---
address:
- |
    Mathematical Institute\
    University of Oslo\
    P. O. Box 1053\
    N–0316 Oslo, Norway
- 'Université Paris 7, UFR de Mathématiques et Institut de Mathématiques de Jussieu, Case Postale 7012, 2, Place Jussieu, F–75251 Paris Cedex 05'
- |
    Mathematical Institute\
    University of Bergen\
    Allég 55\
    N–5007 Bergen, Norway
author:
- Geir Ellingsrud
- Joseph Le Potier
- 'Stein A. Str[ø]{}mme'
title: ' Some Donaldson invariants of $\CC\PP^2$'
---

*In memory of the victims of the Kobe earthquake*

Introduction {#introduction .unnumbered}
============

For an integer $n\ge2$, let $q_{4n-3}$ be the coefficient of the Donaldson polynomial of degree $4n-3$ of $P=\CC\PP^2$. An interpretation of $q_{4n-3}$ in an algebro-geometric context is the following. Let $M_n$ denote the Gieseker-Maruyama moduli space of semistable coherent sheaves on $P$ with rank 2 and Chern classes $c_1=0$ and $c_2=n$. For such a sheaf $F$, the Grauert-Mülich theorem implies that the restriction of $F$ to a general line $L\sub P$ splits as $F_L \iso \OO_L\dsum\OO_L$, and that the exceptional lines form a curve $J(F)$ of degree $n$ in the dual projective plane $P\v$. The association $F\mapsto J(F)$ is induced from a morphism of algebraic varieties, called the Barth map, $f_n\: M_n \to P_n$. Here $P_n=\PP^{n(n+3)/2}$ is the linear system parameterizing all curves of degree $n$ in $P\v$. Let $H\in\Pic(P_n)$ be the hyperplane class and let $\alpha = f_n^*H$. The interpretation of the Donaldson invariant is: $$q_{4n-3} = \int_{M_n} \alpha^{4n-3}.$$ Thus $q_{4n-3}$ is the degree of $f_n$ times the degree of its image. From [@Bart-2] it follows that $f_n$ is generically finite for all $n\ge2$, that $f_2$ is an isomorphism and $q_5=1$, and that $f_3$ is of degree 3 and $q_9=3$. Le Potier [@LePo] proved that $f_4$ is birational onto its image and that $q_{13}=54$. The value of $q_{13}$ has also been computed independently by Tikhomirov and Tyurin [@Tyur-1 prop. 4.1] and by Li and Qin [@Li-Qin thm. 6.29].

The main result in the present note is the following

\[thm1\] $q_{17}=2540$ and $q_{21}=233208$.

The proof consists of two parts. The first part, treated in this note, is to express $q_{4n-3}$ in terms of certain classes on the Hilbert scheme of length-$(n+1)$ subschemes of $P$. This is theorems \[thm2\] and \[thm3\] below. The second part is to evaluate these classes numerically. This has been carried out in [@Elli-Stro-5 prop. 4.2].

Let $H_{n+1}=\Hilb^{n+1}_P$ denote the Hilbert scheme parameterizing closed subschemes of $P$ of length $n+1$. There is a universal closed subscheme $\Z\sub H_{n+1}\x P$. Consider the vector bundles $$\E = R^1{p_1}_* (\I_{\Z}\*{p_2}^* \OO_{P}(-1))\text{ and }
\G = R^1{p_1}_* \I_{\Z}$$ on $H_{n+1}$ of ranks $n+1$ and $n$, respectively, and the linebundle $$\L = \det(\G) \* \det(\E)\i.$$

\[thm2\] Let the notation be as above. Then $$q_{17} = \int_{H_6} s_{12}(\E\*\L) \quad\text{and}\quad
q_{21} = \dfrac25 \int_{H_7} s_{14}(\E\*\L).$$

This result was obtained both by Tikhomirov and Tyurin [@Tyur-Tikh], using the method of “geometric approximation procedure” and by Le Potier [@LePo-3], using “coherent systems”. We present in this note what we believe is a considerably simplified proof, which is strongly hinted at on the last few pages of [@Tyur-Tikh].

The formula for $q_{17}$ is a special case of the following formula:

\[thm3\] For $2\le n\le 5$, we have $$q_{4n-3} =
\dfrac1{2^{5-n}}\int_{H_{n+1}} c_1(\L)^{5-n} s_{3n-3}(\E\*\L).$$

With this it is also easy to recompute $q_5$, $q_9$, and $q_{13}$ using similar techniques as in [@Elli-Stro-5].

We let $h$, $h\v$, and $H$ be the hyperplane classes in $P$, $P\v$, and $P_n$, respectively. In general, if $\omega$ is a divisor class, we denote by $\OO(\omega)$ the corresponding linebundle and its natural pullbacks.

This work is heavily inspired by conversations with A. Tyurin, and we thank him for generously sharing his ideas. We would also like to express our gratitude towards the Taniguchi Foundation.

Hulsbergen sheaves
==================

Barth [@Bart-2] used the term Hulsbergen bundle to denote a stable rank-2 vector bundle $F$ on $P$ with $c_1(F)=0$ and $H^0(P,F(1))\ne0$. We modify this definition a little as follows:

A *Hulsbergen sheaf* is a coherent sheaf $F$ on $P$ which admits a non-split short exact sequence (*Hulsbergen sequence*) $$\label{Hulsbergen}
0 \to \OO_P \to F(1) \to \I_Z(2) \to 0,$$ where $Z\sub P$ is a closed subscheme of finite length (equal to $c_2(F)+1$).

Note that a Hulsbergen sheaf is not necessarily semistable or locally free. However:

\[GM\] Let $F$ be a Hulsbergen sheaf with $c_2(F)=n>0$. Then the set $J(F)\sub P\v$ of exceptional lines for $F$ is a curve of degree $n$, defined by the determinant of the bundle map $$m\: H^1(P,F(-2))\*\OO_{P\v}(-1) \to H^1(P,F(-1))\*\OO_{P\v}$$ induced by multiplication with a variable linear form.

First note from the Hulsbergen sequence that the two cohomology groups have dimension $n$. It is easy to see that any Hulsbergen sheaf is slope semistable, in the sense that it does not contain any rank-1 subsheaf with positive first Chern class. Thus by [@Bart-1 thm. 1], $F_L \iso
\OO_L \dsum \OO_L$ for a general line $L$. On the other hand, it is clear that a line $L$ is exceptional if and only if $m$ is not an isomorphism at the point $[L]\in P\v$.

It is straightforward to construct a moduli space for Hulsbergen sequences. For any length-$(n+1)$ subscheme $Z\sub P$, the isomorphism classes of extensions are parameterized by $\PP(\Ext^1_P(\I_Z(2),\OO_P)\v)$. By Serre duality, $$\Ext^1_P(\I_Z(2),\OO_P)\v \iso H^1(P,\I_Z(-1)).$$ For varying $Z$, these vector spaces glue together to form the vector bundle $\E$ over $H_{n+1}$, hence $D_n=\PP(\E)$ is the natural parameter space for Hulsbergen sequences. Let $\OO(\tau)$ be the associated tautological quotient linebundle. For later use, note that for any divisor class $\omega$ on $H_{n+1}$, we have $\pi_*(\tau+\pi^*\omega)^{k+n} = s_k(\E(\omega))$, where $\pi\: D_n \to H_{n+1}$ is the natural map [@IT].

The tautological quotient $\pi^*\E \to \OO(\tau)$ gives rise to a short exact sequence on $D_n\x P$: $$0 \to \OO(\tau) \to \F(h) \to (\pi\x1)^*\I_{\Z}(2h) \to 0$$ which defines a complete family $\F$ of Hulsbergen sheaves.

As we noted earlier, a Hulsbergen sheaf is not necessarily semistable. On the other hand, the *generic* Hulsbergen sheaf is stable if $n\ge 2$. It follows that the family $\F$ induces a *rational* map $g_n\:
D_n \to M_n$. By [lemma \[GM\]]{} above, there is also a Barth map $b_n\: D_n \to P_n$, defined everywhere, and by construction, the following diagram commutes: $$\begin{CD}
             D_n  @>b_n>> P_n \\
             @V{g_n}VV    @VV{||}V \\
             M_n  @>>f_n> P_n
\end{CD}$$

Put $\lambda=c_1(\pi^*\L)$. Then $b_n^*H = \tau+\lambda$.

Let $L\sub P$ be a line. Twist the universal Hulsbergen sequence by $-2h$ and $-3h$ respectively. Multiplication by an equation for $L$ gives rise to the vertical arrows in a commutative diagram with exact rows on $D_n\x P$: $$\begin{CD}
0 @>>> \OO(\tau-3h) @>>> \F(-2h) @>>> (\pi\x1)^*\I_Z(-h)
@>>>0 \\
@. @VVV @VVV @VVV @.\\
0 @>>> \OO(\tau-2h) @>>> \F(-h) @>>> (\pi\x1)^*\I_Z @>>>0
\end{CD}$$ Pushing this down via the first projection, we get the following exact diagram on $D_n$: $$\begin{CD}
0@>>>R^1{p_1}_* \F(-2h) @>>> \pi^*\E @>>> \OO(\tau)@>>> 0\\
@.    @Vm_LVV                  @VVV        @VVV\\
0@>>>     R^1{p_1}_* \F(-h) @>\iso>> \pi^*\G  @>>> 0
\end{CD}$$ Here the last map of the top row is nothing but the tautological quotient map on $\PP(\E)$. Let $A(L)\sub D_n$ be the set of Hulsbergen sequences where $L$ is an exceptional line for the middle term. Clearly, $A(L)$ is the degeneration locus of the left vertical map $m_L$ above. Hence the divisor class of $A(L)$ is $$\begin{aligned}
[A(L)]&=  c_1(R^1{p_1}_* \F(-h)) - c_1(R^1{p_1}_* \F(-2h)) \\
      &=  \pi^*c_1(\G) - \pi^*c_1(\E) + \tau  \\
      &=  \tau+\lambda.
\end{aligned}$$ On the other hand, $A(L)$ is the inverse image of a hyperplane in $P_n$ under $b_n$, so its divisor class is $b_n^*H$.

The case $n\le 5$
=================

For $2\le n\le 5$, the rational map $g_n$ is dominating, and the general fiber is isomorphic to $\PP^{n-5}$. For $n\ge 5$, the map $g_n$ is generically injective with image of codimension $n-5$. In particular, $g_5$ is birational.

Everything follows from the observation that the fiber over a point $[F]\in M_n$ in the image of $g_n$ is the projectivization of $H^0(P,F(1))$, and that for general such $F$, this vector space has dimension $h^0(F(1))=\max(1,6-n)$, which is easily seen from . The assertion about the codimension follows from a dimension count: $\dim(M_n)=4n-3$ and $\dim(D_n)=3n+2$.

The first half of [theorem \[thm2\]]{} now follows: First of all, since $g_5$ is birational, the two morphisms $f_5$ and $b_5$ have the same image and the same degree. Therefore $q_{17}$ can be computed as $$q_{17} = \int_{D_5} H^{17} =\int_{D_5} (\tau+\lambda)^{17}
       = \int_{H_6} s_{12}(\E\*\L).$$

For [theorem \[thm3\]]{}, let $L_1,\dots,L_{5-n}$ be general lines in $P$, and let $B_n\sub D_n$ be the locus of Hulsbergen sequences where the closed subscheme $Z$ meets all these $5-n$ lines. The cohomology class of $B_n$ in $H^*(D_n)$ is $\lambda^{5-n}$.

\[cover\] Let $2\le n\le5$. The general nonempty fiber of $g_n$ meets $B_n$ in $2^{5-n}$ points, hence the rational map $g_n|_{B_n}\: B_n \to M_n$ is dominating and generically finite, of degree $2^{5-n}$.

The general nonempty fiber is of the form $\PP(H^0(P,F(1))\v)$. It suffices to show that the restriction of $\L$ to this fiber has degree 2 (if $n<5$). For this, it suffices to consider a linear pencil in the fiber. So let $\sigma_0$ and $\sigma_1$ be two independent global sections of $F(1)$, and consider the pencil they span. Now $\sigma_0\wedge \sigma_1 \in
H^0(P,\wedge^2F)=H^0(P,\OO_P(2))$ is the equation of a conic $C\sub P$ which contains the zero scheme $V(t_0\sigma_0 + t_1\sigma_1)$ of each section in the pencil, $(t_0,t_1)\in\PP^1$. Since $C$ meets a general line in two points, it follows that there are exactly two members of the pencil whose zero set meets a general line.

To complete the proof of [theorem \[thm3\]]{}, by [lemma \[cover\]]{} we now have for $2\le n\le5$: $$\begin{aligned}
2^{5-n}\,q_{4n-3} &= 2^{5-n}\int_{M_n} H^{4n-3} \\
        &=\int_{B_n} (\tau+\lambda)^{4n-3} \\
        &=\int_{D_n}
            \lambda^{5-n}\,(\tau+\lambda)^{4n-3} \\
        &=\int_{H_{n+1}}c_1(\L)^{5-n}
        \,s_{3n-3}(\E\*\L).
\end{aligned}$$ This completes the proof of the theorems for $n\le 5$.

The case $n=6$
==============

For $n\ge6$ the techniques above will say something about the restriction of the Barth map to the Brill-Noether locus $B\sub M_n$ of semistable sheaves whose first twist admit a global section. For general $n$ this locus is too small to carry enough information about $M_n$, but in the special case $n=6$, it is actually a divisor, whose divisor class $\beta=[B]$ we can determine. Now $\Pic(M_n)\*\QQ$ has rank 2, generated by $\alpha$ and $\delta=[\Delta]$, the class of the locus $\Delta\sub M_n$ corresponding to non-locally free sheaves [@LePo-1].

In $\Pic(M_6)\*\QQ$, the following relation holds: $$\beta = \frac52 \,\alpha - \frac12\,\delta.$$

Let $\xi\:X\to M_6$ be a morphism induced by a flat family $\F$ of semistable sheaves on $P$, parameterized by some variety $X$. For certain divisor classes $a$ and $d$ on $X$, the second and third Chern classes of $\F$ can be written in the form $$c_2(\F) = a\,h+6\,h^2, \quad c_3(\F) = d\,h^2$$ modulo higher codimension classes on $X$. The Grothendieck Riemann-Roch theorem for the projection $p\: X\x P \to X$ easily gives (for example using [@schubert]) that $$-c_1(p_!\F(h)) = \frac52\, a- \frac12\, d.$$ The locus $\xi\i B\sub X$ is set-theoretically the support of $R^1 p_*\F(h)$. It is not hard to see that one can take the family $X$ in such a way that the 0-th Fitting ideal of $R^1 p_*\F(h)$ is actually reduced. Therefore the left hand side of the equation above is $\xi^*\beta$. On the other hand, $a=\xi^*\alpha$ by the usual definition of the $\mu$ map of Donaldson [@Dona-1], and $d=\xi^*\delta$. Since the family $\F/X$ was arbitrary, the required relation is actually universal, and so holds also in $\Pic(M_6)\*\QQ$. (It suffices to take a family with the properties that (i) $\xi^*\:\Pic_\QQ(M_6)
\to \Pic_\QQ(X)$ is injective, (ii) the Fitting ideal above is reduced, and (iii) the general non-locally free sheaf in the family has colength 1 in its double dual.)

With this, we complete the proof of the second part of [theorem \[thm2\]]{} in the following way. The general fiber of $f_6$ restricted to $\Delta$ has dimension 1, so $f_6(\Delta)$ has dimension 19, see e.g. [@Stro-1]. Therefore we get $$\begin{aligned}
\int_{H_7} s_{14}(\E\*\L) &= \int_{D_6}(\lambda+\tau)^{20} \\
&= \int_{M_6} \beta\, \alpha^{20} \\
&= \int_{M_6} (\frac52\, \alpha - \frac12\,\delta)\,\alpha^{20} \\
&= \frac52\int_{M_6} \alpha^{21} -\frac12\int_{\Delta}\alpha^{20}
= \frac52\, q_{21}.
\end{aligned}$$

A geometric interpretation
==========================

A *Darboux configuration* in $P\v$ consists of a pair $(\Pi,C)$ where $\Pi\sub P\v$ is the union of $n+1$ distinct lines, no three concurrent, and $C\sub P\v$ is a curve of degree $n$ passing through all the nodes of $\Pi$.

If we let $Z\sub P$ consist of the $n+1$ points dual to the components of $\Pi$, we have by Hulsbergen’s theorem [@Bart-2 thm. 4] a natural 1-1 correspondence between Hulsbergen sequences and Darboux configurations $(\Pi,C)$, by letting $C=J(F)$. Therefore $D_n$ can be used as a compactification of the set of Darboux configurations, and the intersection number $$\int_{D_n} \lambda^i (\tau+\lambda)^{3n+2-i} =
\int_{H_{n+1}} c_1(\L)^i s_{2n+2-i}(\E\*\L)$$ can be interpreted as the number of Darboux configurations $(\Pi,C)$ where $\Pi$ passes through $i$ given points and $C$ passes through $3n+2-i$ given points.

It is not known whether the Barth map has degree 1 for $n\ge5$. A related question is the following: Let $(\Pi,C)$ be a general Darboux configuration ($n\ge 5$). Is the inscribed polygon $\Pi$ uniquely determined by $C$?

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