On the double tangent of projective closed curves

Thomas Blomme thomas.blomme@unige.ch Université de Neuchâtel, rue Émile Argan 11, Neuchâtel 2000, Switzerland

Abstract

We generalize a previous result by Fabricius-Bjerre [FB62] from curves in $\mathbb{R}^{2}$ to curves in $\mathbb{R}P^{2}$ . Applied to the case of real algebraic curves, this recovers the signed count of bitangents of quartics introduced by Larson-Vogt [LV21] and proves its positivity, conjectured by Larson-Vogt. Our method is not specific to quartics and applies to algebraic curves of any degree.

:

14N10

keywords:

Enumerative geometry, bitangents, plane algebraic curves

1 Introduction

We consider curves in $\mathbb{R}^{2}$ and $\mathbb{R}P^{2}$ . By curve we mean a closed immersed $C^{1}$ curve which is the union of finitely many strictly convex arcs. Such a curve has a finite number of flexes (points where the curve passes through its tangent), nodes (also called double points or self-intersection) and double tangents, also called bitangents, which are lines tangent at two points of the curve. Throughout the paper, we assume all our curves to be generic in the sense that double points are simple, no tangent at a flex is tangent elsewhere, and a double tangent has only two tangency points. The normalization of the curve (desingularizing the double points) may not be connected, and its connected components are called components of the curve.

In 1962, Fabricius-Bjerre [FB62] introduced a signed count of bitangents for such curves in $\mathbb{R}^{2}$ . Let $C\subset\mathbb{R}^{2}$ be a curve and $D$ be a bitangent to $C$ . There are two types of bitangents according to the relative position of the curve near the tangency points (See Figure 1):

$\circ$

exterior (type $T$ ) if the arcs of $C$ near the tangency points lie on the same side of $D$ ,
$\circ$

interior (type $S$ ) if the arcs of $C$ near the tangency points lie on opposite sides of $D$ .

The Fabricius-Bjerre sign of a bitangent is $+$ for type $T$ and $-$ for type $S$ . If $C$ has $i(C)$ flexes and $n(C)$ self-intersections, Fabricius-Bjerre [FB62] states that the signed count $\sigma(C)$ of bitangents satisfies

\sigma(C)=n(C)+\frac{i(C)}{2}.

Although the proof of [FB62] seems to assume the curve has only one component, such an assumption is actually not necessary.


$(T)$ , exterior bitangent, $+$	$(S)$ , interior bitangent, $-$

Figure 1: Types of bitangents

Our main result is to provide a generalization of the Fabricius-Bjerre count to the projective setting. To extend the sign definition, we choose a line $L_{\infty}\subset\mathbb{R}P^{2}$ and consider curves generically transverse to $L_{\infty}$ : no line passing through a point of $C\cap L_{\infty}$ is a bitangent or tangent at a flex, which means $L_{\infty}$ is transverse to $C$ union its bitangents and tangent at flexes. Let $\mathscr{T}_{\infty}(C)$ be the set of tangents to $C$ at points of $C\cap L_{\infty}$ . As $\mathbb{R}P^{2}\backslash L_{\infty}=\mathbb{R}^{2}$ , we choose as sign for the bitangents the well-defined Fabricius-Bjerre sign in this affine chart. Notice it depends on the choice of $L_{\infty}$ .

Theorem.

3.2 Let $C$ a curve in $\mathbb{R}P^{2}$ transverse to $L_{\infty}$ with $i(C)$ flexes, $n(C)$ nodes and $a$ intersection points with $L_{\infty}$ . Then, we have the following signed count of bitangents:

\sigma(C)=n(C)+\frac{i(C)}{2}+\frac{a(a-2)}{2}-\sum_{T\in\mathscr{T}_{\infty}(C)}(|C\cap T|-1).

The case of [FB62] corresponds to curve such that $C\cap L_{\infty}=\emptyset$ (i.e. $a=0$ ).

If $\mathscr{C}$ is a real algebraic curve, Plücker [Plü34, Plü39] already knew the number of complex bitangents, which only depends on the degree. However, the number of real bitangents depends on the curve. One may hope that there exists some suitable signed counts of the bitangents that is to some extent invariant. In [BBG24], Blomme-Brugallé-Garay proved the existence of a signed count only depending on the topology of the real part for curves of even degree. Prior to the latter, Larson-Vogt proved the existence of a different signed count for quartics, with some conjectures concerning so-called quadratic enrichment to more general fields. The Fabricius-Bjerre sign actually coincides with the Larson-Vogt sign [LV21] used to count real bitangents to a real quartic curve.

In the real algebraic setting, bitangents can be split or non-split according to whether the tangency points are real or complex conjugate. Given a smooth curve $\mathscr{C}$ , Klein’s formula [Kle76] provides a relation between the number of non-split bitangents $t_{0}(\mathscr{C})$ and the number of real flexes. It is possible to apply Theorem 3.2 coupled to Klein’s formula to a real algebraic curve transverse to a line $L_{\infty}$ and get a signed count of the bitangents: $\rho(\mathscr{C})=t_{0}(\mathscr{C})+\sigma(\mathbb{R}\mathscr{C})$ .

Theorem.

4.1 Let $\mathscr{C}$ be a smooth real algebraic curve of degree $d$ in $\mathbb{R}P^{2}$ intersecting $L_{\infty}$ transversally at exactly $a$ points. Then, we have the following signed count:

\rho(\mathscr{C})=\frac{d(d-2)}{2}+\frac{a(a-2)}{2}-\sum_{T\in\mathscr{T}_{\infty}(\mathbb{R}\mathscr{C})}(|\mathbb{R}\mathscr{C}\cap T|-1).

Furthermore, the integer $\rho(\mathscr{C})$ is even and non-negative

In the case of quartics non-intersecting $L_{\infty}$ , we recover the signed count from [LV21]. Furthermore, the non-negativity statement proves [LV21, Conjecture 2] for any degree, already proven for quartics by Kummer-McKean [KM24] through different methods. It would be very interesting to see if quadratic enrichments exist for other fields. Theorem 4.1 suggests such a count should relate a quadratic count of bitangents to the intersection points between the curves and the tangents to points at infinity.

Acknowledgments. The author is supported by the SNF grant 204125 and would like to thank Erwan Brugallé for suggesting to work on bitangents and write this paper in the first place, as well as commenting on a first version.

2 Classical Fabricius-Bjerre Theorem

Let $C$ be a curve in $\mathbb{R}^{2}$ with components $C_{j}$ . For each component, we choose a regular parametrization $\gamma_{j}:I_{j}\to\mathbb{R}^{2}$ inducing an orientation. We denote the tangent at $\gamma_{j}(t_{j})$ by $D(\gamma_{j}(t_{j}))=\gamma_{j}(t_{j})+\mathbb{R}\gamma^{\prime}_{j}(t_{j})$ . We also consider the half-tangents

D_{+}(\gamma_{j}(t_{j}))=\gamma_{j}(t_{j})+\mathbb{R}_{>0}\gamma^{\prime}_{j}(t_{j})\text{ and }D_{-}(\gamma_{j}(t_{j}))=\gamma_{j}(t_{j})+\mathbb{R}_{<0}\gamma^{\prime}_{j}(t_{j}).

They are switched when the orientation of $C$ is reversed. Let $t(C)$ (resp. $s(C)$ ) be the number of bitangents of type $T$ (resp. $S$ ), so that the signed count is $\sigma(C)=t(C)-s(C)$ .

The following Theorem is first due to Fabricius-Bjerre [FB62] in the case where the curve $C$ has a unique component. For convenience of the reader, we recall its proof. We also notice that the proof also works for curves with possibly several components.

Theorem 2.1.

Let $C$ be a curve in $\mathbb{R}^{2}$ with $n(C)$ nodes and $i(C)$ flexes. We have the following identity:

\sigma(C)=n(C)+\frac{i(C)}{2}.

Proof.

On each component $C_{j}$ of $C$ , we consider the function $f_{j}(t_{j})=|D_{+}(\gamma_{j}(t_{j}))\cap C|-|D_{-}(\gamma_{j}(t_{j}))\cap C|$ giving the difference between the number of intersection points of the curve and the two half-tangents.

If $D(\gamma_{j}(t_{j}))$ is transverse to $C$ outside its tangency point $\gamma_{j}(t_{j})$ , intersection points deform continously in a neighborhood of $t_{j}$ . It follows that $f_{j}$ is locally constant near $t_{j}$ . Thus, $f_{j}$ is locally constant outside the values of $t_{j}$ where $\gamma_{j}(t_{j})$ is a flex, a node, or $D(\gamma_{j}(t_{j}))$ is a bitangent. Let $J_{j}(t)=f_{j}(t_{j}^{+})-f(t_{j}^{-})$ be the value of the jump at $t_{j}$ , where $t_{j}^{\pm}$ denotes the limit on the two sides of $t$ . The jump function $J_{j}$ takes the following values:

$\circ$

Assume $\gamma_{j}(t_{j})$ is a node. When crossing the node, passing from $t_{j}^{-}$ to $t_{j}^{+}$ , one intersection point passes from $D_{+}(\gamma_{j}(t_{j}^{-}))$ to $D_{-}(\gamma_{j}(t_{j}^{+}))$ . Thus, we have $J_{j}(t_{j})=-2$ .
$\circ$

Assume $\gamma_{j}(t_{j})$ is a flex of $C_{j}$ . Here, when crossing the flex, we also have an intersection point passing from $D_{+}(\gamma_{j}(t_{j}^{-}))$ to $D_{-}(\gamma_{j}(t_{j}^{+}))$ , so that $J_{j}(t_{j})=-2$ .
$\circ$
Assume that the tangent $D(\gamma_{j}(t_{j}))$ is a bitangent: $D(\gamma_{j}(t_{j}))=D(\gamma_{k}(t^{\prime}_{k}))$ for some $k$ and $t^{\prime}_{k}$ . Assume this bitangent is of type $T$ . When passing from $t_{j}^{-}$ to $t_{j}^{+}$ , we have two possibilities:
- -
  
  If $\gamma_{k}(t^{\prime}_{k})\in D_{+}(\gamma_{j}(t_{j}))$ , we gain two intersection points in $D_{+}(\gamma_{j}(t_{j}))$ .
- -
  
  If $\gamma(t^{\prime}_{k})\in D_{-}(\gamma_{j}(t_{j}))$ , we lose two intersection points in $D_{-}(\gamma_{j}(t_{j}))$ .
In any case, we get that $J_{j}(t_{j})=2$ .
$\circ$

Assuming the bitangent to be of type $S$ , we get similarly $J_{j}(t_{j})=-2$ .

As for each component we have $f_{j}(0)=f_{j}(1)$ , we have

0=\sum_{j}f_{j}(1)-f_{j}(0)=\sum_{j}\sum_{t_{j}:J_{j}(t_{j})\neq 0}J_{j}(t_{j}).

Grouping together the parameters giving the same node or bitangent, we deduce the following relation:

4t(C)-4s(C)-4n(C)-2i(C)=0,

which is the desired identity. ∎

3 A projective Fabricius-Bjerre Theorem

We now aim to generalize the result by considering curves in $\mathbb{R}P^{2}$ . To do so, let $L_{\infty}$ be a line in $\mathbb{R}P^{2}$ and consider curves generically transverse to $L_{\infty}$ , which means the following: no line passing through a point of $C\cap L_{\infty}$ is a bitangent or tangent at a flex. Recall that $\mathscr{T}_{\infty}(C)$ is the set of tangents to $C$ at points of $C\cap L_{\infty}$ . As $\mathbb{R}P^{2}\backslash L_{\infty}=\mathbb{R}^{2}$ , we can use the Fabricius-Bjerre sign in this affine chart and make the associated signed count of bitangents. All constructions depend on the choice of this line $L_{\infty}$ . If $L$ is a line distinct from $L_{\infty}$ , we denote by $p_{L}$ the unique intersection point between $L$ and $L_{\infty}$ .

Let $L$ be a second line in $\mathbb{R}P^{2}$ , different from $L_{\infty}$ . Before getting to the generalization of Theorem 2.1, we provide a signed count of the tangents to $C$ passing through $p_{L}$ . Let $D$ be a tangent to $C$ passing through $p_{L}$ . The set $\mathbb{R}P^{2}\setminus(D\cup L_{\infty}\cup L)$ has three connected components, each one adjacent to two of the lines. Let $U$ the connected component in which $C$ is contained near its tangency point with $D$ : $\overline{U}\supset D$ . We set:

$\circ$

if $\overline{U}\supset L$ (the concavity of $C$ is toward $L$ ), the sign is $\epsilon_{L}(D)=+1$ ;
$\circ$

if $\overline{U}\supset L_{\infty}$ (the concavity of $C$ is toward $L_{\infty}$ ), the sign is $\epsilon_{L}(D)=-1$ .

If $L$ is itself tangent to $C$ , we set $\epsilon_{L}(L)=0$ . We consider the signed count of tangents to $C$ passing through $p_{L}$ :

\sigma_{L}(C)=\sum_{D\ni p_{L}}\epsilon_{L}(D).

Lemma 3.1.

Assume $L\cap C\cap L_{\infty}=\emptyset$ . The signed count of tangents to $C$ passing through $p_{L}$ has the following value:

\sigma_{L}(C)=|C\cap L|-|C\cap L_{\infty}|.

Furthermore, if $L\in\mathscr{T}_{\infty}(C)$ is tangent of $C$ at some point of $C\cap L_{\infty}$ , then

\sigma_{L}(C)=1+|C\cap L|-|C\cap L_{\infty}|.

Proof.

First assume that $p_{L}\notin C$ . We consider the projection from $C$ to the pencil $\mathcal{P}\simeq\mathbb{R}P^{1}$ of lines passing through $p_{L}$ :

\pi:p\in C\longmapsto(pp_{L})\in\mathcal{P},

where $(pp_{L})$ is the unique line passing through $p$ and $p_{L}$ . The pencil $\mathcal{P}$ contains the lines $L$ and $L_{\infty}$ , which split $\mathcal{P}$ into segments $\mathcal{P}_{0}$ and $\mathcal{P}_{1}$ , which we both orient from $L$ to $L_{\infty}$ .

A coordinate function on $\mathcal{P}_{0}$ (resp. $\mathcal{P}_{1}$ ) induces a Morse function on $\pi^{-1}(\mathcal{P}_{0})\subset C$ . Its critical points correspond precisely to points in $C$ whose tangent belongs to $\mathcal{P}$ , and the sign $\epsilon_{L}$ actually coincides with the index. Therefore, for $\mathcal{P}_{0}$ , we have that

|\pi^{-1}(L)|-|\pi^{-1}(L_{\infty})|=2\sum_{D\in\mathcal{P}_{0}}\epsilon_{L}(D),

where the sum is over the tangent to $C$ belonging to $\mathcal{P}_{0}$ . As $\pi^{-1}(L)=C\cap L$ and $\pi^{-1}(L_{\infty})=C\cap L_{\infty}$ , adding the identities for $\mathcal{P}_{0}$ and $\mathcal{P}_{1}$ yields the count $\sigma_{L}(C)$ and its value.

If $L=T$ is the tangent line to a point $p_{T}\in C\cap L_{\infty}$ , the projection $\pi$ is not a priori defined at $p_{T}$ . As $C$ is tangent to $T$ at $p_{T}$ , $\pi$ extends by continuity with $\pi(p_{T})=T\in\mathcal{P}$ . We then proceed as before, with the only difference that $\pi^{-1}(L_{\infty})=(C\cap L_{\infty})\backslash\{p_{T}\}$ , yielding the $1$ discrepancy. ∎

We can now give the projective version of Fabricius-Bjerre.

Theorem 3.2.

Let $C$ be a curve in $\mathbb{R}P^{2}$ transverse to $L_{\infty}$ with $a$ intersection points. We have the following signed count of bitangents:

\sigma(C)=n(C)+\frac{i(C)}{2}+\frac{a(a-2)}{2}-\sum_{T\in\mathscr{T}_{\infty}(C)}(|C\cap T|-1).

Proof.

We proceed as in the affine case, considering a parametrization $\gamma_{j}:I_{j}\to\mathbb{R}P^{2}$ of each component of $C$ and the functions

f_{j}(t_{j})=|D_{+}(\gamma_{j}(t_{j}))\cap C|-|D_{-}(\gamma_{j}(t_{j}))\cap C|.

Outside a finite number of values, $f_{j}$ is locally constant. Let $J_{j}(t_{j})=f(t_{j}^{+})-f(t_{j}^{-})$ be the jump function. On one side, we have the same jumps as for the compact curves:

$\circ$

$J_{j}(t_{j})=2$ if $\gamma_{j}(t_{j})$ is node or a flex, or if $D(\gamma_{j}(t_{j}))$ is a bitangent of type $T$ ;
$\circ$

$J_{j}(t_{j})=-2$ if $D(\gamma_{j}(t_{j}))$ is a bitangent of type $S$ .

However, we have now new jump points when $\gamma_{j}(t_{j})\in L_{\infty}$ or if $D(\gamma_{j}(t_{j}))\ni p_{T}$ for $T\in\mathscr{T}_{\infty}(C)$ .

$\circ$

Assume $\gamma_{j}(t_{j})\in L_{\infty}$ . In particular, $D(\gamma_{j}(t_{j}))=T\in\mathscr{T}_{\infty}(C)$ . Going from $t_{j}^{-}$ to $t_{j}^{+}$ , all intersection points between $C$ and the tangent beside the tangency point move from $D_{-}$ to $D_{+}$ . Therefore, we have $J_{j}(t_{j})=2(|C\cap T|-1)$ . The $-1$ is for the tangency point, not counted in the process.
$\circ$

Assume that $D(\gamma_{j}(t_{j}))\ni p_{T}$ for some $T\in\mathscr{T}_{\infty}(C)$ . Depending on the concavity of $C$ at $\gamma_{j}(t_{j})$ , the intersection point near $p_{T}$ passes from $D_{+}(\gamma_{j}(t_{j}^{-}))$ to $D_{-}(\gamma_{j}(t_{j}^{+}))$ or conversely: we have $J_{j}(t_{j})=2\epsilon_{T}(D(\gamma_{j}(t_{j})))=\pm 2$ .

As $0=f_{j}(1)-f_{j}(0)=\sum_{t_{j}:J_{j}(t)\neq 0}J_{j}(t_{j})$ , summing over the various components and grouping together the parameters corresponding to bitangents and nodes, we get the following identity:

4t(C)-4s(C)-2i(C)-4n(C)+2\sum_{T\in\mathscr{T}_{\infty}(C)}(|T\cap C|-1)+\sum_{T\in\mathscr{T}_{\infty}(C)}\sum_{D\ni p_{T}}2\epsilon_{T}(D)=0.

We recognize the signed count $\sigma_{T}(C)$ computed in Lemma 3.1. Replacing its expression, we get that

4\sigma(C)-2i(C)-4n(C)+4\sum(|T\cap C|-1)-2a(a-2)=0.

∎

4 Applications to real algebraic curves

Let $\mathscr{C}$ be a generic smooth real algebraic curve in $\mathbb{R}P^{2}$ of degree $d\geqslant 2$ . In particular, the real part $\mathbb{R}\mathscr{C}$ is a curve in our sense. If $L$ is a real line tangent to $\mathscr{C}$ at two points, there are two possibilities:

$\circ$

the split bitangents if the tangency points are real,
$\circ$

the non-split bitangents if the tangency points are complex conjugate.

Actually, the number of non-split bitangents $t_{0}(\mathscr{C})$ is related to the number of real flexes $i(\mathbb{R}\mathscr{C})$ via Klein’s formula [Kle76]:

t_{0}(\mathscr{C})+\frac{i(\mathbb{R}\mathscr{C})}{2}=\frac{d(d-2)}{2}.

We consider the following signed count of bitangents to $\mathscr{C}$ :

\rho(\mathscr{C})=t_{0}(\mathscr{C})+\sigma(\mathbb{R}\mathscr{C}).

The first term deals with non-split bitangents (thus counted with a positive sign), and the second term with the split bitangents, counted with the Fabricius-Bjerre sign. The curve being smooth, we have $n(\mathbb{R}\mathscr{C})=0$ . Coupling Klein’s formula with Theorem 3.2, we get the following.

Theorem 4.1.

Let $\mathscr{C}$ be a generic smooth real algebraic curve of degree $d$ in $\mathbb{R}P^{2}$ with $a$ intersection points with $L_{\infty}$ . Then, we have the following signed count:

\rho(\mathscr{C})=\frac{d(d-2)}{2}+\frac{a(a-2)}{2}-\sum_{T\in\mathscr{T}_{\infty}(\mathbb{R}\mathscr{C})}(|\mathbb{R}\mathscr{C}\cap T|-1).

Furthermore, this count is even and non-negative.

Theorem 4.1 actually proves [LV21, Conjecture 2] and generalizes it for curves of any degree. This conjecture was already proved by Kummer-McKean [KM24] by a different method in the case of quartics.

Proof.

As advertised, the signed count is a consequence of Theorem 3.2 and Klein’s formula. We now prove the count is even and non-negative. By Bezout’s theorem, for each tangent $T$ , we have that

|T\cap\mathbb{R}\mathscr{C}|-1\leqslant d-2.

Summing over all the tangents of $\mathscr{T}_{\infty}(\mathbb{R}\mathscr{C})$ , we have the following lower bound:

\rho(\mathscr{C})\geqslant\frac{d(d-2)}{2}+\frac{a(a-2)}{2}-a(d-2)=\frac{(d-a)(d-a-2)}{2}\geqslant 0,

since by Bezout’s theorem, $d-a\neq 1$ . For the parity, Bezout’s Theorem also tells us that $d\equiv a\mod 2$ and $|T\cap\mathbb{R}\mathscr{C}|-1\equiv d\ \mathrm{mod}\ 2$ since the tangency points count twice. Therefore,

	$\displaystyle\rho(\mathscr{C})\equiv$	$\displaystyle\frac{a^{2}+d^{2}}{2}-a-d-ad\ \mathrm{mod}\ 2$
	$\displaystyle\equiv$	$\displaystyle\frac{a^{2}-d^{2}}{2}\ \mathrm{mod}\ 2.$

As $a\equiv d\ \mathrm{mod}\ 2$ , we get $a^{2}\equiv d^{2}\ \mathrm{mod}\ 4$ and we conclude. ∎

Assuming $\mathscr{C}$ has no real point on $L_{\infty}$ , i.e. $a=0$ , the signed count in this particular case is

\rho(\mathscr{C})=\frac{d(d-2)}{2}.

Example 4.2.

For conics, i.e. $d=2$ , the tangents at $\mathbb{R}\mathscr{C}\cap L_{\infty}$ have no further intersection points, so that the signed count is $0+0-0$ , compatible with the fact there is no bitangent.

We now assume $\mathscr{C}$ is a cubic curve: the degree is $3$ . Klein’s formula tells us that there are $3$ real flexes. The number of tangents at $\mathbb{R}\mathscr{C}\cap L_{\infty}$ is either $1$ or $3$ . Furthermore, for any real tangent line $L$ , by Bezout’s theorem, $L\cap\mathscr{C}$ has three points counted with multiplicity. The tangency point counts for $2$ . Thus, the remaining point is real, and we get $|L\cap\mathbb{R}\mathscr{C}|-1=1$ . In the end, if $a=1$ we get $\frac{3}{2}-\frac{1}{2}-1$ , and if $a=3$ we get $\frac{3}{2}+\frac{3}{2}-3$ , which is compatible with the absence of bitangent in this case as well. $\lozenge$

Example 4.3.

The first interesting case is the case of quartic curves. Assuming transversality, a quartic may have $a=0,2$ or $4$ real intersection points with $L_{\infty}$ .

$\triangleright$

If $a=0$ , the real locus is compact in $\mathbb{R}^{2}$ , and we recover the signed count from [LV21].
$\triangleright$

If $r=2$ , we have tangents at points of $L_{\infty}$ . The signed count is

$\rho(\mathscr{C})=4-(|T_{1}\cap\mathbb{R}\mathscr{C}|-1)-(|T_{2}\cap\mathbb{R}\mathscr{C}|-1).$

Thanks to Bezout’s theorem, we have that $|T\cap\mathbb{R}\mathscr{C}|-1\leqslant 2$ for every tangent $T$ to $C$ . Thus, $\rho(\mathscr{C})\geqslant 0$ and the count may take the values $0,2,4$ .

\triangleright

If $r=4$ , we now a have $4$ asymptotic lines, and we get

\rho(\mathscr{C})=4+\frac{4(4-2)}{2}-\sum_{1}^{4}(|T_{j}\cap\mathbb{R}\mathscr{C}|-1)=8-\sum_{1}^{4}(|T_{j}\cap\mathbb{R}\mathscr{C}|-1).

We conclude similarly than $\rho(\mathscr{C})\geqslant 0$ and can take the values $0,2,4,6,8$ .

$\lozenge$

Using the generalization of Klein’s formula [Sch04] to curves with nodes and cusp, we can generalize the signed count to real algebraic curves with nodes.

Proposition 4.4.

Let $\mathscr{C}$ be a degree $d$ real nodal algebraic curve in $\mathbb{P}^{2}$ with $N$ nodes (both real and complex), $n_{\mathbb{R}}$ real nodes, $i$ real flexes and $a$ intersection points with $L_{\infty}$ . Then, we have the following signed count:

\rho(\mathscr{C})=n_{\mathbb{R}}-N+\frac{d(d-2)}{2}+\frac{a(a-2)}{2}-\sum_{T}(|C\cap T|-1)

Proof.

Let $n_{0}$ be the number of real isolated nodes and $n_{2}$ the number of hyperbolic nodes, so that $n_{\mathbb{R}}=n_{0}+n_{2}$ . Klein’s formula for nodal curves [Sch04] states as follows:

d+i+2t_{0}=d(d-1)-2N+2n_{0},

while Theorem 3.2 gives

\sigma(\mathbb{R}\mathscr{C})=n_{2}+\frac{i}{2}+\frac{a(a-2)}{2}-\sum_{T}(|C\cap T|-1).

We conclude as in Theorem 4.1. ∎

References

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[Kle76] F Klein. Eine neue Relation zwischen den Singularitäten einer algebraischen Curve. Math. Ann., 10(2):199–209, 1876.
[KM24] Mario Kummer and Stephen McKean. Bounding the signed count of real bitangents to plane quartics. manuscripta mathematica, 173(3):1003–1013, 2024.
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