Log canonical thresholds of certain Fano hypersurfaces

Abstract

We study log canonical thresholds on quartic threefolds, quintic fourfolds, and double spaces. As an important application, we show that they have Kähler–Einstein metrics if they are general.

Appendix

Let $X_F$ be a smooth quintic hypersurface in $\mathbb{P }^{5}$ that is given by zeroes of a section $F\in H^{0}(\mathbb{P }^5,\mathcal{O }_{\mathbb{P }^{5}}(5))$. It follows from Proposition 2.4 that there exists a non-empty Zariski open subset $U_{G1}\in H^{0}(\mathbb{P }^5,\mathcal{O }_{\mathbb{P }^{5}}(5))$ such that $X_F$ is 4-regular whenever $F\in U_{G1}$. Similarly, it follows from Lemma 2.14 that there exists a non-empty Zariski open subset $U_{G2}\in H^{0}(\mathbb{P }^5,\mathcal{O }_{\mathbb{P }^{5}}(5))$ such that for every 3-dimensional linear space $\Pi$ in $\mathbb{P }^5$, the intersection $X_F\cap \Pi$ is irreducible and reduced if $F\in U_{G2}$.

The purpose of this Appendix is to prove Lemma 2.15, i.e., to prove the existence of a non-empty Zariski open subset $U_{G3}\in H^{0}(\mathbb{P }^5,\mathcal{O }_{\mathbb{P }^{5}}(5))$ such that for each $F\in U_{G3}$ the hypersurface $X_F$ satisfies the condition G 3 (see Sect. 2.3). Indeed, we prove the statement as follows:

For each $a (=0,1,2,3)$ and $b (=1,2,\ldots ,6)$, there exists a non-empty Zariski open subset $U$ in $H^{0}(\mathbb{P }^5,\mathcal{O }_{\mathbb{P }^{5}}(5))$ such that if $F\in U$, then for each point $P\in X$ and each 3-dimensional linear space $\Pi _3$ contained in the tangent hyperplane at $P$ and containing the point $P$, the surface $Z:=X\cap \Pi _3$ satisfies the condition G 3.$a$.$b$.

Since we use the same method in order to prove the statement for each $a$ and $b$, we first explain how the proof goes and then show the required computations in each case G 3.$a$.$b$.

The proof goes as follows.

First we consider the space

$$\begin{aligned} \mathcal{S }=\mathcal{F }\times H^{0}\left( \mathbb{P }^{5}, \mathcal{O }_{\mathbb{P }^{5}}\left( 5\right) \right) \end{aligned}$$

with the natural projections $p:\mathcal{S }\rightarrow H^{0}(\mathbb{P }^5,\mathcal{O }_{\mathbb{P }^{5}}(5))$ and $q:\mathcal{S }\rightarrow \mathcal{F }$. Here, $\mathcal{F }$ is a suitable flag variety in $\mathbb{P }^5$. Depending on the case, the flag $\mathcal{F }$ will be $Flag(0,1,2,3,4),\,Flag(0,2,3,$ $4),\,Flag(0,1,3,4)$ or $Flag(0,3,4)$, where $Flag(n_1,\ldots , n_k)$ is the flag variety that parametrizes $k$-tuples $(\Pi _{n_1},\ldots , \Pi _{n_k})$ of $n_i$-dimensional linear spaces with $\Pi _{n_1}\subset \cdots \subset \Pi _{n_k}\subset \mathbb{P }^5$. A $0$-dimensional linear space will be denoted by $P$ and a four dimensional linear space will be denoted by $T$.

We then put

$$\begin{aligned} \mathcal{I }=\left\{ \left( (P,\Pi _{n_2},\ldots ,\Pi _{n_{k-1}},T), F\right) \in \mathcal{S }\quad \left| \, \begin{array}{l} F(P)=0;\\ T\,\text{ is } \text{ the } \text{ tangent } \text{ hyperplane } \text{ to }\,X_F\,\text{ at }\,P;\\ X_F\,\text{ satisfies } \text{ the } \text{ properties }\,\mathcal{P }_{G3.a.b}. \end{array}\right. \!\!\right\} , \end{aligned}$$

where the properties $\mathcal{P }_{G3.a.b}$ will be specified in the individual proofs. Then in each case, we will see that it is easy to check that the morphism $q |_\mathcal{I } :\mathcal{I }\rightarrow \mathcal{F }$ is surjective.

With this set up, we compute the codimension $c$ of $q |_\mathcal{I }^{-1}(P,\Pi _{n_2},\ldots ,\Pi _{n_{k-1}},T)$ in $H^{0}(\mathbb{P }^5,\mathcal{O }_{\mathbb{P }^{5}}(5))$ for a point $(P,\Pi _{n_2},\ldots ,\Pi _{n_{k-1}},T)\in \mathcal{F }$. We may always assume that $T$ is defined by $x=0,\,\Pi _3$ is defined by $x=y=0,\,\Pi _2$ by $x=y=z=0,\,\Pi _1$ by $x=y=z=u=0$ and $P=[0:0:0:0:0:1]$. We write the quintic polynomial $F$ as

$$\begin{aligned}&w^5q_0+w^{4}q_{1}\left( x,y,z,u,v\right) +w^{3}q_{2} \left( x,y,z,u,v\right) +w^{2}q_{3}\left( x,y,z,u,v\right) \\&\quad +wq_{4}\left( x,y,z,u,v\right) +q_{5}\left( x,y,z,u,v\right) , \end{aligned}$$

where $q_{i}$ is a homogeneous polynomial of degree $i$.

The condition $F(P)=0$ is equivalent to $q_0=0$. The condition that $T$ is the tangent hyperplane to $X_F$ at $P$ is equivalent to $q_1=\lambda x$ for some $\lambda \in \mathbb{C }^*$. These two conditions contribute to the codimension $c$ by 5. For each $a$ and $b$, we will show that that the properties $\mathcal{P }_{G3.a.b}$ makes another contribution to the codimension $c$ by more than $\dim \mathcal{F }-5$.

These altogether show that the codimension of $q |_\mathcal{I }^{-1}(P,\Pi _{n_2},\ldots ,\Pi _{n_{k-1}}, T)$ in $H^{0}(\mathbb{P }^5,$ $\mathcal{O }_{\mathbb{P }^{5}}(5))$ is more than $\dim \mathcal{F }$. These implies that the morphism $p|_{\mathcal{I }}$ cannot be surjective. Taking the properties $\mathcal{P }_{G3.a.b}$ into consideration, we can immediately notice that this non-surjectivity implies the statement.

Therefore, to prove the statement for each case, it is enough to

• specify the flag $\mathcal{F }$ with its dimension;

• specify the property $\mathcal{P }_{G3.a.b}$;

• show that the properties $\mathcal{P }_{G3.a.b}$ makes another contribution to the codimension $c$ by more than $\dim \mathcal{F }-5$.

Now we do these jobs for each case.

Lemma 1

The statement holds for G 3.0.1.

Proof

The flag $\mathcal{F }$ is $Flag(0,1,3,4)$. It is of dimension 14. Put

$$\begin{aligned} \mathcal{P }_{G3.0.1}=\left\{ X_F\cap \Pi _3\,\text{ is } \text{ singular } \text{ along }\,\Pi _2.\right\} . \end{aligned}$$

The condition that $X_F\cap \Pi _3$ contains the line $\Pi _1$ is equivalent to the condition that for each $i=2,3,4,5$, the polynomial $q_i$ contains no $v^i$. For $X_F\cap \Pi _3$ in order to be singular along $L$, for each $i=2,3,4,5$, the polynomial $q_i$ must not contain the monomials $zu^rv^{i-r-1},\,r=0,1,\ldots , i-1$. These altogether show that the properties $\mathcal{P }_{G3.0.1}$ is of codimension $> 9$. $\square$

Lemma 2

The statement holds for G 3.0.2.

Proof

The flag $\mathcal{F }$ is $Flag(0,3,4)$. It is of dimension 12. Put

$$\begin{aligned} \mathcal{P }_{G3.0.2}=\left\{ X_F\cap \Pi _3 \text{ contains } \text{ four } \text{ lines. }\right\} . \end{aligned}$$

Since we may assume that $q_1, q_2, q_3, q_4$ forms a regular sequence, $X_F\cap \Pi _3$ containing four lines is equivalent to $q_5(x,y,z,u,v)$ vanishing at four given points in $q_1(x,y,z,u,v)=q_2(x,y,z,u,v)=q_3(x,y,z,u,v)=q_4(x,y,z,u,v)=0$ in $\mathbb{P }^4$ and $q_4(0,0,z,u,v)$ vanishing at four given points in $q_2(0,0,z,u,v)=q_3(0,0,z,u,v)=0$ in $\mathbb{P }^2$. These altogether show that the properties $\mathcal{P }_{G3.0.2}$ is of codimension 8. $\square$

Lemma 3

The statement holds for G 3.1.1.

Proof

The flag $\mathcal{F }$ is $Flag(0,1,3,4)$. It is of dimension 14.

Put

$$\begin{aligned} \mathcal{P }_{G3.1.1}=\left\{ \begin{array}{l} X_F\cap \Pi _3 \text{ contains } \Pi _1;\\ \Pi _1 \text{ meets } \text{ its } \text{ residual } \text{ curve } \text{ by } \text{ a } \text{ general } \text{ hyperplane }\\ \text{ section } \text{ of }\,X_F\cap \Pi _3\,\text{ in }\,\Pi _3\,\text{ only } \text{ at } \text{ singular } \text{ points; }\\ X_F\cap \Pi _3 \text{ has } \text{ at } \text{ most } \text{ one } \text{ ordinary } \text{ double } \text{ point } \text{ on }\,\Pi _1.\\ \end{array}\!\right\} . \end{aligned}$$

We write

$$\begin{aligned} q_i(0,0,z,u,v)=\sum _{r+s+t=i}A_{rst}z^ru^sv^t, \end{aligned}$$

where $A_{rst}$’s are constants.

The condition that $X_F\cap \Pi _3$ contains the line $\Pi _1$ is equivalent to $A_{00t}=0$ for $t=2,\,3,\,4$ and 5 since the line $\Pi _1$ is defined by $x=y=z=u=0$.

The surface $X_F\cap \Pi _3$ has singular points on the line $\Pi _1$ exactly where the polynomials $A_{101}vw^3+A_{102}v^2w^2+A_{103}v^3w+A_{104}v^4$ and $A_{011}vw^3+A_{012}v^2w^2+A_{013}v^3w+A_{014}v^4$ have common zeros in $\mathbb{P }^1$. The zero given by $v=0$ corresponds to the singular point $P$. To see this, put $\bar{F}(z,u,v,w)=F(0,0,z,u,v,w)$. Since $A_{00t}=0$ for $t=2,\,3,\,4$ and 5, we always have $\frac{\partial \bar{F}}{\partial v}(0,0,v,w)=\frac{\partial \bar{F}}{\partial w}(0,0,v,w)=0$. The common zeros of

$$\begin{aligned} \frac{\partial \bar{F}}{\partial z}(0,0,v,w)&= v(A_{101}w^3+A_{102}vw^2+A_{103}v^2w+A_{104}v^3), \\ \frac{\partial \bar{F}}{\partial u}(0,0,v,w)&= v(A_{011}w^3+A_{012}vw^2+A_{013}v^2w+A_{014}v^3) \end{aligned}$$

are the singular points of $X_F\cap \Pi _3$ on the line $\Pi _1$. Note that $\Pi _1$ and its residual curve by a general hyperplane meet at every singular point of $X_F\cap \Pi _3$ on the line $\Pi _1$. Therefore, the second condition is equivalent to the condition that the polynomials $A_{101}vw^3+A_{102}v^2w^2+A_{103}v^3w+A_{104}v^4$ and $A_{011}vw^3+A_{012}v^2w^2+A_{013}v^3w+A_{014}v^4$ have four common zeros in $\mathbb{P }^1$ with counting multiplicity, i.e., these two polynomials are proportional. This imposes three additional independent conditions on the coefficients of $F$.

The condition that the polynomial $A_{101}vw^3+A_{102}v^2w^2+A_{103}v^3w+A_{104}v^4$ has $k$ zeros without counting multiplicity imposes $4-k$ additional independent conditions on the coefficients of $F$. Note that $1\le k\le 4$.

We claim that the last condition imposes $k-1$ independent conditions on the coefficients of $F$. Here we verify the claim only for the case with $k=4$. The other cases with $k=3$ and 2 can be verified in the same way.

We write the homogenized Hessian matrix of the polynomial $q_2(0,0,z,u,v)+q_3(0,0,z,u,v)+q_4(0,0,z,u,v)+q_5(0,0,z,u,v)$ along the line $\Pi _1$ as follows:

Let $H(v,w)$ be the determinant of the homogenized Hessian matrix. The condition that three of the four singular points on $\Pi _1$ is not ordinary double points is equivalent to the condition that $H(v,w)$ vanishes at three points out of the four points defined by $A_{101}w^3v+A_{102}v^2w^2+A_{103}v^3w+A_{104}v^4=0$ and $A_{011}vw^3+A_{012}v^2w^2+A_{013}v^3w+A_{014}v^4=0$ in $\mathbb{P }^1$. We claim that it imposes three additional independent conditions on the coefficients of $F$. To verify the claim, we put

$$\begin{aligned} A_{110}&= 0, \quad A_{111}=0, \quad A_{112}=0, \quad A_{113}=0, \quad A_{102}=0, \quad A_{103}=0 \\ A_{012}&= 0, \quad A_{013}=0 \quad A_{201}=0, \quad A_{202}=0, \quad A_{021}=0, \quad A_{022}=0. \end{aligned}$$

Since $A_{101}w^3v+A_{104}v^4=0$ and $A_{011}vw^3+A_{014}v^4=0$ defines four points in $\mathbb{P }^1$, we have $[\lambda :\mu ]\in \mathbb{P }^1$ with $\lambda (A_{101}, A_{104})=\mu (A_{011}, A_{014})$. We then see that in our restricted situation, the condition is equivalent to the condition that $A_{101}w^3v+A_{104}v^4=0$ has three common points with

$$\begin{aligned} \left( A_{101}w^3+4A_{104}v^3\right) ^2 \left\{ \lambda ^2\left( A_{200}w^3+A_{203}v^3\right) +\mu ^2 \left( A_{020}w^3+A_{023}v^3\right) \right\} =0 \end{aligned}$$

in $\mathbb{P }^1$. Since this is a condition of codimension 3 in the restricted situation, it verifies the claim.

These altogether show that the properties $\mathcal{P }_{G3.1.1}$ is of codimension $> 9$. $\square$

Lemma 4

The statement holds for G 3.2.1.

Proof

The flag $\mathcal{F }$ is $Flag(0,1,2,3,4)$. It is of dimension 15. Put

$$\begin{aligned} \mathcal{P }_{G3.2.1}\!=\!\left\{ \begin{array}{l} X_F\,\text{ contains }\,\Pi _1; \quad X_F\cap \Pi _2\,\text{ contains } \text{ a } \text{ line } \text{ other } \text{ than }\,\Pi _1\,\text{ passing } \text{ through }\,P;\\ X_F\cap \Pi _3\,\text{ contains } \text{ four } \text{ singular } \text{ points } \text{ other } \text{ than }\,P\,\text{ on } \text{ the } \text{ two } \text{ lines } \text{ on }\,X\cap \Pi _2\\ \text{ passing } \text{ through } \text{ the } \text{ point }\,P.\\ \end{array}\right\} . \end{aligned}$$

The condition that $X_F$ contains the line $\Pi _1$ is equivalent to the condition that for each $i=2,3,4,5$, the polynomial $q_i$ contains no $v^i$. The condition that $X_F\cap \Pi _2$ contains a line other than $\Pi _1$ passing through $P$ is equivalent to the condition that $q_3(0,0,0, u,v),\,q_4(0,0,0,u,v)$ and $q_5(0,0,0,u,v)$ vanish at the point other than the point given by $u=0$ in $\mathbb{P }^1$ where $q_2(0,0,0,u,v)$ vanishes. For $X_F\cap \Pi _3$ in order to have four singular points other than $P$ on the two lines on $X_F\cap \Pi _2$ passing through the point $P$ is a condition of codimension 4. These altogether show that the properties $\mathcal{P }_{G3.2.1}$ is of codimension 11. $\square$

Lemma 5

The statement holds for G 3.2.2.

Proof

The flag $\mathcal{F }$ is $Flag(0,1,2,3,4)$. It is of dimension 15. Put

$$\begin{aligned} \mathcal{P }_{G3.2.2}=\left\{ \begin{array}{l} X_F\,\text{ contains }\,\Pi _1;\quad X_F\cap \Pi _2\,\text{ contains } \text{ two } \text{ lines } \text{ passing } \text{ through }\,P;\\ X_F\cap \Pi _3\,\text{ has } \text{ three } \text{ singular } \text{ points } \text{ other } \text{ than }\,P\,\text{ on }\,\Pi _1;\\ X_F\cap \Pi _3\,\text{ has } \text{ at } \text{ least } \text{ one } \text{ singular } \text{ point } \text{ on }\,\Pi _1\,\text{ that } \text{ is } \text{ not } \text{ an } \text{ ordinary } \text{ double } \text{ point. } \end{array}\right\} . \end{aligned}$$

The condition that $\Pi _1\subset X_F$ is equivalent to the fact that each $q_{i}(x,y,z,u,v)$ does not have $v^i$ monomial, which is condition of codimension 4. The condition that $X_F\cap \Pi _2$ contains another line passing through the point $P$ is equivalent to the condition that either $q_3(0,0,0,u,v),\,q_4(0,0,0,u,v)$ and $q_5(0,0,0,u,v)$ vanish at the points in $\mathbb{P }^1$ where $q_2(0,0,0,u,v)/u$ vanishes, or $q_2(0,0,0,u,v)$ is a zero polynomial and $q_3(0,0,0,u,v),\,q_4(0,0,0,u,v)$ and $q_5(0,0,0,u,v)$ have common root in $\mathbb{P }^1$. Thus, the condition that $X_F\cap \Pi _2$ contains another line passing through the point $P$ is a condition of codimension 3. For the surface $X_F\cap \Pi _3$ to have three singular points on $\Pi _1$ other than $P$ is a condition of codimension 3. Arguing as in the proof of Lemma 3, we can see that the condition that one of the singular points of $X_F\cap \Pi _3$ on $\Pi _1$ is not an ordinary double point is a condition of codimension 1. These altogether show that the properties $\mathcal{P }_{G3.2.2}$ is of codimension $> 10$. $\square$

Lemma 6

The statement holds for G 3.2.3.

Proof

The flag $\mathcal{F }$ is $Flag(0,1,2,3,4)$. It is of dimension 15. Put

$$\begin{aligned} \mathcal{P }_{G3.2.3}=\left\{ \begin{array}{l} X_F\,\text{ contains }\,\Pi _1;\\ X_F\cap \Pi _2\,\text{ contains } \text{ a } \text{ line } \text{ other } \text{ than }\,\Pi _1\,\text{ passing } \text{ through }\,P;\\ X_F\cap \Pi _3\,\text{ contains } \text{ two } \text{ non-ordinary } \text{ singular } \text{ points } \text{ on }\,\Pi _1.\\ \end{array}\right\} . \end{aligned}$$

The condition that $X_F$ contains the line $\Pi _1$ is equivalent to the condition that for each $i=2,3,4,5$, the polynomial $q_i$ contains no $v^i$. The condition that $X_F\cap \Pi _2$ contains a line other than $\Pi _1$ passing through $P$ is equivalent to the condition that $q_3(0,0,0, u,v),\,q_4(0,0,0,u,v)$ and $q_5(0,0,0,u,v)$ vanish at the point other than the point given by $u=0$ in $\mathbb{P }^1$ where $q_2(0,0,0,u,v)$ vanishes. As in the proof of Lemma 3, we can see that for $X_F\cap \Pi _3$ to have two non-ordinary singular points on $\Pi _1$ is a condition of codimension 4. These altogether show that the properties $\mathcal{P }_{G3.2.3}$ is of codimension $> 10$. $\square$

Lemma 7

The statement holds for G 3.2.4.

Proof

The flag $\mathcal{F }$ is $Flag(0,1,2,3,4)$. It is of dimension 15. Put

$$\begin{aligned} \mathcal{P }_{G3.2.4}=\left\{ \begin{array}{l} X_F\,\text{ contains }\,\Pi _1; \quad X_F\cap \Pi _2 \text{ contains } \text{ a } \text{ line } \text{ other } \text{ than }\,\Pi _1\,\text{ passing } \text{ through }\,P;\\ X_F\cap \Pi _3\,\text{ contains } \text{ two } \text{ singular } \text{ points } \text{ other } \text{ than }\,P\,\text{ on } \text{ the } \text{ line }\,\Pi _1;\\ \Pi _1\,\text{ meets } \text{ its } \text{ residual } \text{ curve } \text{ by } \text{ a } \text{ general } \text{ hyperplane } \text{ section } \text{ of }\,X_F\cap \Pi _3\,\text{ in }\,\Pi _3\\ \text{ only } \text{ at } \text{ three } \text{ points. }\\ \end{array}\right\} . \end{aligned}$$

We write

$$\begin{aligned} q_i(0,0,z,u,v)=\sum _{r+s+t=i}A_{rst}z^ru^sv^t, \end{aligned}$$

where $A_{rst}$’s are constants.

The condition that $X_F\cap \Pi _3$ contains the line $\Pi _1$ is equivalent to $A_{00t}=0$ for $t=2,\,3,\,4$ and 5. The condition that $X_F\cap \Pi _2$ contains a line other than $\Pi _1$ passing through $P$ is equivalent to the condition that $q_3(0,0,0, u,v),\,q_4(0,0,0,u,v)$ and $q_5(0,0,0,u,v)$ vanish at the point other than the point given by $u=0$ in $\mathbb{P }^1$ where $q_2(0,0,0,u,v)$ vanishes. For $X_F\cap \Pi _3$ in order to have two singular points other than $P$ on the line $\Pi _1$ and for $\Pi _1$ to meet its residual curve by a general hyperplane section of $X_F\cap \Pi _3$ in $\Pi _3$ only at three point are equivalent to the condition that the polynomials $A_{101}vw^3+A_{102}v^2w^2+A_{103}v^3w+A_{104}v^4$ and $A_{011}vw^3+A_{012}v^2w^2+A_{013}v^3w+A_{014}v^4$ have four common zeros in $\mathbb{P }^1$ with counting multiplicity and the polynomial $A_{101}vw^3+A_{102}v^2w^2+A_{103}v^3w+A_{104}v^4$ has three zeros without counting multiplicity. This condition is of codimention 4. These altogether show that the properties $\mathcal{P }_{G3.2.4}$ is of codimension 11. $\square$

Lemma 8

The statement holds for G 3.2.5.

Proof

The flag $\mathcal{F }$ is $Flag(0,1,2,3,4)$. It is of dimension 15. Put

$$\begin{aligned} \mathcal{P }_{G3.2.5}=\left\{ \begin{array}{l} X_F\,\text{ contains }\,\Pi _1; \quad X_F\cap \Pi _2\,\text{ contains } \text{ a } \text{ line } \text{ other } \text{ than }\,\Pi _1\,\text{ passing } \text{ through }\,P;\\ X_F\cap \Pi _3\,\text{ contains } \text{ one } \text{ singular } \text{ points } \text{ other } \text{ than }\,P\,\text{ on } \text{ the } \text{ line }\,\Pi _1;\\ \text{ either }\,\Pi _1\,\text{ meets } \text{ its } \text{ residual } \text{ curve } \text{ by } \text{ a } \text{ general } \text{ hyperplane } \text{ section } \text{ in }\,\Pi _3\\ \text{ only } \text{ at } \text{ singular } \text{ points } \text{ or }\\ \Pi _1\,\text{ meets } \text{ its } \text{ residual } \text{ curve } \text{ by } \text{ a } \text{ general } \text{ hyperplane } \text{ section } \text{ in }\,\Pi _3\\ \text{ only } \text{ at } \text{ three } \text{ points } \text{ and }\,X_F\cap \Pi _3\,\text{ has } \text{ a } \text{ non-ordinary } \text{ singular } \text{ point }\,P.\\ \end{array}\right\} . \end{aligned}$$

The first two conditions imposes seven independent conditions on the coefficients of $F$ as before. For the surface $X_F\cap \Pi _3$ to have a singular point on $\Pi _1$ other than $P$ and plus for $\Pi _1$ to meet its residual curve by a general hyperplane section in $\Pi _3$ only at singular points impose at least four independent conditions on the coefficients of $F$. Meanwhile, for the surface $X_F\cap \Pi _3$ to have a singular point on $\Pi _1$ other than $P$ and plus for $\Pi _1$ to meet its residual curve by a general hyperplane section in $\Pi _3$ only at three points impose at least four independent conditions on the coefficients of $F$. However, the condition that $X_F\cap \Pi _3$ has a non-ordinary singular point $P$ is of codimension 1. Therefore, the properties $\mathcal{P }_{G3.2.5}$ is of codimension 11.

These altogether show that the properties $\mathcal{P }_{G3.2.5}$ is of codimension 11. $\square$

Lemma 9

The statement holds for G 3.2.6.

Proof

The flag $\mathcal{F }$ is $Flag(0,1,2,3,4)$. It is of dimension 15. Put

$$\begin{aligned} \mathcal{P }_{G3.2.6}=\left\{ \begin{array}{l} X_F\,\text{ contains }\,\Pi _1; \quad X_F\cap \Pi _2\,\text{ contains } \text{ a } \text{ line } \text{ other } \text{ than }\,\Pi _1\,\text{ passing } \text{ through }\,P;\\ \Pi _1\,\text{ meets } \text{ its } \text{ residual } \text{ curve } \text{ by } \text{ a } \text{ general } \text{ hyperplane } \text{ section } \text{ of }\,X_F\cap \Pi _3\,\text{ in }\,\Pi _3\\ \text{ at } \text{ at } \text{ most } \text{ two } \text{ points. }\\ \end{array}\right\} . \end{aligned}$$

The first two conditions imposes seven independent conditions on the coefficients of $F$ as before.

For the last condition, we write

$$\begin{aligned} q_i(0,0,z,u,v)=\sum _{r+s+t=i}A_{rst}z^ru^sv^t, \end{aligned}$$

where $A_{rst}$’s are constants.

The last condition is equivalent to the condition that either the polynomials $A_{101}vw^3+A_{102}v^2w^2+A_{103}v^3w+A_{104}v^4$ and $A_{011}vw^3+A_{012}v^2w^2+A_{013}v^3w+A_{014}v^4$ have a common zero at $v=0$ with multiplicity at least 3 or they are proportional and have only two zeros (without counting multiplicities). The former and the latter are both a condition of codimension at least 4.

Therefore, the properties $\mathcal{P }_{G3.2.6}$ is of codimension at least 11. $\square$

Lemma 10

The statement holds for G 3.3.1.

Proof

The flag $\mathcal{F }$ is $Flag(0,2,3,4)$. It is of dimension 14. Put

$$\begin{aligned} \mathcal{P }_{G3.3.1}{=}\left\{ \begin{array}{l} X_F\cap \Pi _2\,\text{ contains } \text{ three } \text{ lines } \text{ passing } \text{ through }\,P;\\ \text{ either }\,X_F\cap \Pi _3\,\text{ has } \text{ a } \text{ singular } \text{ point } \text{ on }\,\Pi _2\,\text{ other } \text{ than }\,P\,\text{ or } \text{ one } \text{ of } \text{ the } \text{ lines }\,L_i\,\text{ meets }\\ \text{ its } \text{ residual } \text{ curve } \text{ by } \text{ a } \text{ general } \text{ hyperplane } \text{ section } \text{ in }\,\Pi _3\,\text{ at } \text{ at } \text{ most } \text{ three } \text{ points. }\\ \end{array}\right\} . \end{aligned}$$

The condition that $X_F\cap \Pi _2$ contains three lines passing through the point $P$ is equivalent to the condition that $q_2(0,0,0,u,v)$ is identically zero; $q_4(0,0,0,u,v)$ and $q_5(0,0,0,u,v)$ vanish at the three points in $\mathbb{P }^1$ where $q_3(0,0,0,u,v)$ vanishes. For the surface $X_F\cap \Pi _3$ to have a singular point on $\Pi _2$ other than $P$ is a condition of codimension 1. For one of the lines $L_i$ to meet its residual curve by a general hyperplane section in $\Pi _3$ at at most three points is also a condition of codimension 1. These altogether show that the properties $\mathcal{P }_{G3.3.1}$ is of codimension $> 9$. $\square$

Lemma 11

The statement holds for G 3.3.2.

Proof

The flag $\mathcal{F }$ is $Flag(0,3,4)$. It is of dimension 12. Put

$$\begin{aligned} \mathcal{P }_{G3.3.2}=\left\{ \begin{array}{l} X_F\cap \Pi _3\,\text{ contains } \text{ three } \text{ lines } \text{ passing } \text{ through }\,P;\\ \text{ either } \text{ one } \text{ of } \text{ the } \text{ lines }\,L_i\,\text{ meets } \text{ its } \text{ residual } \text{ curve } \text{ by } \text{ a } \text{ general } \text{ hyperplane } \text{ section }\\ \text{ in }\,\Pi _3\,\text{ at } \text{ at } \text{ most } \text{ one } \text{ smooth } \text{ point } \text{ or }\\ \text{ one } \text{ of } \text{ the } \text{ lines }\,L_i\,\text{ meets } \text{ its } \text{ residual } \text{ curve } \text{ by } \text{ a } \text{ general } \text{ hyperplane } \text{ section } \text{ in }\,\Pi _3\\ \text{ at } \text{ at } \text{ most } \text{ two } \text{ smooth } \text{ points } \text{ and }\,X_F\cap \Pi _3\,\text{ has } \text{ a } \text{ non-ordinary } \text{ singular } \text{ point } \text{ at }\,P.\\ \end{array}\right\} . \end{aligned}$$

The condition that $X_F\cap \Pi _3$ contains three lines passing through the point $P$ is equivalent to the condition that $q_4(0,0,z,u,v)$ and $q_5(0,0,z,u,v)$ vanish at three points in $\mathbb{P }^2$ where both $q_2(0,0,z,u,v)$ and $q_3(0,0,z,u,v)$ vanish.

For one of the lines $L_i$ to meet its residual curve by a general hyperplane section in $\Pi _3$ at at most one smooth is also a condition of codimension at least 2. For one of the lines $L_i$ to meet its residual curve by a general hyperplane section in $\Pi _3$ at at most two smooth is also a condition of codimension at least 1. The condition that $X_F\cap \Pi _3$ has a non-ordinary singular point at $P$ is equivalent to the condition that the quadratic polynomial $q_2(0,0,z,u,v)$ is singular in variables $z,\,u,\,v$. This is a condition of codimension 1. These altogether show that the properties $\mathcal{P }_{G3.3.2}$ is of codimension $> 7$. $\square$