}}}}{3}\right)
   (b-1)c;$$ where ${\widetilde N}={\widetilde{N_{1}}} 
 +{\widetilde{N_{ 2,2}}}+{\widetilde{N_{3}}}$ and a combinatorical definition of ${\widetilde{N_{1}}}, {\widetilde{N_{2,2}}}, {\widetilde{N_{3}}}$ is given in [@HMo Proposition 2.9].

Following [@HMo] we blow-up $\epsilon\colon A\to Y\times{\mathbb{P}}^{1}$ at the points of the set $[I_{Y,(1)}]$ and we construct a finite cover $\pi\colon G\to A$ and a semistable fibration $f\colon F\to Y$ see [@HMo Diagram (2.2), page 338] such that over the complement $Y'$ of $[J_{Y}]$ in $Y$ the surface $G$ is obtainable by the pull-back of the smooth universal admissible cover $\theta\colon{\mathcal{U}}_{{\mathcal{H}}_{k,b}}\subset{\mathbb{P}}^{1}\times 
   {\mathcal{H}}_{k,b}\to{\mathcal{H}}_{k,b}$ whose existence is shown in [@HM]. We recall that over $Y'$, $G$ is a stable fibration $G_{Y'}\to 
   Y'\subset Y$, it coincides with $f\colon F\setminus f^{-1}([J_{Y}])=F_{Y'}\to Y'$. Moreover $A$ coincides with $Y\times{\mathbb{P}}^{1}$ over $Y'$ and there exists a [*[finite]{}*]{} morphism $G_{Y'}=F_{Y'}\to 
   A_{Y'}={\mathbb{P}}({\mathcal{O}}_{Y'}\oplus{\mathcal{O}}_{Y'})$.

By the very explicit local analytic description of the [*[finite]{}*]{} cover $\pi\colon G\to A$ and of the blow-up $\epsilon\colon A\to Y\times{\mathbb{P}}^{1}$ done in [@HMo from page 343 to the top of page 347] we obtain an explicit description of the smooth surface $F$ and of the semistable fibration $f\colon 
   F\to Y$ inducing the desired moduli map $\rho\colon Y\to{\overline{{\mathcal{M}}_{g}}}$. We set $Z:=\rho(Y)$. We sum up the above construction in the following diagram:

$$\label{diag}
\xymatrix{
 S \ar[dr]^h \ar@{<-}[d]^u &  &&\\
G  \ar[r]^\zeta \ar[d]_\pi^{k:1}  & F\ar[dr]^f \ar@{-->}[d]&& \qquad Z \subset {{\overline {\mathcal  M}}_{g}}\ar@{<-}[dl]^\rho\\
 A \ar[r]^{\epsilon } & Y\times{\mathbb{P}}^{1}\ar[r]^{\pi_Y} \ar[d] _\nu^{\widetilde N:1} & Y \ar[d]_\mu ^{\widetilde N:1}\ar[r]& \overline{H}_{k,b}\ar[d]^{\overline\beta}\\
  &  X\times{\mathbb{P}}^{1} \ar[r]^{\pi_X} & X \ar[r]^\alpha &\overline {{\mathcal{M}}_{0,b}}\\
}$$

What makes this families very interesting is that thanks to the work done in [@HMo] the dominant morphism $\zeta\colon G\to F$ is very explicit and hence the nature of the fibers of $f\colon F\to Y$ over $[J_{Y}]$ is evident.

If $y\in Y$ we denote by $F_{y}$ and $G_{y}$ respectively the $f$-fiber over $y$ and the $(f\circ \zeta)$-fiber over $y$.

The analysis of fibers of type $(1)$, that is $y\in [J_{Y, 
    (1)}]$, splits into four cases: $(1_{0})$, $(1_{j, 0})$ $(1_{j, g})$, $(1_{j, i})$ where $j>0$ and $1\leq i\leq 
    g/2$; see [@HMo Diagram (2.12)] and with obvious notation we write $y\in [J_{Y,\Pi}]$ if $y\in [J_{Y, 
    (1)}]$ and $y$ is of type $\Pi$.

If $y\in [J_{Y, (1_{0})}]$ then $F_{y}=F'_{y}\cup E$ where $F'_{y}$ is a genus $g-1$ curve and $E$ is a rational $(-2)$ curve; see the top of diagram [@HMo Diagram (2.12)]. The corresponding fiber $G_{y}$ on $G$ is $$G_{y}=G''_{y}\cup E'\cup\bigcup_{j=1}^{k-2}E_{j,y}$$ where $G''_{y}$ is a genus $g-1$ curve, $E_{j,y}^{2}=-1$ and $E'$ is a -$2$ curve. Over a neighbouhood of $y$ the morphism $\zeta\colon G\to F$ consists on the contraction of the $k-2$ rational curves $E_{j,y}$.

If $y\in [J_{Y, (1_{j, 0})}]$ or $y\in [J_{Y, 
    (1_{j, g})}]$ then $F_{y}$ is a [*[smooth]{}*]{} semistable fiber and over a neighbouhood of $y$ the morphism $\zeta\colon G\to F$ consists on a [*[ordered contraction]{}*]{} of $k$ rational curves. More precisely if $y\in [J_{Y, (1_{j, 0})}]\cup [J_{Y, 
    (1_{j, g})}]$ then $$G_{y}=G'_{y}\cup E'_{y}\cup 
    G''_{y}\cup \bigcup_{i=1}^{k-2}E_{i,y}$$ where $G'_{y}$ is a rational curve, $E'_{y}$ is a $(-2)$-rational curve, $E'_{y}\cdot G'_{y}=1$, $G''_{y}$ is a smooth curve of genus $g$, $E'_{y}\cdot G''_{y}=1$, $G''_{y}$ is a smooth curve of genus $g$ and $E_{i,y}^{2}=-1$. The map $\zeta$ first contracts $E_{i,y}$ where $i=1,\ldots ,k-2$. After this contraction process the image of $G'_{y}$ is a $-1$ curve which intersects the image of $E'_{y}$ in a unique point. Hence to get the semistable reduction $F$ we need [*[first]{}*]{} to contract the image of $G'_{y}$ and then the image of $E'_{y}$.

If $y\in [J_{Y, (1_{j, i})}]$ where $j>0$ and $1\leq i\leq [g/2]$ then $F_{y}=F_{y,i}\cup E_{y}\cup F_{y,g-i}$ where $F_{y,i}\cap F_{y,g-i}=\emptyset$, $E_{y}^{2}=-2$, $F_{y,i}\cdot E_{y}= F_{y,g-i}\cdot E_{y}=1$ and $$G_{y}= 
    G'_{y}\cup E'_{y}\cup G''_{y}\cup\bigcup_{i=1}^{k-2}E_{i,y},$$ where $E_{i,y}$ are $-1$-rational curves, $E_{y}$ is a $(-2)$ curve. Moreover $F_{y,i}$ is smooth of genus $i$ and admits a $j$ covering over the fiber ${\mathbb{P}}^{1}_{y}\subset Y\times{\mathbb{P}}^{1}$ and $F_{y,g-i}$ is smooth of genus $g-i$ and admits a $k-j$ covering over ${\mathbb{P}}^{1}_{y}$. Note that if $e_{y}\subset A$ is the exceptional curve over $y$ arised in the blow-up $A\to Y\times{\mathbb{P}}^{1}$ then the restriction of $\pi\colon G\to A$ to $E'_{y}\cup\left(\bigcup_{i=1}^{k-2}E_{i,y}\right)$ gives the $k$-cover of $e_{y}$ contained in $G$. Over a neighbouhood of $y$ the morphism $\zeta\colon G\to F$ consists on the contraction of the $k-2$ rational curves $E_{i,y}$.

\[tipodueduetre\] If $y\in [J_{Y, (2,2)}]\cup [J_{Y, (3)}]$ then the fiber $F_{y}$ of $f\colon F\to Y$ over $y$ is a smooth semistable curve. The rational map $F\dashrightarrow Y\times{\mathbb{P}}^{1}$ is finite in a neighbourhood of $F_{y}$. Moreover the surfaces $G$ and $F$ coincide over an analytic neighboorhood of $y\in Y$.

See [@HMo Pages 343-346].

\[daGaS\] Over each of the points of $[I_{X}]=\pi_{X}([J_{X}])\subset X$ there are ${\widetilde{N}}_{{\rm{sing}}}$ points $y\in Y$ such that the fiber $F_{y}\subset F$ is singular. Any singular fiber of $f\colon 
        F\to Y$ contains a unique $(-2)$ rational curve.

This is a restatement of [@HMo Theorem 3.1] and it easily follows by the above description of the singularities of the fibration $f\colon F\to Y$.

A semistable fibration $f\colon F\to Y$ as the one of Proposition \[daGaS\] is called a Harris-Morrison family.

Note that by Harris Morrison construction it follows

\[harris\] Let $X$ be any smooth complete curve of genus $g(X)$. Let $g\geq 3$ be any natural number. If $g$ is odd let $k=\frac{g+3}{2}$ and set $k=\frac{g+2}{2}$ if $g$ is even. Then the family of curves $Z\subset {\overline {\mathcal M}_g}$ constructed by Harris and Morrison varying the curve $X$ forms a sweeping family.

The chosen number $k$ is the maximal gonality for a curve of genus $g$. Hence the claim follows by a standard result of Brill-Noether theory. See also [@HMo page 350].

Notice that, even without the assumption of maximal gonality, the families of curves $Z\subset {\overline {\mathcal M}_g}$ constructed by Harris and Morrison varying the curve $X$ depend freely on $g(X)$. In particular we will study those families such that the parameter $g(X)>>0$.

Numerical invariants.
---------------------

We recall that $\epsilon\colon A\to Y\times{\mathbb{P}}^{1}$ is the blow-up of $[I_{Y, (1)}]$ which is a reduced scheme of length $(b-1)c{\widetilde{N_{1}}}$. Let $E_{A}$ be the exceptional divisor of $\epsilon\colon A\to Y\times{\mathbb{P}}^{1}$. Hence

$$\label{autointe}
    E_{A}^{2}=-I_{Y, (1)}=-(b-1)c{\widetilde{N_{1}}}.$$

Let $\widetilde\tau$ be the strict transform of $\tau$, that is $\widetilde\tau=\epsilon_{A}^{\star}(\tau)-2E_{A}$. Since $\tau=\nu^{\star}(\sigma)=\nu^{\star}(bs+cf)$ then

$$\label{tautildequadro}
    (\widetilde\tau)^{2}=2bc{\widetilde{N}}-4(b-1)c{\widetilde{N_{1}}}.$$

Let $R$ be the ramification divisor of $\pi\colon G\to A$. The following relation holds:

$$\label{pistarR}
    \pi_{\star}R=\widetilde\tau.$$

Since we have simple branch on the fibers the local analysis shows that $$\label{pistabassopistaralto}
    \pi^{\star}\pi_{\star}R=2R+\widetilde R$$

and that:

$$\label{erreerretilde}
R\cdot\widetilde R=(b-1)c\left(2{\widetilde{N}}_{2,2}+\frac{4}{3}{\widetilde{N}}_{3}\right).$$

\[erreerre\] The ramification divisor $R$ of the finite cover $\pi\colon G\to A$ satisfies:

$$R^{2}=\left({\widetilde N}+(b-1)\left(\frac{1}{3}{\widetilde N_{3}}-
    {\widetilde N_{1}}\right)\right)c$$

It is as in [@HMo Theorem 2.15].

By construction we have seen that over any point $y\in [J_{Y,(1)}]$ there exist $k-2$ rational curves $E_{i,y}\subset G_{y}$ such that $E_{i,y}^{2}=-1$. In particular there are $(k-2){\widetilde{N_{1}}}(b-1)c$ exceptional curves contained inside fibers of $f_{G}\colon G\to Y$ over $J_{Y,(1)}$ .

\[laesse\] Let $u\colon G\to S$ be the contraction of these $(k-2){\widetilde{N_{1}}}(b-1)c$ rational curves. Then $S$ is a smooth surface. Moreover $K_{S}^{2}=K_{G}^{2}+ (k-2){\widetilde{N_{1}}}(b-1)c$.

It follows straightly from Castelnuovo contraction theorem.

Now we want to compute the invariants of $S$.

\[eccez=0\] Let $E$ be any $-1$ rational curve which is contracted by $u\colon G\to S$. Then $R\cdot E=0$.

We use the ramification divisor formula: $K_{G|Y}=\pi^{\star}K_{A|Y}+R$. Let $E$ be any $-1$ rational curve which is contracted by $u\colon G\to S$ then the finite morphism $\pi\colon G\to A$ restricts to an isomorphism $\pi_{|E}\colon 
    E\to L\subset A$ where $L$ is an exceptional curve produced via the blow-up $\epsilon\colon A\to Y\times{\mathbb{P}}^{1}$. In particular $\pi_{\star}E=L$. Then $R\cdot 
     E=-1-\pi^{\star}(K_{A|Y})E=-1-K_{A|Y}\cdot L=-1+1=0$.

\[erretildeconmenouno\] Let $E$ be any $-1$ rational curve which is contracted by $u\colon G\to S$. Then $E\cdot{\widetilde R}=2$.

By Equation \[pistabassopistaralto\] and by Proposition \[eccez=0\] we have: $0=E\cdot 
    R=E\cdot\frac{1}{2}(\pi^{\star}\pi_{\star}R-\widetilde R)=
    \frac{1}{2}(L\cdot\widetilde\tau-E\cdot\widetilde R)$. Since $\widetilde\tau=\epsilon^{\star}(\tau)-2E_{A}$ and $E_{A}\cdot 
    L=-1$ then the claim follows.

We now consider the $(-2)$ rational curves of $G$. We notice that if $y\in[J_{Y,{1_{(2,0)}}}]\cup [J_{Y,{1_{(2k-2,g)}}}]$ then there is a $(-2)$ rational curve $G'_{y}\subset G_{y}$ which is mapped to a fiber of the canonical projection $\pi_{Y}\colon Y\times{\mathbb{P}}^{1}\to Y$; that is: $\epsilon\circ\pi(G'_{y})=\pi_{Y}^{-1}(y)\subset Y\times{\mathbb{P}}^{1}$. Next lemma deals with the $(-2)$ curves which are mapped to the exceptional curves of the morphism $\epsilon\colon A\to 
    Y\times{\mathbb{P}}^{1}$.

\[erretildeconmenodue\] Let $E$ be any $(-2)$ rational curve which is contained in a fiber of $f_{G}\colon G\to Y$ and which is mapped $2$-to-$1$ to a cuve $L\subset A$ such that $\epsilon(L)\in [I_{Y}]$. Then $R\cdot E=2$. Moreover $E\cdot\widetilde R=0$

If $E$ is a $(-2)$ curve contained in a fiber of $f_{G}\colon G\to Y$ which is mapped to an exceptional curve $L$ of the morphism $\epsilon\colon A\to Y\times{\mathbb{P}}^{1}$ then $\pi_{|_E}\colon E\to L$ is a $2$-to-$1$ covering. Hence $\pi_{\star}E=2L$. Then $0=E\cdot K_{G|Y}=E\cdot 
    (\pi^{\star}K_{A|Y}+R)=2L^{2}+E\cdot R$. This implies $R\cdot 
    E=2$. Then $E\cdot{\widetilde{R}}=E\cdot\pi^{\star}\pi_{\star}R-4=
    2L\cdot\widetilde\tau-4=0$.

By the local description of $f\colon S\to Y$ we know that there are $(b-1)c\left[{\widetilde{N}}_{{\rm{sing}}}
+\sum_{j=0}^{[\frac{k}{2}]}({\widetilde{M_{j,0}}}+{\widetilde{M_{j,g}}})\right]$ mutually disjoint $(-2)$ rational curves, where the $[M_{j,i}]$ are given in [@HMo Formula 1.35].

\[combinatorics\] $\sum_{j=0}^{[\frac{k}{2}]}({\widetilde{M_{j,0}}}
    +{\widetilde{M_{j,g}}})\leq {\widetilde{N_{1}}}$.

By [@HMo Formula 1.30] $[N_{1}]=\cup_{i=0}^{[\frac{k}{2}]}[M_{j}]$ where $[M_{j}]$ is defined in [@HMo Formula 1.28]. Since by [@HMo Formula 1.36] $[M_{j}]=\cup_{i=1}^{g}[M_{j,i}]$ the claim follows.

Moreover let $$\label{discrepanza}
    e:=\sum_{j=0}^{[\frac{k}{2}]}(\widetilde{M_{j,0}}+{\widetilde{M_{j,g}}}).$$ Obviously $(b-1)ce=\sum _{j=0}^{[\frac{k}{2}]}(J_{Y,1_{j,0}}+J_{Y,1_{j,g}})$. We set $$\bigcup _{j=0}^{[\frac{k}{2}]}([J_{Y,1_{(j,0)}}]\cup [J_{Y,1_{(j,g)}}])=:\{a_{1},\ldots 
,a_{(b-1)ce}\}\subset Y.$$ We recall that the morphism $h\colon S\to F$ contracts two rational curves of any fiber of $f_{S}\colon S\to Y$ over $\{a_{1},\ldots 
,a_{(b-1)ce}\}\subset Y$. Moreover the fiber $F_{a_{i}}$ $i=1,\ldots 
,(b-1)ce$ of $f\colon F\to Y $is smooth, but over a neighbourhood $U_{i}$ of $a_{i}\in Y$ the map $\rho_{F}\colon F_{U}\dashrightarrow U\times{\mathbb{P}}^{1}$ is [*[only a rational one]{}*]{}. We know that the fiber $f_{S}^{\star}(a_{i}):=S_{a_{i}}=
F'_{a_{i}}+E_{a_{i}}+E'_{a_{i}}\subset S$ where $F'_{a_{i}}$ is a smooth curve of genus $g$, $E_{a_{i}}$ is a $(-2)$ rational curve such that $E_{a_{i}}\cdot F'_{a_{i}}=1$. Moreover we know that $E'_{a_{i}}$ is a $-1$-rational curve and $E_{a_{i}}\cdot E'_{a_{i}}=1$.

\[calcolodiscrepanza\] Let $h\colon S\to F$ be the morphism which factorises the morphism $\zeta\colon G\to F$. Then $K_{F}^{2}=K_{S}^{2}+2(b-1)ce$.

By Harris Morrison construction the contraction of $k-2$ rational curves in each one of the $(b-1)c{\widetilde N}_{1}$ singular fibers of $f_{G}\colon G\to Y$ gives the morphism $u\colon G\to S$. To reach $F$ we need to contract first $E'_{a_{i}}$ and then the image of $E_{a_{i}}$ for every $i=1,\ldots, (b-1)ce$. Hence $\zeta\colon G\to F$ is given by $h\zeta=h\circ u$ where $h\colon S\to F$ is a composition of $2(b-1)ce$ simple contractions. Hence $K_{F}^{2}=K_{S}^{2}+2(b-1)ce$.

By Lemma \[tipodueduetre\] the points of type $(2,2)$ and the points of type $(3)$ give a smooth semistable fiber of $f\colon F\to Y$. By Proposition \[daGaS\] there are ${\widetilde{N}}_{{\rm{sing}}}(b-1)c$ singular fibers of $f\colon F\to 
    Y$. Set $$r:={\widetilde{N}}_{{\rm{sing}}}(b-1)c.$$ Let $\{y_{i}\}_{i=1}^{r}={\rm{Sing}}(f)$ be the subset of $Y$ given by the points $y\in Y$ such that $f^{-1}(y)=F_{y}$ is a singular fiber. By the local description of the singular fibers of $f\colon F\to 
        Y$ it follows that for any $l=1,\ldots, r$, $f^{-1}(y_{l})=F_{y_{l}}=F_{y_{l},i}\cup E_{y_{l}}\cup 
        F_{y_{l},g-i}$ where $E_{y_{l}}$ is a $(-2)$ curve, $F_{y_{l},i}\cdot E_{y_{l}}=
        F_{y_{l},g-i}\cdot E_{y_{l}}=1$ and obviously $F_{y_{l},i}\cdot
        F_{y_{l},g-i}=0$, or $F_{y_{l}}=F'_{y_{l}}\cup E$ where $F'_{y_{l}}$ is a genus $g-1$ curve and $E$ is a rational $(-2)$ curve.

We study the invariants of the surface $F$.

\[eulerodif\]Let $f\colon F\to Y$ be a Harris Morrison family. Then the topological Euler characteristic of $F$ is: $$e(F)=4(g-1)(g(Y)-1)+2r.$$

By a standard topological argument it follows that $e(F)=4(g-1)(g(Y)-1)+\sum_{i=1}^{r} (e(F_{y_{i}})-(2-2g))$. By the analysis of singular fibers, see Proposition \[daGaS\], it follows that for every $i=1,\ldots, r$ $e(F_{y_{i}})-(2-2g)=2$. Then $e(F)=4(g-1)(g(Y)-1)+2r$.

We point out the reader the very important relation: $$\label{tuttoli}
{\widetilde{N}}_{1}=e+{\widetilde{N}}_{{\rm{sing}}}$$where $e$ is defined in Equation \[discrepanza\].

\[kappaquadroeffe\] Let $f\colon F\to Y$ be an Harris Morrison semistable fibration. Then $$K^{2}_{F}=c\left[(b-1)\left[2e 
    +{\widetilde{N}}_{1}+(8g-7)\frac{{\widetilde{N}}_{3}}{3}
    \right]-3{\widetilde{N}}\right] +8{\widetilde{N}}(g(X)-1)(g-1)$$

We notice that by Corollary \[calcolodiscrepanza\] $K^{2}_{F}=2e(b-1)c+K^{2}_{S}$. Then by Lemma \[laesse\] $K^{2}_{F}=2e(b-1)c+(k-2)(b-1)c{\widetilde{N}}_{1}+K^{2}_{G}$. By definition of $R$ we have $K_{G}^{2}=R^{2}+2R\cdot\pi^{\star}K_{A}+(\pi^{\star}K_{A})^{2}$. By Equation \[autointe\] we have $$(\pi^{\star}K_{A})^{2}=kK^{2}_{A}=-k\left[8(g(Y)-1)+(b-1)c{\widetilde{N}}_{1}\right].$$ By projection formula and by Equation \[pistarR\] we have: $$2R\cdot\pi^{\star}K_{A}=2{\widetilde{\tau}}K_{A}=
   \left[ 2b(g(X)-1)-2c\right]{\widetilde{N}}+b(b-1)c\frac{2{\widetilde{N}}_{3}}{3}.$$ Now the claim follows by Proposition \[erreerre\] taking into account that $b=2g+2k-2$ and $$\label{generedi Y}
     2g(Y)-2={\widetilde{N}}(2g(X)-2)+\frac{2}{3}(b-1)c{\widetilde{N}}_{3}.$$ since Riemann-Hurwitz formula.

We put $$\label{alpha}
    \alpha:=(b-1)\left[2e 
    +{\widetilde{N}}_{1}+(8g-7)\frac{{\widetilde{N}}_{3}}{3}
    \right]-3{\widetilde{N}}$$

\[notdependence\] For every fibration constructed as in [@HMo Theorem 2.5] we can write by Proposition \[kappaquadroeffe\] and by Equation \[alpha\]: $$\label{facile}
    K^{2}_{F}=c\alpha+8{\widetilde{N}}(g(X)-1)(g-1)$$ where the coefficient $\alpha$ does not depend on $g(X)$.

\[kappaquadroeffehaarris\] If $f\colon F\to Y$ is the fibration constructed in [@HMo Theorem 2.5] then $$\chi({\mathcal{O}}_{F})=\frac{c}{12}\left[(b-1)\left[ 
    +3{\widetilde{N}}_{1}+(12g-11)\frac{{\widetilde{N}}_{3}}{3}
    \right]-3{\widetilde{N}}\right] 
    +{\widetilde{N}}(g(X)-1)(g-1) .$$

By Noether Identity and by Proposition \[eulerodif\] we have $$12\chi({\mathcal{O}}_{F})=K^{2}_{F}+4(g-1)(g(Y)-1) 
     +2{\widetilde{N}}_{{\rm{sing}}}(b-1)c.$$ By Proposition \[kappaquadroeffe\] we can write $$12\chi({\mathcal{O}}_{F})=c\left[(b-1)\left(2e 
    +{\widetilde{N}}_{1}+(8g-7)\frac{{\widetilde{N}}_{3}}{3}
    \right)-3{\widetilde{N}}\right] +$$ $$\qquad \qquad \qquad \qquad +8{\widetilde{N}}(g(X)-1)(g-1)+4(g-1)(g(Y)-1) 
     +2{\widetilde{N}}_{{\rm{sing}}}(b-1)c.$$ By Equation \[tuttoli\] and by Equation \[generedi Y\] we next obtain: $$12\chi({\mathcal{O}}_{F})=c\left[(b-1)\left( 
    3{\widetilde{N}}_{1}+(12g-11)\frac{{\widetilde{N}}_{3}}{3}
    \right)-3{\widetilde{N}}\right] 
    +12{\widetilde{N}}(g(X)-1)(g-1)$$ and the claim follows.

We put $$\label{alphaprimo}
 \alpha':=\frac{1}{12}\left[(b-1)\left(
    3{\widetilde{N}}_{1}+(12g-11)\frac{{\widetilde{N}}_{3}}{3}
    \right)-3{\widetilde{N}}\right].$$ In particular we have:

\[notdependencebis\] For every fibration constructed as in [@HMo Theorem 2.5] we can write by Proposition \[kappaquadroeffehaarris\] and by Equation \[alphaprimo\]:

$$\label{facilefacile}
\chi({\mathcal{O}}_{F})=\alpha'c+{\widetilde{N}}  (g(X)-1)(g-1).$$

where the coefficient $\alpha'$ does not depend on $g(X)$.

\[primadifferenza\] Let $X$ be any smooth complete curve with genus $g(X)$. Let $f\colon F\to Y$ be the fibration constructed in [@HMo Theorem 2.5] then $$K^{2}_{F}-8\chi({\mathcal{O}}_{F})=c\left[(b-1)
     \left[2e-{\widetilde{N}}_{1}+\frac{{\widetilde{N}}_{3}}{9}\right]
     -{\widetilde{N}}\right].$$

It follows by Corollary \[kappaquadroeffehaarris\] and by Proposition \[kappaquadroeffe\].

The proof of the Theorem
------------------------

Now we want to analyse the relation between the fiber degrees $c_{i}$ for the divisors $\sigma_{i}\in |s+\pi_X ^\star C_i|$ and the genus $g(X)$ in order that a family $f\colon F\to Y$ as in [@HMo Theorem 2.5] exists if we start from a curve $X$. Since we are working over a product surface $X\times{\mathbb{P}}^{1}$ a divisor $L$ which is numerically equivalent to $s+c_{i}f$ is very ample iff $L_{|(s=0)}$ is very ample; hence a necessary condition is that $c_{i}$ is the degree of a very ample divisor on $X$. On the other hand if ${\mathcal{O}}_{X}(l)$ is a very ample sheaf on $X$ then $s+\pi_{X}^{\star}(l)$ is a very ample divisor on $X\times{\mathbb{P}}^{1}$. For a general $X$ if ${\mathcal{O}}_{X}(l)$ is a nonspecial very ample divisor then by Halphen theorem [@Ha Proposition 6.1] it follows that $c_{i}\geq g(X)+3$. Then if we want to construct families $f\colon F\to Y$ as in [@HMo Theorem 2.5] starting from a curve $X$ where $c$ is small with respect to $g(X)$, definitely we need to consider curves $X$ with very ample special divisors.

\[secondaadifferenza\] Let $g,k\in\mathbb N$ with $3\le k \le \lfloor {(g+3)\over 2} \rfloor$. For every real number $\epsilon >0$, there exists a real number $\Delta(\epsilon)\geq 0$ such that there are families $f\colon F\to Y$ obtained by the Harris Morrison construction starting from any plane curve $X$ of genus $g(X)\geq \Delta(\epsilon)$ such that the following holds: $$8-\epsilon\leq \frac{K^{2}_{F}}{\chi({\mathcal{O}}_{F})}\leq 
       8+\epsilon.$$

Since $X$ is a plane curve of genus $g(X)$ then by Clebsh formula its degree is $d(X)=\frac{3+\sqrt{8g(X)+1}}{2}$. We can consider $c_{i}=d(X)$ where $i=1,\ldots, b$ where $b=2(g+k-1)$. Then $c$ can be taken equal to $bd(X)$. For every Harris-Morrison family we can write:

$$\label{facilefacilefacile}
      \frac{K^{2}_{F}}{\chi({\mathcal{O}}_{F})}=
      \frac{c\alpha+8{\widetilde{N}}(g(X)-1)(g-1)}{c\alpha'+{\widetilde{N}}
      (g(X)-1)(g-1)}.$$

Since the parameters $\alpha$, $\alpha'$ [*[do not depend]{}*]{} on $g(X)$ since $g,k$ are fixed and since $d(X)=\frac{3+\sqrt{8g(X)+1}}{2}$ then we can find $\frac{2g(X)-2}{3+\sqrt{8g(X)+1}}\geq  
\frac{b||\alpha-8\alpha'|-\epsilon\alpha'|}{\epsilon(g-1)}$ to obtain $|\frac{K^{2}_{F}}{\chi({\mathcal{O}}_{F})}-8|\le \epsilon$.

\[pianodifferenza\] Theorem \[secondaadifferenza\] can be extended easily to subcanonical curves $X$ where the subcanonical degree is sufficiently small with respect to $g(X)$.

We point out the reader that the invariants of surfaces of general type which supports fibrations as those of Proposition \[pianodifferenza\] are strongly influenced by the base $Y$ of the fibration, in a way which is quite new for the theory of surfaces of general type, as far as we know.

We have shown Theorem stated in the Introduction.

Maximal gonality and surfaces of positive index.
================================================

In [@HMo] the genus $g(X)$ plays no role, see: [@HMo Corollary 3.15]. In this work it plays an essential role due to the Equations \[facile\] and Equation \[facilefacile\].

We consider the expressions of $K^{2}_{F}$ and $\chi({\mathcal{O}}_{F})$ as (linear) polynomials in the variable $g(X)$. In this section we consider the Harris Morrison families obtained in [@HM] and with [*[maximal]{}*]{} gonality.

In particular in this section we have:

$$\label{casodisparo}
    g=2n+1, k=n+2, b=6n+4, m=(6n+3)c$$

or

$$\label{casoparo}
    g=2n, k=n+1, b=6n, m=(6n-1)c$$

Let us see how $k$ influences $K^{2}_{F}$ and $8\chi({\mathcal{O}}_{F})$. We will use the following

\[tecnico\] Let $f\colon F\to Y$ be an Harris Morrison genus-$g$ fibration as in [@HMo Theorem 2.5]. Assume that the gonality $k$ is maximal, that is $k=\frac{g+3}{2}$ if $g$ is odd and $k=\frac{g+2}{2}$ if $g\geq 4$ is even. Assume that the conjectured estimate in [@HMo p. 351-352] is true. Then if $g>>0$, we have $\alpha> 8\alpha'$.

By Equation \[facile\], by Equation \[facilefacile\] and by Corollary \[primadifferenza\] we have $$\alpha-8\alpha'=(b-1)\left[2e-{\widetilde{N}}_{1}+
      \frac{{\widetilde{N}}_{3}}{9}\right]-{\widetilde{N}}.$$ Since $e\geq 0$ and $b=2(g+k-1)>0$ it is sufficient to show that $(b-1)\left[ 
-{\widetilde{N}}_{1}+\frac{{\widetilde{N}}_{3}}{9}\right]-{\widetilde{N}}\geq 0$. In the case of maximal gonality up to the first factor by [@HM bottom of the page 351] if $k>>0$ it is conjectured that ${\widetilde{N}}_{3}\simeq (k-2){\widetilde{N}}_{1}$, ${\widetilde{N}}\simeq (k)(k-1)\frac {{\widetilde{N}}_{1}} {2}$. Finally assume that $g=2n+1$. By Equation \[casodisparo\] we have $b-1=6n+3$ then up to the first order $(b-1)\left[ 
-{\widetilde{N}}_{1}+\frac{{\widetilde{N}}_{3}}{9}\right]-{\widetilde{N}}\simeq 
{\widetilde{N}}_{1}\left[ 
(6n+3)\left[ -1+\frac{n}{9}\right]-(n+2)(n+1)\frac{1}{2}\right]=
{\widetilde{N}}_{1}\frac{n^{2}-43n-18}{6}>0$ if $n\geq 44$, that is $k\ge 46$. By the same argument if $g=2n$ we have that $\alpha\geq 8\alpha'$ if $3n^{2}-131n+20\geq 0$ that is if $n\geq 43$ and then $k\ge 44$.

\[secondo\] Let $f\colon F\to Y$ be an Harris Morrison genus-$g$ fibration starting from any plane curve $X$ of genus $g(X)>>0$. Assume that the gonality $k$ is maximal and that the conjectured estimates in [@HMo p. 351-352] are true. If $g$ is big enough, then $F$ is a surface of positive index i.e. $K^{2}_{F}> 8\chi({\mathcal{O}}_{F})$. Moreover if $f\colon F\to Y$ is general in its class then the irregularity of $F$ is $g(Y)$. In particular, $f\colon F\to Y$ is the Albanese morphism of $F$.

For every Harris Morrison family as in the statment:

$$\label{nonfacilefacilefacile}
      \frac{K^{2}_{F}}{\chi({\mathcal{O}}_{F})}=
      \frac{c\alpha+8{\widetilde{N}}(g-1)(g(X)-1)}{c\alpha'+{\widetilde{N}}
      (g-1)(g(X)-1)}$$

since Equation \[facile\] and Equation \[facilefacile\]. Then the first claim is equivalent to show that $\alpha\geq 8\alpha'$ and under our assumption this follows by Lemma \[tecnico\]. Let $q(F)$ be the irregularity of $Y$. By contradiction assume that $q(Y)>g(Y)$. Then by universal property of the Albanese morphism it follows that the Jacobian of the general fiber of $f\colon F\to Y$ contains an Abelian subvariety; but the locus of such curves in $\overline{\mathcal{M}}_{g}$ is a proper closed. Hence the family is not a sweeping one: a contradiction to Proposition \[harris\].

We have shown the Proposition stated in the Introduction.

We conclude by observing that if the conjectured estimates of Harris Morrison hold, then the families $F$ of Proposition \[secondo\] furnish an intriguing example of surfaces with ratio $\frac{K^{2}_{F}}{\chi({\mathcal{O}}_{F})}$ asymptotically $8$, which are minimal as semistable models, but with slope of the supported fibration aymptotically equal to $12$. We think that this kind of divergence between the two fundamental ratios among the invariants of a fibered surface which is minimal as a semistable model is worthy to be studied in the light of the recent results of Urz[ú]{}a quoted in the Introduction of this paper.

[Muk04]{}

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