V. L. Popov, “On the variety of flexes of plane cubics”, Russian Math. Surveys, 79:6 (2024), 1107

Let $V:=\mathbb C^3$, and let $U$ be the space ${\sf S}^3(V^*)$ of degree $3$ forms on $V$. Let $x_0$, $x_1$, $x_2$ be a basis of $V^*$, and let $\alpha_{i_0 i_1 i_2}$ be the elements of the basis of $U^*$ dual to a basis of $U$ consisting of monomials $x_0^{i_0}x_1^{i_1}x_2^{i_2}$. The sets of forms $\{x_j\}$ and $\{\alpha_{i_0 i_1 i_2}\}$ are projective coordinate systems on the projective spaces $\mathbb{P}(V)$ and $\mathbb{P}(U)$ associated with $V$ and $U$. The variety $X=\{a\in \mathbb{P}(U)\times \mathbb{P}(V)\mid f(a)=h(a)=0\}$, where $f:=\sum_{i_0+i_1+ i_2=3} \alpha_{i_0 i_1 i_2} x_0^{i_0}x_1^{i_1}x_2^{i_2}$ and $h:=\det(\partial^2f/\partial x_i\,\partial x_j)$, is called the variety of flexes of plane cubics because it is a compactification of the set of all pairs $(C, c)$ where $C$ is a smooth plane cubic and $c$ is a flex of $C$. The monodromy of the projection $\pi\colon X\to \mathbb{P}(U)$ and the cohomological properties of $X$ were explored in [1]–[4]. Here we present results on other properties of $X$, with comments to some proofs; see details in [7]. Obtaining them was related for the author to his work on [5] and [6]. In what follows, for an action of a group $R$ on a set $A$ and for a subset $B$ of $A$ we set $N_{R, B}:=\{r\in R\mid r\cdot B\subseteq B\}$. By a variety we mean an algebraic variety.

To the proof of Theorem 1. The group $G:=\operatorname{PSL}_3(\mathbb C)$ acts naturally on $U$, $\mathbb{P}(V)$, $\mathbb{P}(U)$, and $\mathbb{P}(U)\times \mathbb{P}(V)$, and $X$ is an invariant subset of $\mathbb{P}(U)\times \mathbb{P}(V)$. The non-existence of irreducible components $Y$ of $X$ such that $\overline{\pi(Y)}\neq \mathbb{P}(U)$ follows from an analysis of the orbital decompositions determined by these actions. The uniqueness of an irreducible component $Y$ such that $\overline{\pi(Y)}= \mathbb{P}(U)$ follows from the transitivity of the action of the monodromy group of the projection $\pi$ on its fibres in general position, which was proved in [1]. $\Box$

For $(\alpha, \beta)\in\mathbb C^2$ we put $f_{\alpha, \beta}=\alpha(x_0^3+x_1^3+x_2^3)+ \beta x_0x_1x_2$. In $U$ we consider the two-dimensional linear subspace $L:=\{f_{\alpha, \beta}\mid (\alpha, \beta)\in\mathbb C^2\}$, and in $\mathbb{P}(U)$ the projective line $\ell:=\mathbb{P}(L)$. The cubic in $\mathbb{P}(V)$ defined by the equation $f_{\alpha, \beta}=0$, where $(\alpha, \beta)\neq (0,0)$, depends only on the ratio $\beta/\alpha\in \mathbb C\cup\infty$; we denote it by $C_{\beta/\alpha}$. The Hesse pencil $\mathcal H$ and the Hesse group ${H}$ are defined, respectively, as the set $\{C_{\lambda}\mid \lambda\in \mathbb C\cup \infty\}$ and, for the action of $G$ on the set of all cubics in $\mathbb{P}(V)$, the group $N_{G, \mathcal H}$. The smoothness of the cubic $C_\lambda$ is equivalent to the condition $\lambda\neq \infty, -3, -3\varepsilon, -3\varepsilon^2$, where $\varepsilon^3=1$, $\varepsilon\neq 1$. All smooth cubics of the form $C_\lambda$ have the same set $\mathcal F$ of flexes, containing precisly nine points $t_1,\ldots, t_9$.

The groups $N_{G, \ell}$ and $N_{G, \mathcal F}$ coincide with ${H}$. The action of $H$ on $\mathcal F$ is transitive, and the $H$-stabilizer ${H}_i$ of any point $t_i\in \mathcal F$ is isomorphic to the binary tetrahedral group $\operatorname{SL}_2(\mathbb F_3)$. The action of ${H}_i$ on $\ell$ defines the homogeneous fibre space over $G/{{H}_i}$ with fibre $\ell$; we denote its total space, as customary (see [8], § 3.2), by $G \times^{{H}_i}\ell$.

Theorem 2. (i) The varieties $X$ and $G \times^{H_i}\ell$ are equivariantly birationally isomorphic.

(ii) The homogeneous fibre space over $G/{H}_i$ with fibre $\ell$ is the projectivization of a homogeneous vector bundle of rank $2$ over $G/{H}_i$.

To the proof of Theorem 2. This is based on the following general considerations.

Let an algebraic group $R$ act algebraically on an irreducible variety $M$. Let $S$ be a closed subset of $M$, and let $S_1,\ldots, S_d$ be all of its irreducible components.

Definition 1. $S$ is called a relative section for the action of $R$ on $M$ if the following conditions hold: (a) $\overline{R\cdot S_j}=M$ for every $j$; (b) there is a dense open subset $S^0$ of $S$ such that, for every $r\in R$, from $r\cdot S^0\cap S^0\neq \varnothing$ it follows that $r\in N_{R, S}$.

In what follows let $S$ be a relative section for the action of $R$ on $M$ such that $R\times^{N_{R, S}}S$ exists (for instance, this is so if $S$ is quasiprojective: see [8], § 3.2).

Lemma 1. (a) $N_{R, S}$ permutes $S_1,\ldots, S_d$ transitively. (b) Every $S_i$ is a relative section for the same action. (c) $N_{R, S_i}=N_{N_{R, S}, S_i}$. (d) $R \times^{N_{R, S}} S$ is irreducible and $R$-equivariantly birationally isomorphic to $M$.

Let $\widetilde M$ be another irreducible variety endowed with an algebraic $R$-action, let $\tau\colon \widetilde{M}\to M$ be a dominant $R$-equivariant morphism, and let $T_1,\ldots, T_k$ be all the irreducible components of the set $\tau^{-1}(S)$.

Definition 2. If $\overline{\tau(T_i)}=S_j$ for some $j$ and $\dim (T_i\cap \tau^{-1}(m))=\dim \widetilde{M}-\dim M$ for points $m$ in general position in $S_j$, then $T_i$ is called a regular component.

Lemma 2. Regular components exits, their union $\widetilde S$ is a relative section for the action of $R$ on $\widetilde M$, and $N_{R, \widetilde{S}}=N_{R, S}$.

An analysis of the properties of $\mathcal H$ shows that $\ell$ is a relative section for the action of $G$ on $\mathbb{P}(U)$. Using Lemma 2, as applied to $\pi$, and Lemma 1 (b), we deduce that $\ell\times t_i$ is a relative section for the action of $G$ on $X$, and $N_{G, \ell\times t_{i}}={H}_i$. This and Lemma 1 (d) yield (i). The projectivization of $G \times^{{H}_i}L$ is $G \times^{{H}_i}\ell$, which yields (ii). $\Box$

Proof of Theorem 3. By Theorem 2 (i) the rationality of $X$ is equivalent to the rationality of $G \times^{{H}_i} \ell$. Since vector bundles are locally trivial in the Zariski topology (see [8], § 5.4 and Theorem 2), Theorem 2 (ii) implies that $G \times^{{H}_i} \ell$ and $G/{{H}_i} \times \ell$ are birationally isomorphic. Because of the rationality of $\ell$, the finiteness of ${{H}_i}$, and Theorem 4 that follows, this completes the proof of Theorem 3. $\Box$

Theorem 4. For any finite subgroup $K$ of the group $R:=\operatorname{SL}_3(\mathbb C)$ the eight-dimensional homogeneous space $R/K$ is a rational variety.

To the proof of Theorem 4. This is based on the fact that a maximal proper parabolic subgroup of $R$ is special in the sense of Serre ([8], § 4.1) and has codimension 2 in $R$. $\Box$

Citation: V. L. Popov, “On the variety of flexes of plane cubics”, Russian Math. Surveys, 79:6 (2024), 1107–1109

Citation in format AMSBIB

\Bibitem{Pop24}

\by V.~L.~Popov

\paper On the variety of flexes of plane cubics

\jour Russian Math. Surveys

\yr 2024

\vol 79

\issue 6

\pages 1107--1109

\mathnet{http://mi.mathnet.ru/eng/rm10211}

\crossref{https://doi.org/10.4213/rm10211e}

\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4867096}

\adsnasa{https://adsabs.harvard.edu/cgi-bin/bib_query?2024RuMaS..79.1107P}

\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=001443210000012}

Linking options:

https://www.mathnet.ru/eng/rm10211

https://doi.org/10.4213/rm10211e

https://www.mathnet.ru/eng/rm/v79/i6/p169

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

Bibliography