The Hesse curve of a Lefschtz pencil of plane curves

TL;DR

This paper introduces the Hesse curve associated with a generic Lefschetz pencil of plane curves, detailing its degree, genus, singularities, and its relation to inflection points and base points of the pencil.

Contribution

It establishes the existence and properties of the Hesse curve for a Lefschetz pencil of plane curves of degree d ≥ 3, including its degree, genus, and singularity structure.

Findings

Hesse curve has degree 6(d-1) and specific genus.

H has d^2 singular points of multiplicity three at base points.

H intersects each pencil member at inflection points and base points.

Abstract

We prove that for a generic Lefschetz pencil of plane curves of degree $d\geq 3$ there exists a curve $H$ (called the Hesse curve of the pencil) of degree $6(d-1)$ and genus $3(4d^2-13d+8)+1$, and such that: $(i)$ $H$ has $d^2$ singular points of multiplicity three at the base points of the pencil and $3(d-1)^2$ ordinary nodes at the singular points of the degenerate members of the pencil; $(ii)$ for each member of the pencil the intersection of $H$ with this fibre consists of the inflection points of this member and the base points of the pencil.

Equations

$${\mathcal{H}}_d=\{(\overline{a},\overline{z})\in {\mathbb{P}}^{K_d}\times {\mathbb{P}}^2\mid \det (\frac{{\partial }^{2}F(\overline{a},\overline{z})}{\partial z_i\partial z_j})=0\}.$$

$$2(g_d-1)=-2\mathrm{deg}{\tilde{h}}_d+m_d+4\mathrm{deg}{S}_d=-6d(d-2)+m_d+12(d-1{)}^2.$$

Full text

1. Let $F(\overline{a},\overline{z})=\displaystyle\sum_{k+m+n=d}a_{k,m,n}z_{1}^{k}z_{2}^{m}z_{3}^{n}$ be the homogeneous polynomial of degree $d$ in variables $z_{1},z_{2},z_{3}$ and of degree one in variables $a_{k,m,n}$, $k+m+n=d$. Denote by $\mathcal{C}_{d}\subset\mathbb{P}^{K_{d}}\times\mathbb{P}^{2}$, where $K_{d}=\frac{d(d+3)}{2}$, the complete family of plane curves of degree $d$ given by equation $F(\overline{a},\overline{z})=0$. Let ${\mathcal{I}_{d}}=\mathcal{C}_{d}\cap\mathcal{H}_{d}$, where

[TABLE]

Denote by $f_{d}:\mathcal{C}_{d}\to\mathbb{P}^{K_{d}}$ the restrictions of the projection $\text{pr}_{1}:\mathbb{P}^{K_{d}}\times\mathbb{P}^{2}\to\mathbb{P}^{K_{d}}$ to $\mathcal{C}_{d}$. It is well-known (see, for example, [1]) that for a generic point $\overline{a}_{0}\in\mathbb{P}^{K_{d}}$ the intersection of the curve $C_{\overline{a}_{0}}=f_{d}^{-1}(\overline{a})$ and its Hessian curve $H_{C_{\overline{a}_{0}}}$ given by $\frac{\partial^{2}F(\overline{a}_{0},\overline{z})}{\partial z_{i}\partial z_{j}})=0$ is the set of the inflection points of $C_{\overline{a}_{0}}$ containing $3d(d-2)$ points. Therefore for $d\geqslant 3$ the morphism $h_{d}=f_{d|\mathcal{I}_{d}}:{\mathcal{I}_{d}}\to\mathbb{P}^{K_{d}}$ has degree $\deg h_{d}=3d(d-2)$.

Let $\mathcal{S}_{d}$ be a subvariety of $\mathbb{P}^{K_{d}}$ consisting of the points $\overline{a}$ such that the curves $C_{\overline{a}}$ are singular and let $\mathcal{M}_{d}$ be a subvariety of $\mathbb{P}^{K_{d}}$ consisting of the points $\overline{a}$ such that for $\overline{a}\in\mathcal{M}_{d}$ the curve $C_{\overline{a}}$ has a $r$-tuple inflection point with $r\geqslant 2$. Let $\mathcal{B}_{d}=\mathcal{S}_{d}\cup\mathcal{M}_{d}$ (if $d=3$ then $\mathcal{M}_{3}=\emptyset$). It is easy to show ([5]) that $\mathcal{M}_{d}$ is an irreducible hypersurface in $\mathbb{P}^{K_{d}}$ if $d\geqslant 4$. It is well-known also that $\mathcal{S}_{d}$ is an irreducible hypersurface in $\mathbb{P}^{K_{d}}$, $\deg\mathcal{S}_{d}=3(d-1)^{2}$.

2. Remind that, by definition, a Lefschetz pencil is a fibration $f_{d}:C_{L}=f_{d}^{-1}(L)\to L$ of curves over $L$ (and also the linear system $\{C_{\overline{a}}\}_{\overline{a}\in L}$ of curves of degree $d$ in $\mathbb{P}^{2}$), where $L$ is a line in $\mathbb{P}^{K_{d}}$ in general position with respect to the divisor $\mathcal{S}_{d}$. The body $C_{L}$ of the Lefschetz pencil is a non-singular surface. The linear system $\{C_{\overline{a}}\}_{\overline{a}\in L}$ has $d^{2}$ base points and the restriction of the projection $pr_{2}$ to $C_{L}$ is the composition of $d^{2}$ $\sigma$-processes with centers at the base points. We say that the Lefschtz pencil $f_{d}:C_{L}\to L$ is generic if $L$ and $\mathcal{M}_{d}\setminus\mathcal{S}_{d}$ meet at $m_{d}=\deg\mathcal{M}_{d}$ different points.

[TABLE]

On the other hand, the curve $I_{L}\subset L\times\mathbb{P}^{2}$ is a complete intersection of $H_{C_{L}}$ and the smooth surface $C_{L}$. The Picard group $Pic(L\times\mathbb{P}^{2})$ is a free abelian group generated by divisirs $A=pr_{1}^{-1}(t)$ and $B=pr_{2}^{-1}(\mathbb{P}^{1})$, where $t$ is a point in $L$ and $\mathbb{P}^{1}$ is a line in $\mathbb{P}^{2}$. The canonical class $K_{L\times\mathbb{P}^{2}}=-2A-3B$, $C_{L}\in|A+dB|$, and $H_{C_{L}}\in|3A+3(d-2)B|$. Let us restrict the divisors $A$ and $B$ to the surface $C_{L}$. It follows from the adjunction formula that $K_{C_{L}}=-A+(d-3)B$. Besides, we have $I_{L}\in|3A+3(d-2)B|$, $(A,A)_{C_{L}}=(A,A,A+dB)_{L\times\mathbb{P}^{2}}=0,\quad(A,B)_{C_{L}}=(A,B,A+dB)_{L\times\mathbb{P}^{2}}=d$, and $(B,B)_{C_{L}}=(B,B,A+dB)_{L\times\mathbb{P}^{2}}=1$.

Therefore we have $2(g_{d}-1)=(I_{L},I_{L}+K_{C_{L}})_{C_{L}}-2\deg\mathcal{S}_{d}=6(4d^{2}-13d+8)$. Сomparing this equality with (1), we obtain that $m_{d}=6(d-3)(3d-2)$. ∎

Let $L$ be a line in $\mathbb{P}^{K_{d}}$, $L\not\subset\mathcal{S}_{d}$, and $I_{L}=h_{d}^{-1}(L)$. We call $H_{L}=pr_{2}(I_{L})\subset\mathbb{P}^{2}$ the Hesse curve of the pencil $\{C_{\overline{a}}\}_{\overline{a}\in L}$.

Since for a generic Lefschetz pencil $\{C_{\overline{a}}\}_{\overline{a}\in L}$ the Hesse curve $H_{L}$ is irreducible, simple calculations show that the number of elements of a generic Lefschetz pencil having an inflection point at a base point of the pencil is less than or equal to three. Besides, we have $\deg H_{L}=(H_{L},\mathbb{P}^{1})_{\mathbb{P}^{2}}=(I_{L},B)_{C_{L}}=6(d-1)$. Therefore we have

• $(i)$

$\deg H_{L}=6(d-1)$* and its genus is equal to $3(4d^{2}-13d+8)+1$;*

• $(ii)$

$H_{L}$* has $d^{2}$ singular points of multiplicity three at the base points of the pencil $\{C_{\overline{a}}\}_{\overline{a}\in L}$ and $3(d-1)^{2}$ ordinary nodes at the singular points of the degenerate fibres of the pencil;*

• $(iii)$

for each $\overline{a}\in L$ the intersection $H_{L}\cap C_{\overline{a}}$ consists of the inflection points of $C_{\overline{a}}$ and the base points of the pencil, and if $p\in H_{L}\cap C_{\overline{a}}$ is the $2$-tuple inflection point of $C_{\overline{a}}$, then $H_{L}$ and $C_{\overline{a}}$ touch each other at $p$.