Quasi complete intersections and global Tjurina number of plane curves

TL;DR

This paper studies quasi complete intersections in the projective plane, providing bounds on their degree based on syzygies, and recovers known results on the Tjurina number of plane curves.

Contribution

It establishes bounds on the degree of quasi complete intersections using syzygies and generalizes results related to the Tjurina number of plane curves.

Findings

Bounds on deg(T) in terms of a, b, c, and syzygy degree r

Recovery of du Plessis-Wall theorem on Tjurina number

Extension of results to related invariants of plane curves

Abstract

A closed subscheme of codimension two $T \subset P^2$ is a quasi complete intersection (q.c.i.) of type $(a,b,c)$ if there exists a surjective morphism $\mathcal{O} (-a) \oplus \mathcal{O} (-b) \oplus \mathcal{O} (-c) \to \mathcal{I} _T$. We give bounds on deg$(T)$ in function of $a,b,c$ and $r$, the least degree of a syzygy between the three polynomials defining the q.c.i. As a by-product we recover a theorem of du Plessis-Wall on the global Tjurina number of plane curves and some other related results.

Equations

$$0\to E\to \mathcal{O}(c-a)\oplus \mathcal{O}(c-b)\oplus \mathcal{O}\to {\mathcal{I}}_T(c)\to 0$$

$$c(a+b-c-r)\le t\le r^2+r(c-a-b)+ab$$

$$t\le r(c-a-b)+ab+r^2-\frac{(c-a-b+2r+1)(c-a-b+2r)}{2}.$$

$$0\to \mathcal{O}(r-a)\to E(r)\to {\mathcal{I}}_{\Gamma }(c-b+r)\to 0$$

$$0\to \mathcal{O}(r+c-a-b-2)\to \mathcal{O}(-r)\oplus 2.\mathcal{O}(r+c-a-b-1)\to E\to 0$$

$$0\to 2.\mathcal{O}(-c-r+1)\oplus \mathcal{O}(r-a-b)\to \mathcal{O}(-c-r+2)\oplus \mathcal{O}(-a)\oplus \mathcal{O}(-b)\oplus \mathcal{O}(-c)\to {\mathcal{I}}_T\to 0$$

$$0\to \mathcal{O}(r+c-a-b-2)\to \mathcal{O}(-r)\oplus 2.\mathcal{O}(r+c-a-b-1)\to E\to 0$$

$$0\to 2.\mathcal{O}(-c-r+1)\oplus \mathcal{O}(r-a-b)\to \mathcal{O}(-c-r+2)\oplus \mathcal{O}(-a)\oplus \mathcal{O}(-b)\oplus \mathcal{O}(-c)\to {\mathcal{I}}_T\to 0$$

$$0\to \mathcal{O}(c-a-b+2r-2)\to \mathcal{O}\oplus 2.\mathcal{O}(c-a-b+2r-1)\to E(r)\to 0$$

$$0\to \mathcal{O}(-a+r-2)\oplus \mathcal{O}(b-a-c)\to \mathcal{O}(-r-c+b)\oplus 2.\mathcal{O}(-a+r-1)\to {\mathcal{I}}_{\Gamma }\to 0$$

$$0\to E_C\to 3.\mathcal{O}\stackrel{\partial f}{\to }{\mathcal{I}}_{\Sigma }(d-1)\to 0$$

$$(d-1)(d-r-1)\le \tau \le (d-1)(d-r-1)+r^2$$

$$\tau \le (d-1)(d-r-1)+r^2-\frac{1}{2}(2r+1-d)(2r+2-d)$$

$$\begin{array}[]{ccccccccc}0&\to&E_{C}(-d+1)&\to&3.\mathcal{O}(-d+1)&\to&\mathcal{I}_{\Sigma}&\to&0\\ &&\downarrow\psi&&\downarrow\varphi&&||&&\\ 0&\to&\mathcal{O}(-d)&\to&\mathcal{O}(-d+1)\oplus\mathcal{O}(-1)&\to&\mathcal{I}_{\Sigma}&\to&0\end{array}$$

$$0\to S(-d+r-1)\to S(-r)\oplus 2.S(-d+r)\to H_{\ast }^0(E_C)\to 0.$$

Full text

1. Introduction.

Let $C\subset\mathbb{P}^{2}$ be a reduced, singular curve, of degree $d$, of equation $f=0$. The partials of $f$ determine a morphism: $3.\mathcal{O}\stackrel{{\scriptstyle\partial f}}{{\to}}\mathcal{O}(d-1)$, whose image is a, twisted, ideal sheaf, $\mathcal{I}_{f}(d-1)$. From the assumption on $C$, it follows that $\mathcal{I}_{f}$ is the ideal sheaf of a zero-dimensional subscheme, $\Sigma$, of $C$, called the jacobian scheme of $C$. (Hence $\mathcal{I}_{f}=\mathcal{I}_{\Sigma}$.) The support of $\Sigma$ is the singular locus of $C$, but the scheme structure is rather mysterious. The global Tjurina number of $C$ is $\tau(C):=h^{0}(\mathcal{O}_{\Sigma})$ (we will often write $\tau$ instead of $\tau(C)$ if no confusion can arise).

It is natural to ask for some bound on $\tau$ in function of $d$ (and of some other natural invariants).

The kernel of the morphism $\partial f$ is a rank two reflexive sheaf, $E_{C}$. Since we are on a smooth surface, $E_{C}$ is in fact locally free. Let $r$ denotes the minimal twist of $E_{C}$ having a section. In other words $r$ is the least degree of a syzygy between the partials of $f$. Then a very nice result of du Plessis-Wall (see Theorem 20) gives bounds on $\tau$ in function of $d$ and $r$. This result is proved in the framework of singularity theory. The proof is hard to follow for those who, like me, are far from being experts in this field. So I tried to find another proof. It turns out that the theorem of du Plessis and Wall is a direct consequence of a general statement about quasi-complete intersections in $\mathbb{P}^{2}$ (see Theorem 6), whose proof requires only notions of projective geometry (vector bundles, liaison). So, at this point, one doesn’t even need to know what is a partial to prove du Plessis-Wall’s theorem ! This is amazing and the first reaction is to think, that using the specific assumption (i.e. the three polynomials giving the quasi-complete intersection are the partials of a single polynomial $f$), one could improve the bounds given by the theorem. Alas this is not the case, examples (see 22) show that, in some sense, du Plessis and Wall’s theorem is sharp. This is the content of the first sections of this paper. In the last section, I give some related results, some known, some new, but all in the framework of projective geometry.

2. Quasi-complete intersections of codimension two in $\mathbb{P}^{2}$.

Let us start with a definition:

Definition 1**. **

Let $F_{a},F_{b},F_{c}\in S:=k[x_{0},...,x_{n}]$ be three homogeneous polynomials of degrees $a,b,c$. The ideal $J=(F_{a},F_{b},F_{c})$ is said to be a quasi complete intersection (q.c.i.) if the morphism $\mathcal{O}(-a)\oplus\mathcal{O}(-b)\oplus\mathcal{O}(-c)\stackrel{{\scriptstyle\rho}}{{\to}}\mathcal{O}$, defined by these polynomials, has for image the ideal sheaf, $\mathcal{I}_{T}$, of a codimension two subscheme $T$.

Remark 2**. **

Sometimes one says, and we will do it, that the subscheme $T$ is a q.c.i. of type $(a,b,c)$ if there exists a surjective morphism $\mathcal{O}(-a)\oplus\mathcal{O}(-b)\oplus\mathcal{O}(-c)\stackrel{{\scriptstyle\rho}}{{\to}}\mathcal{I}_{T}$. Observe that $T$ does not determine $(a,b,c)$. For example a point in $\mathbb{P}^{2}$ is q.c.i. $(a,b,c)$ for any $c\geq b\geq a\geq 1$. However if $n\geq 3$ and if $T$ is locally a complete intersection (l.c.i.), then it is true that $T$ determines $(a,b,c)$, see Proposition 1 of [2].

The kernel of a surjective morphism $\mathcal{O}(-a)\oplus\mathcal{O}(-b)\oplus\mathcal{O}(-c)\stackrel{{\scriptstyle\rho}}{{\to}}\mathcal{I}_{T}$ is a rank two reflexive sheaf (Prop. 1 of [9]), $\mathcal{F}$. Clearly the graded module $H^{0}_{*}(\mathcal{F})$ is the module of syzygies between $F_{a},F_{b}$ and $F_{c}$.

Definition 3**. **

The q.c.i. $T$ (or better the ideal $J=(F_{a},F_{b},F_{c}))$ is said to be an almost complete intersection (a.c.i.) if $\mathcal{F}$ splits: $\mathcal{F}=\mathcal{O}(-p)\oplus\mathcal{O}(-m)$ (i.e. the module $H^{0}_{*}(\mathcal{F})$ is free).

Remark 4**. **

So $J$ is an a.c.i. if and only if it is saturated ($J=J^{sat}=H^{0}_{*}(\mathcal{I}_{T})$).

In terms of $T$ the definition can be unfortunate: one expect a c.i. to be an a.c.i. If $T$ is a point in $\mathbb{P}^{2}$ and if $c\geq b\geq 2$, then $T$ is q.c.i. of type $(a,b,c)$ but the corresponding $\mathcal{F}$ never splits ($J$ is not saturated). So the c.i. $T$ yields a q.c.i. which is not an a.c.i., this can be confusing.

Observe, by the way, that this cannot happen if $n>2$. If $T$ is a c.i. then $\mathcal{F}$ splits by Horrocks’ theorem. (If $n\geq 4$ any q.c.i. $T$ which is integral and subcanonical is a c.i. [1]).

From now on we will assume $n=2$. In this case, since we are on a smooth surface, reflexive implies locally free so $\mathcal{F}$ is a rank two vector bundle.

Lemma 5**. **

Let $J=(F_{a},F_{b},F_{c})\subset S=k[x,y,z]$ be a q.c.i. ideal of type $(a,b,c)$, $a\leq b\leq c$. So we have an exact sequence:

[TABLE]

where $T\subset\mathbb{P}^{2}$ is a closed subscheme of codimension two and where $E$ is a rank two vector bundle.

Then we have $c_{1}(E)=c-a-b$ and $c_{2}(E)=ab-t$, where $t:=\deg(T)=h^{0}(\mathcal{O}_{T})$.

Proof.

If $T\subset\mathbb{P}^{2}$ is zero-dimensional, the Chern classes of $\mathcal{O}_{T}$ are $(0,-t)$. This can be seen by starting with one point (see Section 2 of [9] for similar computations). From the exact sequence $0\to\mathcal{I}_{T}(c)\to\mathcal{O}(c)\to\mathcal{O}_{T}\to 0$, we get that the Chern classes of $\mathcal{I}_{T}(c)$ are $(c,t)$. We conclude with the exact sequence (1). ∎

For later use we have twisted the previous exact sequence by $c$, so $E=\mathcal{F}(c)$ with the previous notations.

Here is the main result:

Theorem 6**. **

Let $J=(F_{a},F_{b},F_{c})\subset S=k[x,y,z]$ be a q.c.i. ideal of type $(a,b,c)$, $a\leq b\leq c$. So we have an exact sequence:

[TABLE]

where $T\subset\mathbb{P}^{2}$ is a closed subscheme of codimension two and where $E$ is a rank two vector bundle.

Set $t=h^{0}(\mathcal{O}_{T})$ and let $r:=min\{k\mid h^{0}(E(k))\neq 0\}$. Then:

(i)

[TABLE]

(ii) If $2r>a+b-c$ then:

[TABLE]

Proof.

(i) Since $\mathcal{I}_{T}(c)$ is generated by global sections, $T$ is contained in a complete intersection $F_{a}\cap G_{c}$ of type $(a,c)$ and we may assume $G_{c}=F_{c}$.

If $T=F_{a}\cap F_{c}$, then $t=ac,r=b-c$ and $E=\mathcal{O}(c-b)\oplus\mathcal{O}(-a)$ and both inequalities are satisfied (in fact they give a single equality !).

So we may assume that $T$ is linked to $\Gamma$ by $F_{a}\cap F_{c}$, where $\gamma:=\deg(\Gamma)=ac-t>0$. Now from the resolution (2), by mapping cone, taking into account that $E^{*}(-a)=E(b-c)$ ($E$ has rank two and $c_{1}(E)=c-a-b$, by Lemma 5), we get, after simplifications:

[TABLE]

Because of the Koszul syzygy: $F_{b}(F_{a})-F_{a}(F_{b})=0$, we have $r\leq a+b-c$. We observe that the inequality $c(a+b-c-r)\leq t$ is clearly satisfied if $r=a+b-c$. Thus from now on we may assume $r<a+b-c\leq a$. The previous exact sequence shows that $h^{0}(\mathcal{I}_{\Gamma}(c-b+r)\neq 0$. So $\Gamma$ is contained in a curve of degree $c-b+r$ and in $F_{a}\cap F_{c}$. We have $c-b+r<a\leq c$. Since $h^{0}(\mathcal{I}_{\Gamma}(c))\geq 2$ and since the base locus of the linear system $|H^{0}(\mathcal{I}_{\Gamma}(c))|$ has dimension zero (because $\dim(F_{a}\cap F_{c})=0$ and $a\leq c$), we conclude that $\Gamma$ is contained in a complete intersection of type $(c-b+r,c)$. It follows that $ac-t=\gamma\leq c(c-b+r)$ and we get the lower bound of (i).

By definition we have $h^{0}(E(r))\neq 0$ and by minimality of $r$, a non zero section of $E(r)$ vanishes in codimension two or doesn’t vanish at all. So we have an exact sequence $0\to\mathcal{O}\to E(r)\to\mathcal{I}_{Z}(c-a-b+2r)\to 0$, where $Z$ is empty or of codimension two, of degree $c_{2}(E(r))$. If $Z$ is empty, then $E(r)=\mathcal{O}\oplus\mathcal{O}(c-a-b+2r)$. In any case $c_{2}(E(r))\geq 0$. From Lemma 5, $c_{2}(E(r))=r(c-a-b)+ab-t+r^{2}$ and we get the upper bound of (i). This concludes the proof of (i).

(ii) Consider again the previous exact sequence $0\to\mathcal{O}\to E(r)\to\mathcal{I}_{Z}(c-a-b+2r)\to 0$. If $Z=\emptyset$, then by minimality of $r$: $c-a-b+2r\leq 0$, against our assumption. So $Z$ is non empty and $0=h^{0}(E(r-1))=h^{0}(\mathcal{I}_{Z}(c-a-b+2r-1)$. This implies: $h^{0}(\mathcal{O}(c-a-b+2r-1))\leq\deg(Z)=c_{2}(E(r))$ and this is the desired inequality.

∎

Remark 7**. **

(i) The bounds in (i) of Theorem 6 are sharp in the sense that both inequalities are equalities if $T$ is a complete intersection $(a,c)$ (then $t=ac,r=b-c$).

(ii) The condition $2r>a+b-c$ is equivalent to require $E$ stable ($E$ is stable if $h^{0}(E_{norm})=0$, where $E_{norm}$ is the twist $E(m)$ such that $-1\leq c_{1}(E(m))\leq 0$).

If $t$ reaches the upper bound in (i) or is right below, we have:

Proposition 8**. **

With notations as in Theorem 6 we have:

(i) If $t=r^{2}+r(c-a-b)+ab$, then $E=\mathcal{O}(-r)\oplus\mathcal{O}(c-a-b+r)$ and $J$ is an a.c.i. (and $2r\leq a+b-c$).

(ii) If $t=r^{2}+r(c-a-b)+ab-1$, there are exact sequences:

[TABLE]

[TABLE]

and $2r\leq a+b-c+1$.

Proof.

(i) If $t=r(c-a-b)+ab+r^{2}$, then $c_{2}(E(r))=0$. Since $h^{0}(E(r))\neq 0$, this implies that $E$ splits ($Z=\emptyset$ with notations as above). From the definition of $r$ we have $E=\mathcal{O}(-r)\oplus\mathcal{O}(c-a-b+r)$ and $r\leq a+b-c-r$.

(ii) We have $c_{2}(E(r))=1$ and we conclude with Lemma 9 below. ∎

Lemma 9**. **

With notations as above assume $c_{2}(E(r))=1$, then we have:

[TABLE]

[TABLE]

Moreover $2r\leq a+b-c+1$.

Proof.

By minimality of $r$, $E(r)$ has a section vanishing along one point: $0\to\mathcal{O}\to E(r)\to\mathcal{I}_{p}(c-a-b+2r)\to 0$. The resolution of $\mathcal{I}_{p}$ yields a surjective morphism: $2.\mathcal{O}(c-a-b+2r-1)\to\mathcal{I}_{p}(c-a-b+2r)$. This morphism can be lifted to $E(r)$ and by completing the diagram we get:

[TABLE]

We observe that $T$ is not a complete intersection $(a,c)$. Indeed if it were we would have $r=c-b$ and $E(r)=\mathcal{O}\oplus\mathcal{O}(b-c-a)$ which has $c_{2}=0$. As seen in the proof of Theorem 6, $T$ is linked to $\Gamma$ by a complete intersection $(a,c)$ and we have $0\to\mathcal{O}(b-a-c)\to E(-c+b)\to\mathcal{I}_{\Gamma}\to 0$. Combining with the resolution of $E$ found before we get:

[TABLE]

Now by mapping cone, using again the complete intersection $(a,c)$, we get the resolution of $\mathcal{I}_{T}$.

The last inequality follows from the definition of $r$ ($h^{0}(E(r-1))=0$). ∎

Remark 10**. **

Of course we have a similar statement if $c_{2}(E(r))=2$ (or if we know the minimal free resolution of $\mathcal{I}_{Z}$).

With our notations we have $a-c\leq r\leq a-c+b$.

Proposition 11**. **

(i) With notations as above, if $r=a-c$, then $T$ is a complete intersection of type $(a,c)$.

(ii) If $r=a-c+1$, then there are three cases:

1. $a=b$, $t=c(a-1)$ and $E(r)$ has a scetion vanishing at one point.

2. $a=b$, $t=c(a-1)+1$ and $E$ splits.

3. $b=a+1$, $t=ac$ and $E(r)$ has a section vanishing at one point.

When $E(r)$ has a section vanishing at one point, we get the resolutions of $E$ and $\mathcal{I}_{T}$ from Lemma 9.

Proof.

(i) We have $0\to E(a-c)\to\mathcal{O}\oplus\mathcal{O}(a-b)\oplus\mathcal{O}(a-c)\to\mathcal{I}_{T}(a)\to 0$, with $h^{0}(E(a-c))\neq 0$. If $b>a$, then $\lambda F_{a}=0$ for some $\lambda\neq 0$, which is absurd. Hence $a=b$. If $c>a$, then $F_{a}$ and $F_{b}$ are linearly dependent and $T=F_{a}\cap F_{c}$. If $a=b=c$, then $F_{a},F_{b},F_{c}$ are linearly dependent, one of them is a linear combination of the other two and $T$ is the complete intersection of these two curves.

(ii) First observe that necessarily $b\leq a+1$. We have $c_{2}(E(a-c+1))=a-b-c+cb+1-t$. From (i) of Theorem 6 we have: $c(a-1)\leq t\leq c(a-1)+1$, if $a=b$ and: $t=ac$ if $b=a+1$.

If $a=b$ and $t=c(a-1)$, then $c_{2}(E(a-c+1))=1$ and $E(r)$ has a section vanishing at one point. The same happens if $b=a+1$, $t=ac$.

If $a=b$ and $t=c(a-1)+1$, then $c_{2}(E(a-c+1))=0$ and $E$ splits. ∎

Now we give criteria for $J$ to be an a.c.i. ideal.

Proposition 12**. **

With the notations of Theorem 6, we have:

(i) $E$ splits if and only if $h^{1}(E(m_{0}))=0$, where $m_{0}=\lfloor\frac{(a+b-c-1)}{2}\rfloor$

(ii) If $H^{0}_{*}(E)$ has two generators of degrees $r_{1},r_{2}$, with $r_{1}+r_{2}\leq a+b-c$, then $E$ splits.

Proof.

(i) This follows from the fact (see [7]) that a normalized rank two vector bundle, $\mathcal{E}$, on $\mathbb{P}^{2}$ splits if and only if $h^{1}(\mathcal{E}(-1))=0$ ($\mathcal{E}$ is normalized if $-1\leq c_{1}(\mathcal{E})\leq 0$).

(ii) Assume $r_{1}\leq r_{2}$. Of course $r\leq r_{1}$. Consider the exact sequence $0\to\mathcal{O}\to E(r)\to\mathcal{I}_{Z}(c-a-b+2r)\to 0$. If $Z=\emptyset$, we are done. Otherwise, since $2r\leq a+b-c$, we have $h^{0}(E(r))=1$. The next generator will come from a section of some twist $\mathcal{I}_{Z}(m)$, $m>0$. The first possibility is $0\to\mathcal{O}(a+b-c-2r+1)\to E(a+b-c-r+1)\to\mathcal{I}_{Z}(1)\to 0$. It follows that $r_{2}\geq-r+a+b-c+1\geq-r_{1}+a+b-c+1$. This implies $r_{1}+r_{2}>a+b-c$, contradiction. Hence $Z=\emptyset$ and $E$ splits. ∎

Remark 13**. **

Condition (ii) can be useful when doing explicit computations. It was first proved in [11] in the case $a=b=c$, by a different method.

3. Global Tjurina number of plane curves.

Let $C\subset\mathbb{P}^{2}$ be a reduced, singular curve, of degree $d$, of equation $f=0$. The partials of $f$ determine an exact sequence:

[TABLE]

The codimension two subscheme $\Sigma$ is the jacobian singular scheme of $C$.

Definition 14**. **

The global Tjurina number of $C$, $\tau(C)$, is defined by $\tau(C):=h^{0}(\mathcal{O}_{\Sigma})$. (If no confusion can arise we will just write $\tau$ instead of $\tau(C)$).

Let $r:=min\{k\mid h^{0}(E_{C}(k))\neq 0\}$. In other words $r$ is the minimal degree of a syzygy between $f_{x},f_{y},f_{z}$.

From Lemma 5 we get:

Lemma 15**. **

With notations as above $c_{1}(E_{C})=-d+1,c_{2}(E_{C})=-\tau+(d-1)^{2}$ ($d=\deg(C)$, $\tau=h^{0}(\mathcal{O}_{\Sigma})$.

Let us first recall the following well known (and easy to prove) fact:

Lemma 16**. **

Let $C\subset\mathbb{P}^{2}$ be a plane curve of equation $f=0$. The partial $f_{x},f_{y},f_{z}$ are linearly dependent if and only if $C$ is a set of lines through a point.

Remark 17**. **

If $C$ is a set of $d$ distinct lines through a point, then $\Sigma$ is the complete intersection of two partials and $\tau(C)=(d-1)^{2}$. Indeed since $C$ is reduced there exist two linearly independent partials and they don’t share any common component. Moreover in this case it is easy to see that $E_{C}=\mathcal{O}\oplus\mathcal{O}(-d+1)$ (combine the exact sequence $0\to\mathcal{O}(-2d+2)\to 2.\mathcal{O}(-d+1)\to\mathcal{I}_{\Sigma}\to 0$ with the exact sequence defining $E_{C}$).

Clearly we have $r\leq d-1$ (Koszul relations). The case $r=0$ is settled by Lemma 16, hence in the sequel we will assume $1\leq r\leq d-1$.

The following definition goes back to Saito [10]:

Definition 18**. **

With notations as above, the divisor $C\subset\mathbb{P}^{2}$ is said to be free if $E_{C}=\mathcal{O}(-a)\oplus\mathcal{O}(-b)$. In this case $(a,b),a\leq b,$ is called the exponent of $C$.

Remark 19**. **

(i) If $E_{C}$ splits then, from the definition of $r$, $E_{C}=\mathcal{O}(-r)\oplus\mathcal{O}(-d+1+r)$. In this case $c_{2}(E_{C})=(d-1)^{2}-\tau=-r(-d+1-r)$. It follows that $\tau=(d-1)(d-1-r)+r^{2}$. If $-r(-d+1-r)=0$, then $\tau=(d-1)^{2}$ and $\Sigma$ is a complete intersection $(d-1,d-1)$. This implies $r=0$ hence $C$ is a set of lines through a point.

(ii) If $a,b$ are two integers such that $a+b=d-1$, then the maximal value of $ab$ is $(d-1)^{2}/4$ if $d$ is odd and $(d-2)d/4$ if $d$ is even. It follows that $(d-1)^{2}-\tau\leq(d-1)^{2}/4$ if $d$ is odd, i.e. $\tau\geq 3(d-1)^{2}/4$; if $d$ is even we get: $\tau\geq 1+\frac{3d(d-2)}{4}$. This has been already observed in [3]. So free curves have a big global Tjurina number (see also the proof of Proposition 28).

(iii) Since a stable vector bundle is indecomposable, if $C$ is free $2r+1\leq d$ and the exponent is $(r,d-1-r)$.

(iv) An important fact about free curves is that they exist! Indeed, as proved in [4], for every $d\geq 3$ and any $r$, $1\leq r<d/2$, there exists a free curve with exponent $(r,d-1-r)$.

4. Bounds on $\tau(C)$ and a theorem of du Plessis-Wall.

It is natural to ask for a bound of $\tau$ in function of $d$ and other invariants of $C$. The following result has been proved by du Plessis and Wall (see Theorem 3.2 of [6]).

Theorem 20**. **

(du Plessis-Wall)

Let $C\subset\mathbb{P}^{2}$ be a reduced, singular curve of degree $d$.

(i) Then:

[TABLE]

(ii) If, moreover, $2r+1>d$, then:

[TABLE]

Proof.

Just put $a=b=c=d-1$ and $t=\tau$ in Theorem 6. ∎

Let us see that the bounds in the first part of the theorem are sharp.

Lemma 21**. **

For every $d\geq 2$, there exists a curve of degree $d$ with $\tau=(d-1)(d-1-r)$.

Proof.

Let $C=X\cup L$ where $X$ is a smooth curve of degree $d-1$ intersecting the line $L$ transversally at $d-1$ distinct points. Clearly $\Sigma=X\cap L$ and $\tau=d-1$. It remains to show that $r=d-2$. We may assume that $L$ has equation $x=0$. Let $g=0$ be an equation of $X$, so that $C$ has equation $f=xg$. Since $\Sigma$ is the complete intersection $X\cap L$, we have $0\to\mathcal{O}(-d)\to\mathcal{O}(-d+1)\oplus\mathcal{O}(-1)\to\mathcal{I}_{\Sigma}\to 0$. We have a commutative diagram:

[TABLE]

where $\varphi$ is given by M=\left(\begin{array}[]{ccc}1&0&0\\ g_{x}&g_{y}&g_{z}\end{array}\right), the expressions of $f_{x},f_{y},f_{z}$ in function of $x,g$. We have $Ker(\varphi)=Ker(\psi)=K$. We observe that $K$ is reflexive ($E_{C}(-d+1)$ is locally free and $Im(\varphi)$ is torsion free), hence locally free. Since $\varphi$, hence also $\psi$ are non zero and since $Im(\psi)$ is torsion free, we conclude that $K$ has rank one, say $K=\mathcal{O}(-c)$. Clearly $c-d+1$ is the least degree of a relation between $g_{y},g_{z}$ (look at $Ker(\varphi)$ and $M$). If we take $g=x^{d-1}+h(y,z)$, $g_{y}=h_{y},g_{z}=h_{z}$, and since $\Sigma$ is smooth, we get $c-d+1=d-2$, hence $c=2d-3$.

The locus where $\varphi$ doesn’t have rank two is defined by the $2\times 2$ minors of $M$, it is the complete intersection $(g_{y})_{0}\cap(g_{z})_{0}$. Since $Coker(\psi)=Coker(\varphi)$, it is also the locus where $\psi$ is not vector-bundle surjective, it is the locus where the section $\mathcal{O}\to E_{C}(c-d+1)$ vanishes. In particular it has codimension two and we have an exact sequence $0\to\mathcal{O}\to E_{C}(-d+1+c)\to\mathcal{I}_{Z}(2c-3(d-1))\to 0$. Since $2c-3(d-1)=d-3$ and since $Z$ is a complete intersection $(d-2,d-2)$, we get $h^{0}(\mathcal{I}_{Z}(2c-3(d-1))=0$, hence $h^{0}(E_{C}(-d+1+c))=1$ and this shows that $r=-d+1+c=d-2$. ∎

Proposition 22**. **

For every $d\geq 3$, there exist $r$ and $\tau$ satisfying the bounds in (i) of Theorem 20.

Proof.

For the bound $(d-1)(d-r-1)\leq\tau$, this follows from Lemma 21. For the bound $\tau\leq(d-1)(d-r-1)+r^{2}$, this follows from [4] (see Remark 19). ∎

Remark 23**. **

For the lower bound we have examples only with $\tau=d-1,r=d-2$. I don’t know if other values are possible.

5. Further topics.

We have the following definition (see [5]):

Definition 24**. **

The curve $C$ is said to be nearly free if we have:

[TABLE]

It can be shown (see [3] or also [7]) that this is equivalent to: $E_{C}(r)$ has a section vanishing at one point (so we are near to the free case where the section doesn’t vanish at all).

If $\tau$ reaches the bound in Theorem 20 (i) or is just below, we have:

Proposition 25**. **

If $\tau=(d-1)(d-r-1)+r^{2}$, then $C$ is free.

If $\tau=(d-1)(d-r-1)+r^{2}-1$, then $C$ is nearly free.

Proof.

Apply Proposition 8 with $a=b=c=d-1$ and $t=\tau$. ∎

The case $r=0$ is completely settled (Lemma 16) and it is natural to investigate the cases where $r$ is small.

Proposition 26**. **

If $r=1$ then: $\tau=(d-1)(d-2)+1$ and $C$ is free, or, $\tau=(d-1)(d-2)$ and $C$ is nearly free.

Proof.

This follows from Proposition 11 (ii). ∎

Remark 27**. **

The last two propositions are already known (except maybe the resolution of $\mathcal{I}_{\Sigma}$), see [3].

Let us conclude with the following:

Proposition 28**. **

Assume $C$ is not a set of lines through a point. Then:

(i) $\tau\leq d^{2}-3d+3$

(ii) Assume $d>7$. If $\tau>d^{2}-4d+5$, then:

(1) $\tau=d^{2}-3d+3$, $r=1$ and $C$ is free.

(2) $\tau=d^{2}-3d+2$, $r=1$ and $C$ is nearly free.

(3) $\tau=d^{2}-4d+7$, $r=2$ and $C$ is free.

(4) $\tau=d^{2}-4d+6$, $r=2$ and $C$ is nearly free.

Proof.

By Theorem 20 we have: $\tau\leq\varphi(r)=r^{2}-r(d-1)+(d-1)^{2}$, if $0\leq r\leq(d-1)/2$ and $\tau\leq\psi(r)=-r^{2}+r(d-2)+d(d-1)/2$, for $(d-1)/2<r\leq d-1$.

The function $\varphi(r)$ on $I=[0,(d-1)/2]$ is decreasing, we have $\varphi(0)=(d-1)^{2}$, $\varphi(1)=d^{2}-3d+3$ and $\varphi(2)=d^{2}-4d+7$. The function $\psi(r)$ reaches its maximal value for $(d-2)/2$ and is decreasing on $[(d-1)/2,d-1]$. It follows that $\psi(r)<\psi(\frac{d-1}{2})=3(d-1)^{2}/4$ for $r\in I_{s}=](d-1)/2,d-1]$.

Observe that $\varphi(\frac{d-1}{2})=\psi(\frac{d-1}{2})=3(d-1)^{2}/4$ (cp. Remark 19).

Since $d^{2}-3d+3\geq 3(d-1)^{2}/4$ for every $d\geq 2$, if $\tau>d^{2}-3d+3$, we have $r\leq(d-1)/2$ and then $r=0$, which is impossible under our assumption. Hence $\tau\leq d^{2}-3d+3$ and this proves (i).

(ii) Observe that $d^{2}-4d+6\geq 3(d-1)^{2}/4$ if $d\geq 7$. So if $\tau\geq d^{2}-4d+6$, we must have $r\leq(d-1)/2$. Moreover since $d^{2}-4d+6>\varphi(3)$, if $d>7$, we have $1\leq r\leq 2$. If $r=1$ we conclude with Proposition 26, obtaining cases (1) and (2).

If $r=2$, by Theorem 20: $(d-1)(d-3)\leq\tau\leq(d-1)(d-3)+4$. We have $c_{2}(E_{C}(2))=(d-1)(d-3)+4-\tau$. Hence if $\tau=d^{2}-4d+7$, $c_{2}(E_{C}(2))=0$ and $E_{C}$ splits. If $\tau=d^{2}-4d+6$, then $c_{2}(E_{C}(2))=1$ hence $E_{C}(r)$ has a section vanishing along one point and $C$ is nearly free. These are cases (3) and (4). ∎

Remark 29**. **

(i) The existence of free and nearly free curves with invariants as in the previous proposition follows from [4].

(ii) For fixed $d$ the possible values of $\tau$ seem sparse.

Let $G(d,3)=1+\frac{d(d-3)-2r(3-r)}{6}$, where $d+r\equiv 0\pmod{3},0\leq r<3$. Then it is known ([8]) that for every $g$, $0\leq g\leq G(d,3)$, there exists a smooth, irreducible, non degenerated curve $X\subset\mathbb{P}^{3}$, of degree $d$, genus $g$. A general projection of $X$ in $\mathbb{P}^{2}$ is a curve of degree $d$ with $\tau=(d-1)(d-2)/2-G(d,3)$ nodes. We conclude that for $(d-1)(d-2)/2-G(d,3)\leq\tau\leq(d-1)(d-2)/2$, there exists a curve, $C$, of degree $d$ with $\tau(C)=\tau$.