On complete intersections containing a linear subspace

TL;DR

This paper studies the geometry of complete intersections containing linear subspaces, showing that for certain cases the locus is irreducible, of codimension t, and the Fano scheme is zero-dimensional with length one, implying rationality.

Contribution

It extends known results to the case t > 0, proving irreducibility, codimension, and rationality of the locus of complete intersections containing linear subspaces.

Findings

The locus W_{d,k} is irreducible and of codimension t.

For general Y in W_{d,k}, the Fano scheme F_k(Y) is zero-dimensional with length one.

The locus W_{d,k} is rational.

Abstract

Consider the Fano scheme $F_k(Y)$ parameterizing $k$-dimensional linear subspaces contained in a complete intersection $Y \subset \mathbb{P}^m$ of multi-degree $\underline{d} = (d_1, \ldots, d_s)$. It is known that, if $t := \sum_{i=1}^s \binom{d_i +k}{k}-(k+1) (m-k)\leqslant 0$ and $\Pi_{i=1}^sd_i >2$, for $Y$ a general complete intersection as above, then $F_k(Y)$ has dimension $-t$. In this paper we consider the case $t> 0$. Then the locus $W_{\underline{d},k}$ of all complete intersections as above containing a $k$-dimensional linear subspace is irreducible and turns out to have codimension $t$ in the parameter space of all complete intersections with the given multi-degree. Moreover, we prove that for general $[Y]\in W_{\underline{d},k}$ the scheme $F_k(Y)$ is zero-dimensional of length one. This implies that $W_{\underline{d},k}$ is rational.

Equations

$$W_{\underline{d},k}:=\left\{u\in S_{\underline{d}}^{\ast }|F_k(Y_u)\ne \varnothing \right\}\subseteq S_{\underline{d}}^{\ast }$$

$$t(m,k,\underline{d}):=\sum _{i=1}^s\left(\genfrac{}{}{0ex}{}{d_i+k}{k}\right)-(k+1)(m-k).$$

$$J:=\left\{\left(\left[\Pi \right],u\right)\in \mathbb{G}\times S_{\underline{d}}^{\ast }|\Pi \subset Y_u\right\}\subset \mathbb{G}\times S_{\underline{d}}^{\ast }$$

$$\mathbb{G}\stackrel{{\pi }_1}{⟵}J\stackrel{{\pi }_2}{⟶}{S}_{\underline{d}}^{\ast }.$$

$$0\to \underset{i=1}{\overset{s}{⨁}}{\mathcal{I}}_{\Pi /{\mathbb{P}}^m}(d_i)\to \underset{i=1}{\overset{s}{⨁}}{\mathcal{O}}_{{\mathbb{P}}^m}(d_i)\to \underset{i=1}{\overset{s}{⨁}}{\mathcal{O}}_{\Pi }(d_i)\cong \underset{i=1}{\overset{s}{⨁}}{\mathcal{O}}_{{\mathbb{P}}^k}(d_i)\to 0,$$

$$\dim (J)=(k+1)(m-k)+\sum _{i=1}^s\left(\genfrac{}{}{0ex}{}{d_i+m}{m}\right)-\sum _{i=1}^s\left(\genfrac{}{}{0ex}{}{d_i+k}{k}\right)=\dim (S_{\underline{d}}^{\ast })-t.$$

$$g_i=\sum _{h=k+1}^m{y}_h{p}_i^{(h)}+r_i,1⩽i⩽s,$$

$$r_i\in (I_{\Pi }^2{)}_{d_i}\mathrm{whereas}{p}_i^{(h)}=\sum _{|\underline{\mu }|=d_i-1}{c}_{i,\underline{\mu }}^{(h)}{\underline{y}}^{\underline{\mu }}\in \mathbb{C}[y_0,y_1,\dots ,y_k{]}_{d_i-1},1⩽i⩽s,k+1⩽h⩽m,$$

$$M=(0\mathbf{P})\text{where}\mathbf{P}:=(p_i^{(h)}{)}_{1⩽i⩽s;k+1⩽h⩽m}$$

$$0\to N_{\Pi /Y_u}\to N_{\Pi /{\mathbb{P}}^m}\cong {\mathcal{O}}_{{\mathbb{P}}^k}(1{)}^{\oplus (m-k)}\to {N_{Y_u/{\mathbb{P}}^m}|}_{\Pi }\cong \underset{i=1}{\overset{s}{⨁}}{\mathcal{O}}_{{\mathbb{P}}^k}(d_i)$$

$${\phi }_h^{\xi }(\underline{y}):=a_{h,0}{y}_0+a_{h,1}{y}_1+\cdots +a_{h,k}{y}_k,k+1⩽h⩽m,$$

$$A_{\xi }:=(a_{h,j}),k+1⩽h⩽m,0⩽j⩽k;$$

$$A_{\xi }\stackrel{\sigma }{⟶}{\left(\sum _{0⩽j⩽k<h⩽m}{a}_{h,j}{y}_j{p}_i^{(h)}\right)}_{1⩽i⩽s}.$$

$$(k+1)(m-k)=h^0\left(\Pi ,N_{\Pi /{\mathbb{P}}^m}\right)<h^0\left(\Pi ,{N_{Y_u/{\mathbb{P}}^m}|}_{\Pi }\right)=\sum _{i=1}^s\left(\genfrac{}{}{0ex}{}{d_i+k}{k}\right).$$

$$\sum _{h=k+1}^m\sum _{j=0}^k{a}_{h,j}{y}_j\left(\sum _{|\underline{\mu }|=d_i-1}{c}_{i,\underline{\mu }}^{(h)}{\underline{y}}^{\underline{\mu }}\right),1⩽i⩽s.$$

$$\sum _{0⩽j⩽k<h⩽m}{c}_{i,\underline{\nu }-{\underline{e}}_j}^{(h)}{a}_{h,j}=0,1⩽i⩽s,$$

$$(h,i,\underline{\mu }),\text{mbox}wherek+1⩽h⩽m,|\underline{\mu }|=d_i-1,1⩽i⩽s,$$

$${\delta }_h(m,k,\underline{d}):=\sum _{i=1}^s\left(\genfrac{}{}{0ex}{}{d_i+k}{k}\right)-\sum _{i=1}^s\left(\genfrac{}{}{0ex}{}{d_i+h}{h}\right)-(k-h)(m+h+1-k).$$

$$t(m,k,\underline{d})⩽0.$$

$$t⩽-\sum _{i=1}^s\left[\frac{h+1}{k-h}\left(\genfrac{}{}{0ex}{}{d_i+k}{k}\right)-\frac{k+1}{k-h}\left(\genfrac{}{}{0ex}{}{d_i+h}{h}\right)\right]+(k+1)(h+1).$$

$$t⩽-\sum _{i=1}^{s}D(d_i)+(k+1)(h+1).$$

$$\begin{split}D(d_{i})&=\frac{(h+1)(d_{i}+1)\cdots(d_{i}+h)}{k!(k-h)}\bigg{(}(d_{i}+h+1)\cdots(d_{i}+k)-(k+1)k\cdots(h+2)\bigg{)},\;1{\leqslant}i{\leqslant}s.\end{split}$$

$$D(2)=\frac{h+1}{k-h}\left(\genfrac{}{}{0ex}{}{k+2}{k}\right)-\frac{k+1}{k-h}\left(\genfrac{}{}{0ex}{}{h+2}{h}\right)=\frac{(h+1)(k+1)}{2}>0.$$

$$t{\leqslant}-\sum_{i=1}^{s}D(d_{i})+(k+1)(h+1){\leqslant}-s\;D(2)+(k+1)(h+1)=(k+1)(h+1)\bigg{(}1-\frac{s}{2}\bigg{)}.$$

$$D(3)=(k+1)(h+1)\frac{k+h+5}{6}$$

$$t⩽-D(d)+(k+1)(h+1)⩽-D(3)+(k+1)(h+1)=(k+1)(h+1)\frac{1-k-h}{6}.$$

$$I:=\left\{\left(\left[{\Pi }_1\right],\left[{\Pi }_2\right],u\right)\in \mathbb{G}\times \mathbb{G}\times S_{\underline{d}}^{\ast }|{\Pi }_1\ne {\Pi }_2,{\Pi }_i\subset Y_u,1⩽i⩽2\right\}\subset \mathbb{G}\times \mathbb{G}\times S_{\underline{d}}^{\ast }.$$

$$\phi \left(\left(\left[{\Pi }_1\right],\left[{\Pi }_2\right],u\right)\right)=\left(\left[{\Pi }_1\right],u\right).$$

$$I_h:=\left\{\left(\left[{\Pi }_1\right],\left[{\Pi }_2\right],u\right)\in I|{\Pi }_1\cap {\Pi }_2\cong {\mathbb{P}}^h\right\},\mathrm{where}-1⩽h⩽k-1$$

$${\mathbb{G}}_h^2:=\left\{\left(\left[{\Pi }_1\right],\left[{\Pi }_2\right])\in \mathbb{G}\times \mathbb{G}\right|{\Pi }_1\cap {\Pi }_2\cong {\mathbb{P}}^h\right\}\subset \mathbb{G}\times \mathbb{G},$$

Full text

This collaboration has benefitted of funding from the research project “Families of curves: their moduli and their related varieties” (CUP: E81-18000100005) - Mission Sustainability - University of Rome Tor Vergata.

1. Introduction

In this paper we will be concerned with the Fano scheme $F_{k}(Y)$, parameterizing $k$–dimensional linear subspaces contained in a subvariety $Y\subset\mathbb{P}^{m}$, when $Y$ is a complete intersection of multi–degree $\underline{d}=(d_{1},\ldots,d_{s})$, with $1\leqslant s\leqslant m-2$. We will assume that $Y$ is neither a linear subspace nor a quadric, cases to be considered as trivial. Thus we will constantly assume that $\Pi_{i=1}^{s}\;d_{i}>2$.

Let $S_{\underline{d}}:=\bigoplus_{i=1}^{s}H^{0}\left(\mathbb{P}^{m},\mathcal{O}_{\mathbb{P}^{m}}(d_{i})\right)$, and consider its Zariski open subset S^{*}_{\underline{d}}:=\bigoplus_{i=1}^{s}\big{(}H^{0}\left(\mathbb{P}^{m},\mathcal{O}_{\mathbb{P}^{m}}(d_{i})\right)\setminus\{0\}\big{)}. For any $u:=(g_{1},\ldots,g_{s})\in S^{*}_{\underline{d}}$, let $Y_{u}:=V(g_{1},\ldots,g_{s})\subset\mathbb{P}^{m}$ denote the closed subscheme defined by the vanishing of the polynomials $g_{1},\ldots,g_{s}$. When $u\in S^{*}_{\underline{d}}$ is general, $Y_{u}$ is a smooth, irreducible variety of dimension $m-s\geqslant 2$. For any integer $k{\geqslant}1$, we define the locus

[TABLE]

and set

[TABLE]

If no confusion arises, we will simply denote $t(m,k,\underline{d})$ by $t$.

First of all, consider the case $t{\leqslant}0$. This is the most studied case in the literature, and it is now well understood (cf. e.g. [2, 3, 6, 7]). In particular, the following holds.

Result 1**.**

Let $m,k,s$ and $\underline{d}=(d_{1},\dots,d_{s})$ be such that $\Pi_{i=1}^{s}d_{i}>2$ and $t{\leqslant}0$. Then:

• (a)

$W_{\underline{d},k}=S_{\underline{d}}^{*}$;

• (b)

for general $u\in S_{\underline{d}}^{*}$, $F_{k}(Y_{u})$ is smooth, of dimension $\dim(F_{k}(Y_{u}))=-t$ and it is irreducible when $\dim(F_{k}(Y_{u})){\geqslant}1$.

The proof of this result can be found e.g. in [2, Prop.2.1, Cor.2.2, Thm. 4.1], for the complex case, and in [3, Thm. 2.1, (b) & (c)], for any algebraically closed field. In addition, in [3, Thm. 4.3] the authors compute $\deg(F_{k}(Y_{u}))$ under the Plücker embedding $F_{k}(Y_{u})\subset\mathbb{G}(k,m)\hookrightarrow\mathbb{P}^{N}$, with $N={{m+1}\choose{k+1}}-1$. Their formulas extend to any $k{\geqslant}1$ enumerative formulas by Libgober in [4], who computed $\deg(F_{1}(Y_{u}))$ when $t(m,1,\underline{d})=0$.

On the other hand, we are interested in the case $t>0$, where the known results can be summarized as follows.

Result 2**.**

Let $m,k,s$ and $\underline{d}=(d_{1},\dots,d_{s})$ be such that $\Pi_{i=1}^{s}d_{i}>2$ and $t>0$. Then:

• (a)

$W_{\underline{d},k}\subsetneq S_{\underline{d}}^{*}$.

• (b)

$W_{\underline{d},k}$* contains points $u$ for which $Y_{u}\subset\mathbb{P}^{m}$ is a smooth complete intersection of dimension $m-s$ if and only if $s{\leqslant}m-2k$.*

• (c)

For $s{\leqslant}m-2k$, set $H_{\underline{d},k}:=\left\{\left.u\in W_{\underline{d},k}\right|Y_{u}\subset\mathbb{P}^{m}\text{\emph{ is smooth, of dimension }}m-s\right\}$. If $d_{i}{\geqslant}2$ for any $1{\leqslant}i{\leqslant}s$, then $H_{\underline{d},k}$ is irreducible, unirational and $\operatorname{codim}_{S^{*}_{\underline{d}}}(H_{\underline{d},k})=t$. Moreover, for general $u\in H_{\underline{d},k}$, $F_{k}(Y_{u})$ is a zero–dimensional scheme.

The proof of Result 2 (a) is contained in [3, Thm. 2.1 (a)], whereas that of assertions (b) and (c) is contained in [5, Cor. 1.2, Rem. 3.4]; both proofs therein hold for any algebraically closed field.

The main result of this paper, which improves on Result 2, is the following.

Theorem 1.1**.**

Let $m,k,s$ and $\underline{d}=(d_{1},\dots,d_{s})$ be such that $\Pi_{i=1}^{s}d_{i}>2$ and $t>0$. Then $W_{\underline{d},k}\subsetneq S^{*}_{\underline{d}}$ is non–empty, irreducible and rational, with $\operatorname{codim}_{S^{*}_{\underline{d}}}(W_{\underline{d},k})=t$. Furthermore, for a general point $u\in W_{\underline{d},k}$, the variety $Y_{u}\subset\mathbb{P}^{m}$ is a complete intersection of dimension $m-s$ whose Fano scheme $F_{k}(Y_{u})$ is a zero–dimensional scheme of length one. Moreover, $Y_{u}$ has singular locus of dimension $\max\{-1,2k+s-m-1\}$ along its unique $k$–dimensional linear subspace (in particular $Y_{u}$ is smooth if and only if $m-s{\geqslant}2k$).

The proof of this theorem is contained in Section 2 and it extends [1, Prop. 2.3] to arbitrary $k{\geqslant}1$. Theorem 1.1 improves, via different and easier methods, Miyazaki’s results in [5, Cor. 1.2], showing that for general $u\in W_{\underline{d},k}$ one has $\deg(F_{k}(Y_{u}))=1$, which implies the rationality of $W_{\underline{d},k}$. Moreover we also get rid of Miyazaki’s hypothesis $m-s{\geqslant}2k$.

2. The proof

This section is devoted to the proof of Theorem 1.1.

Proof of Theorem 1.1.

Let $\mathbb{G}:=\mathbb{G}(k,m)$ be the Grassmannian of $k$-linear subspaces in $\mathbb{P}^{m}$ and consider the incidence correspondence

[TABLE]

with the two projections

[TABLE]

The map $\pi_{1}\colon J\to\mathbb{G}$ is surjective and, for any $[\Pi]\in\mathbb{G}$, one has \pi_{1}^{-1}\left(\left[\Pi\right]\right)=\bigoplus_{i=1}^{s}\big{(}H^{0}\left(\mathcal{I}_{\Pi/\mathbb{P}^{m}}(d_{i})\right)\setminus\{0\}\big{)}, where $\mathcal{I}_{\Pi/\mathbb{P}^{m}}$ denotes the ideal sheaf of $\Pi$ in $\mathbb{P}^{m}$.

Thus $J$ is irreducible with $\dim(J)=\dim(\mathbb{G})+\dim(\pi_{1}^{-1}\left(\left[\Pi\right]\right))=(k+1)(m-k)+\sum_{i=1}^{s}h^{0}\left(\mathcal{I}_{\Pi/\mathbb{P}^{m}}(d_{i})\right)$. From the exact sequence

[TABLE]

one gets

[TABLE]

The next step recovers [5, Cor. 1.2] via different and easier methods, and we also get rid of the hypothesis $m-s{\geqslant}2k$ present there. We essentially adapt the argument in [2, Proof of Prop. 2.1], used for the case $t{\leqslant}0$.

Step 1**.**

*The map $\pi_{2}\colon J\to S^{*}_{\underline{d}}$ is generically finite onto its image $W_{\underline{d},k}$, which is therefore irreducible and unirational. Moreover $\operatorname{codim}_{S^{*}_{\underline{d}}}(W_{\underline{d},k})=t$.

For general $u\in W_{\underline{d},k}$, $F_{k}(Y_{u})$ is a zero–dimensional scheme and $Y_{u}$ has singular locus of dimension $\max\{-1,2k+s-m-1\}$ along any of the $k$–dimensional linear subspaces in $F_{k}(Y_{u})$.*

Proof of Step 1.

One has $W_{\underline{d},k}=\pi_{2}(J)$, hence $W_{\underline{d},k}\subsetneq S_{\underline{d}}^{*}$ is irreducible and unirational, because $J$ is rational, being an open dense subset of a vector bundle over $\mathbb{G}$. Once one shows that $\pi_{2}\colon J\to W_{\underline{d},k}$ is generically finite, one deduces that $\operatorname{codim}_{S^{*}_{\underline{d}}}(W_{\underline{d},k})=t$ from (2.2). Therefore, we focus on proving that $\pi_{2}$ is generically finite, i.e. that if $u\in W_{\underline{d},k}$ is a general point, then $\dim(\pi_{2}^{-1}(u))=0$.

Let $[\Pi]\in\mathbb{G}$ and choose $[y_{0},y_{1},\ldots,y_{m}]$ homogeneous coordinates in $\mathbb{P}^{m}$ such that the ideal of $\Pi$ is $I_{\Pi}:=(y_{k+1},\dots,y_{m})$. For general $\left(\left[\Pi\right],u\right)\in\pi_{1}^{-1}\left(\left[\Pi\right]\right)\subset J$, with $u=(g_{1},\ldots,g_{s})\in W_{\underline{d},k}$, we can write

[TABLE]

with

[TABLE]

where $(I^{2}_{\Pi})_{d_{i}}$ is the homogenous component of degree $d_{i}$ of the ideal $I^{2}_{\Pi}$, $\underline{\mu}:=(\mu_{0},\cdots,\mu_{k})\in\mathbb{Z}^{k+1}_{\geqslant 0}$, $|\underline{\mu}|:=\sum_{r=0}^{k}\mu_{r}$, and $\underline{y}^{\underline{\mu}}:=y_{0}^{\mu_{0}}y_{1}^{\mu_{1}}\cdots y_{k}^{\mu_{k}}$. By the generality assumption on $u$, the polynomials $p_{i}^{(h)}$ and $r_{i}$ are general.

The Jacobian matrix $(\frac{\partial g_{i}}{\partial y_{j}})_{1{\leqslant}i{\leqslant}s;0{\leqslant}j{\leqslant}m}$ computed along $\Pi$ takes the block form

[TABLE]

where the $\bf 0$–block has size $s\times(k+1)$ and ${\bf P}$ has size $s\times(m-k)$, where $m-k{\geqslant}s$ because of course $\dim(Y_{u})=m-s{\geqslant}k$. By the generality of the polynomials $p_{i}^{(h)}$, the locus of $\Pi$ where ${\rm rk}(M)<s$, which coincides with the singular locus of $Y_{u}$ along $\Pi$, has dimension $\max\{-1,2k+s-m-1\}$ and, by Bertini’s theorem, it coincides with the singular locus of $Y_{u}$.

Next we consider the following exact sequence of normal sheaves

[TABLE]

(see [8, Lemma 68.5.6]). Any $\xi\in H^{0}(\Pi,N_{\Pi/\mathbb{P}^{m}})$ can be identified with a collection of $m-k$ linear forms on $\Pi\cong\mathbb{P}^{k}$

[TABLE]

whose coefficients fill up the $(m-k)\times(k+1)$ matrix

[TABLE]

by abusing notation, one may identify $\xi$ with $A_{\xi}$.

Thus the map $H^{0}\left(\Pi,N_{\Pi/\mathbb{P}^{m}}\right)\stackrel{{\scriptstyle\sigma}}{{\longrightarrow}}H^{0}\left(\Pi,\left.N_{Y_{u}/\mathbb{P}^{m}}\right|_{\Pi}\right)$, arising from (2.4) is given by (cf. e.g. [2, formula (4)])

[TABLE]

Notice that the assumption $t>0$ reads as

[TABLE]

Claim 2.1**.**

The map $H^{0}\left(\Pi,N_{\Pi/\mathbb{P}^{m}}\right)\stackrel{{\scriptstyle\sigma}}{{\longrightarrow}}H^{0}\left(\Pi,\left.N_{Y_{u}/\mathbb{P}^{m}}\right|_{\Pi}\right)$ is injective, equivalently $h^{0}(N_{\Pi/{Y_{u}}})=0$. In particular, for a general point $u\in W_{\underline{d},k}$, the Fano scheme $F_{k}({Y_{u}})$ contains $\{[\Pi]\}$ as a zero–dimensional integral component.

Proof of Claim 2.1.

Using (2.3), the polynomials on the right–hand–side of (2.5) read as

[TABLE]

Ordering the previous polynomial expressions via the standard lexicographical monomial order on the canonical basis $\{\underline{y}^{\underline{\mu}}\}$ of $\mathbb{C}[y_{0},y_{1},\ldots,y_{k}]_{d_{i}}=H^{0}(\mathcal{O}_{\mathbb{P}^{k}}(d_{i})),\;\;1{\leqslant}i{\leqslant}s$, the injectivity of the map $\sigma$ is equivalent for the homogeneous linear system

[TABLE]

to have only the trivial solution, where $\underline{\nu}:=(\nu_{0},\nu_{1},\ldots,\nu_{k})\in\mathbb{Z}^{k+1}_{\geqslant 0}$ is such that $|\underline{\nu}|=d_{i}$, $\underline{e}_{j}$ is the $(j+1)$–th vertex of the standard $(k+1)$-simplex in $\mathbb{Z}^{k+1}_{\geqslant 0}\setminus\{\underline{0}\}$, and $c^{(h)}_{i,\underline{\nu}-\underline{e}_{j}}=0$ when $\underline{\nu}-\underline{e}_{j}\notin\mathbb{Z}^{k+1}_{\geqslant 0}$ (this last condition stands for “$\underline{\nu}-\underline{e}_{j}$ improper” as formulated in [2, p.  29]). The linear system (2.6) consists of $\sum_{i=1}^{s}\binom{d_{i}+k}{k}$ equations in the $(k+1)(m-k)$ indeterminates $a_{h,j}$, with coefficients $c^{(h)}_{i,\underline{\mu}}$, $0{\leqslant}j{\leqslant}k<h{\leqslant}m$.

Let $C:=(c^{(h)}_{i,\underline{\nu}-\underline{e}_{j}})$ be the coefficient matrix of (2.6); one is reduced to show that, for general choices of the entries $c^{(h)}_{i,\underline{\nu}-\underline{e}_{j}}$, the matrix $C$ has maximal rank $(k+1)(m-k)$. This can be done arguing as in [2, p. 29]. Namely, row–indices of $C$ are determined by the standard lexicographical monomial order on the canonical basis of $\bigoplus_{i=1}^{s}\mathbb{C}[y_{0},y_{1},\ldots,y_{k}]_{d_{i}}$, whereas column–indices of $C$ are determined by the standard lexicographic order on the set of indices $(h,j)$. If one considers the square sub–matrix $\widehat{C}$ of $C$ formed by the first $(k+1)(m-k)$ rows and by all the columns of $C$, then $\det(\widehat{C})$ is a non–zero polynomial in the indeterminates $c^{(h)}_{i,\underline{\mu}}$. Indeed, take the lexicographic order on the set of indices

[TABLE]

and order the monomials appearing in the expression of $\det(\widehat{C})$ according to the following rule: the monomials $m_{1}$ and $m_{2}$ are such that $m_{1}>m_{2}$ if, considering the smallest index $(h,\;i,\;\underline{\mu})$ for which $c^{(h)}_{i,\underline{\mu}}$ occurs in the monomial $m_{1}$ with exponent $p_{1}$ and in the monomial $m_{2}$ with exponent $p_{2}\neq p_{1}$, one has $p_{1}>p_{2}$. The greatest monomial (in the monomial ordering described above) appearing in $\det(\widehat{C})$ has coefficient $\pm\;1$, since in each column the choice of the $c^{(h)}_{i,\underline{\mu}}$ entering in this monomial is uniquely determined. By maximality of such monomial, it follows that $\det(\widehat{C})\neq 0$, which shows that $C$ has maximal rank $(k+1)(m-k)$, i.e. the map $\sigma$ is injective.

The injectivity of $\sigma$ and (2.4) yield $h^{0}(N_{\Pi/{Y_{u}}})=0$. Since $H^{0}(N_{\Pi/{Y_{u}}})$ is the tangent space to $F_{k}(Y_{u})$ at its point $[\Pi]$, one deduces that $\{[\Pi]\}$ is a zero–dimensional, reduced component of $F_{k}(Y_{u})$, as claimed. ∎

Finally, by monodromy arguments, the irreducibility of $J$ and Claim 2.1 ensure that for general $u\in W_{\underline{d},k}$, the Fano scheme $F_{k}(Y_{u})$ is zero-dimensional and reduced, i.e. $\pi_{2}\colon J\to W_{\underline{d},k}$ is generically finite, and that $Y_{u}$ has a singular locus of dimension $\max\{-1,2k+s-m-1\}$ along any of the $k$–dimensional linear subspaces in $F_{k}(Y_{u})$. This completes the proof of Step 1. ∎

To conclude the proof of Theorem 1.1, we need the following numerical result.

Step 2**.**

For $0{\leqslant}h{\leqslant}k-1$ integers, consider the integer

[TABLE]

If $\delta_{h}(m,k,\underline{d})\leqslant 0$, then

[TABLE]

Proof of Step 2.

In order to ease notation, we set $\delta_{h}:=\delta_{h}(m,k,\underline{d})$. Therefore, the condition $\delta_{h}\leqslant 0$ implies $m{\geqslant}\frac{1}{k-h}\left[\sum_{i=1}^{s}\binom{d_{i}+k}{k}-\binom{d_{i}+h}{h}\right]-(h+1-k)$. Plugging the previous inequality in the expression of $t$, one has

[TABLE]

Set $D(x):=\frac{h+1}{k-h}\binom{x+k}{k}-\frac{k+1}{k-h}\binom{x+h}{h}.$ Thus, (2.7) reads

[TABLE]

The assumption $0{\leqslant}h{\leqslant}k-1$ gives

[TABLE]

The polynomial $D(x)$ vanishes for $x=1$, which is its only positive root. Notice that

[TABLE]

In particular, $D(x)$ is increasing and positive for $x>1$, so from (2.8) it follows that

[TABLE]

Therefore, when $s{\geqslant}2$, we have $t{\leqslant}0$ and we are done in this case.

If $s=1$, set $d:=d_{1}$. In this case (2.8) is $t{\leqslant}-D(d)+(k+1)(h+1)$, where again $D(d)$ is increasing and positive for $d>1$. When $s=1$, we have $d{\geqslant}3$ by assumption. Thus, one computes

[TABLE]

and so, for any $d{\geqslant}3$, one has

[TABLE]

Being $0{\leqslant}h{\leqslant}k-1$, one deduces that $t{\leqslant}0$, completing the proof of Step 2. ∎

The final step of the proof of Theorem 1.1 is the following.

Step 3**.**

For general $u\in W_{\underline{d},k}$, the zero–dimensional Fano scheme $F_{k}(Y_{u})$ has length one. In particular, the map $\pi_{2}\colon J\to W_{\underline{d},k}$ is birational and $W_{\underline{d},k}$ is rational.

Proof of Step 3.

Let us consider the (locally closed) incidence correspondence

[TABLE]

If $I$ is not empty, let $\varphi\colon I\to J$ be the map defined by

[TABLE]

We need to prove that $\varphi$ is not dominant. To do this, consider the (locally closed) subset

[TABLE]

(we set $\mathbb{P}^{-1}=\emptyset$, i.e. the case $h=-1$ occurs when $\Pi_{1}$ and $\Pi_{2}$ are skew). Clearly, one has $I=\bigsqcup_{h=-1}^{k-1}I_{h}$. Setting $\varphi_{h}:=\varphi_{|I_{h}}$, it is sufficient to prove that $\varphi_{h}$ is not dominant, for any $-1{\leqslant}h{\leqslant}k-1$.

So, let $h$ be such that $I_{h}$ is not empty, and let $T_{h}$ be an irreducible component of $I_{h}$. Of course, if $\dim(T_{h})<\dim(J)$, the restriction ${\varphi_{h|T_{h}}}\colon T_{h}\to J$ is not dominant. On the other hand, suppose that $\dim(T_{h})>\dim(J)$. For any such a component, the map ${\varphi_{h|T_{h}}}$ cannot be dominant, otherwise the composition $T_{h}\stackrel{{\scriptstyle{\varphi_{h|T_{h}}}}}{{\longrightarrow}}J\stackrel{{\scriptstyle\pi_{2}}}{{\longrightarrow}}W_{\underline{d},k}$ would be dominant, as $\pi_{2}$ is, which would imply that the general fiber of $\pi_{2}$ is positive dimensional, contradicting Step 1.

Therefore, it remains to investigate the case $\dim(T_{h})=\dim(J)$. We estimate the dimension of $T_{h}$ as follows. Consider

[TABLE]

which is locally closed in $\mathbb{G}\times\mathbb{G}$. The projection

[TABLE]

is surjective onto $\mathbb{G}$ and any $\widehat{\pi}_{1}$–fiber is irreducible, of dimension equal to $\dim\left(\mathbb{G}(h,k)\times\mathbb{G}(k-h-1,m-h-1)\right)=(h+1)(k-h)+(k-h)(m-k)$. Thus

[TABLE]

One has the projection

[TABLE]

which is surjective, because the projective group acts transitively on $\mathbb{G}^{2}_{h}$. Hence $\dim(T_{h})=\dim(\mathbb{G}^{2}_{h})+\dim(\mathfrak{F_{h}})$, where \mathfrak{F_{h}}:=\bigoplus_{i=1}^{s}\Big{(}H^{0}\left(\mathcal{I}_{\Pi_{1}\cup\Pi_{2}/\mathbb{P}^{m}}(d_{i})\right)\setminus\{0\}\Big{)} is the general fiber of ${\psi_{h|T_{h}}}$ and where $\mathcal{I}_{\Pi_{1}\cup\Pi_{2}/\mathbb{P}^{m}}$ denotes the ideal sheaf of $\Pi_{1}\cup\Pi_{2}$ in $\mathbb{P}^{m}$.

Claim 2.2**.**

For every positive integer $d$ one has

[TABLE]

Proof of Claim 2.2.

We have

[TABLE]

Consider the linear system $\Sigma$ cut out on $\Pi_{2}$ by $|\mathcal{I}_{\Pi_{1}/\mathbb{P}^{m}}(d)|$. We claim that $\Sigma$ is the complete linear system of hypersurfaces of degree $d$ of $\Pi_{2}$ containing $\Pi:=\Pi_{1}\cap\Pi_{2}$. Indeed $\Sigma$ contains all hypersurfaces consisting of a hyperplane through $\Pi$ plus a hypersurface of degree $d-1$ of $\Pi_{2}$, which proves our claim. In the light of this fact, and arguing as in (2.1) and (2.2), we deduce that

[TABLE]

which, by (2.9), yields the assertion. ∎

By Claim 2.2 we have

[TABLE]

Hence

[TABLE]

Since $\dim(T_{h})=\dim(J)$, (2.10) implies $\delta_{h}=0$. When $0{\leqslant}h{\leqslant}k-1$, Step 2 gives $t{\leqslant}0$, contrary to our assumption. When $h=-1$, one has $0=\delta_{-1}=t$, again against our assumptions.

Since no component $T_{h}\subset I_{h}$ can dominate $J$, the map $\varphi\colon I\to J$ is not dominant. We conclude therefore that the map $\pi_{2}\colon J\to W_{\underline{d},k}$ is birational, completing the proof of Step 3. ∎

Steps 1–3 prove Theorem 1.1. ∎

Acknowledgements

We would like to thank Enrico Fatighenti and Francesco Russo for helpful discussions.