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Jason Starr
Jason Starr
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I believe this is incorrect. Let $V$ be a $3$-dimensional vector space, so that $\mathbb{P}V$ is $\mathbb{P}^2$. Let $\mathbb{P}(V^\vee)$ be the dual projective space parameterizing lines $L\subset \mathbb{P}V$. Let $\mathbb{P}(S)\subset \mathbb{P} V \times \mathbb{P}(V^\vee)$ denote the universal line, and let $S$ denote the locally free sheaf on $\mathbb{P}(V^\vee)$ that is a subsheaf of $V\otimes\mathcal{O}_{\mathbb{P}(V^\vee)}$ and whose associated projective bundle is $\mathbb{P}(S)$. Let $A$ denote $\mathbb{P}(\text{Sym}^2(S^\vee))$, the parameter space for pairs $([L],[D])$ of a line $L\subset \mathbb{P} V$ and an effective, degree $2$ Cartier divisor $D\subset L$. Finally, denote by $$B \subset A \times \mathbb{P}(\text{Sym}^2(V^\vee)) $$ the closed subscheme parameterizing triples $([L],[D],[C])$ of a line $L\subset \mathbb{P} V$, an effective, degree $2$ Cartier divisor $D\subset L$, and an effective, degree $2$ Cartier divisor $C\subset \mathbb{P} V$ (i.e., a plane conic), such that $D$ is contained in $C$. Since the restriction linear transformation, $$ \text{Sym}^2(V^\vee) \to H^0(L,\mathcal{O}_L(2)),$$ is surjective, the projection morphism, $$\text{pr}_A : B \to A, $$ is a projective bundle of fiber dimension $3$, i.e., a $\mathbb{P}^3$-bundle.

Now denote by $A_3$ the self-fiber product $$ A_3 = A\times_{\mathbb{P}(V^\vee)} A \times_{\mathbb{P}(V^\vee)} A,$$ parameterizing data $([L],[D_1],[D_2],[D_3])$ of a line $L\subset \mathbb{P} V$, and a triple of effective, degree $2$ Cartier divisors $D_i \subset L$. Similarly, denote by $B_3$ the self-fiber product $$ B_3 = B\times_{\mathbb{P}(V^\vee)} B \times_{\mathbb{P}(V^\vee)} B, $$ parameterizing data $([L],[D_1],[D_2],[D_3],[C_1],[C_2],[C_3])$ of a line $L\subset \mathbb{P} V$, a triple of effective, degree $2$ Cartier divisors $D_i \subset L$, and a triple of effective, degree $2$ Cartier divisors $C_i\subset \mathbb{P} V$ (i.e., plane conics) such that $D_i \subset C_i$. Since $\text{pr}_A$ is a $\mathbb{P}^3$-bundle, also the projection $$\text{pr}_{A_3}:B_3 \to A_3, \ ([L],[D_1],[D_2],[D_3],[C_1],[C_2],[C_3]) \mapsto ([L],[D_1],[D_2],[D_3]),$$ is a $\mathbb{P}^3\times \mathbb{P}^3\times \mathbb{P}^3$-bundle. In particular, $\text{pr}_{A_3}$ is faithfully flat of fiber dimension $9$.

Now, consider the locally closed subset $I\subset A_3$ parameterizing data $([L],[D_1],[D_2],[D_3])$ such that the degree $6$ Cartier divisor $E=D_1+D_2+D_3 \subset L$ is invariant under a cyclic subgroup $G\subset \text{Aut}(L)$ of order $5$. Consider the dimension of $I$. First of all, $\mathbb{P}(V^\vee)$ has dimension $2$. For fixed $[L]$ in $\mathbb{P}(V^\vee)$, every cyclic subgroup $G$ of $\text{Aut}(L) \cong \textbf{PGL}_2$ lifts to $\text{Aut}(L,\mathcal{O}(1)) \cong \textbf{SL}_2$, i.e., $G$ is diagonalizable. The eigenspaces are two distinct points of $L$, say $p_0$ and $p_\infty$. Thus, the parameter space for such subgroups $G$ is just $L\times L \setminus \Delta$, which is $2$-dimensional. Finally, for fixed $L$ and fixed $G$, there is a $1$-dimensional parameter space for $G$-invariant, degree $6$ Cartier divisors $E\subset L$, namely the $7$ isolated points $m\underline{p}_0 + n\underline{p}_\infty$ for $(m,n)\in \mathbb{Z}_{\geq 0}\times \mathbb{Z}_{\geq 0}$ with $m+n=6$ together with the two $\mathbb{A}^1$s parameterizing $\underline{p}_0 + \underline{G\cdot q}$, resp. $\underline{p}_\infty + \underline{G\cdot q}$ for $q\in L\setminus\{p_0, p_\infty\}$. Thus $I$ has dimension $2+2+1 = 5$. Since $\text{pr}_{A_3}$ is faithfully flat of fiber dimension $9$, $\text{pr}_{A_3}^{-1}(I)$ has dimension $5+9 = 14$.

Finally, consider the parameter space $$J =\mathbb{P}(\text{Sym}^2(V^\vee)) \times \mathbb{P}(\text{Sym}^2(V^\vee)) \times \mathbb{P}(\text{Sym}^2(V^\vee))$$ of triples $([C_1],[C_2],[C_3])$ of effective, degree $2$ Cartier divisors $C_i\subset \mathbb{P} V$, i.e., plane conics. Of course $J$ is isomorphic to $\mathbb{P}^5 \times \mathbb{P}^5 \times \mathbb{P}^5$, which has dimension $15$. Since $\text{pr}_{A_3}^{-1}(I)$ has dimension $14$, which is strictly less than $\text{dim}(J)$, the projection morphism, $$ \text{pr}_J : \text{pr}_{A_3}^{-1}(I) \to J, $$ cannot be dominant. Thus, the image is not Zariski dense. The Zariski closure of the image is a proper closed subvariety $K\subset J$. For every $([C_1],[C_2],[C_3])$ parameterized by a point of the dense, Zariski open subset $J\setminus K$ of $J$, there exists no $([L],[D_1],[D_2],[D_3])$ in $I$ such that $D_i\subset C_i$ for $i=1,2,3$.

I believe this is incorrect. Let $V$ be a $3$-dimensional vector space, so that $\mathbb{P}V$ is $\mathbb{P}^2$. Let $\mathbb{P}(V^\vee)$ be the dual projective space parameterizing lines $L\subset \mathbb{P}V$. Let $\mathbb{P}(S)\subset \mathbb{P} V \times \mathbb{P}(V^\vee)$ denote the universal line, and let $S$ denote the locally free sheaf on $\mathbb{P}(V^\vee)$ that is a subsheaf of $V\otimes\mathcal{O}_{\mathbb{P}(V^\vee)}$ and whose associated projective bundle is $\mathbb{P}(S)$. Let $A$ denote $\mathbb{P}(\text{Sym}^2(S^\vee))$, the parameter space for pairs $([L],[D])$ of a line $L\subset \mathbb{P} V$ and an effective, degree $2$ Cartier divisor $D\subset L$. Finally, denote by $$B \subset A \times \mathbb{P}(\text{Sym}^2(V^\vee)) $$ the closed subscheme parameterizing triples $([L],[D],[C])$ of a line $L\subset \mathbb{P} V$, an effective, degree $2$ Cartier divisor $D\subset L$, and an effective, degree $2$ Cartier divisor $C\subset \mathbb{P} V$ (i.e., a plane conic), such that $D$ is contained in $C$. Since the restriction linear transformation, $$ \text{Sym}^2(V^\vee) \to H^0(L,\mathcal{O}_L(2)),$$ is surjective, the projection morphism, $$\text{pr}_A : B \to A, $$ is a projective bundle of fiber dimension $3$, i.e., a $\mathbb{P}^3$-bundle.

Now denote by $A_3$ the self-fiber product $$ A_3 = A\times_{\mathbb{P}(V^\vee)} A \times_{\mathbb{P}(V^\vee)} A,$$ parameterizing data $([L],[D_1],[D_2],[D_3])$ of a line $L\subset \mathbb{P} V$, and a triple of effective, degree $2$ Cartier divisors $D_i \subset L$. Similarly, denote by $B_3$ the self-fiber product $$ B_3 = B\times_{\mathbb{P}(V^\vee)} B \times_{\mathbb{P}(V^\vee)} B, $$ parameterizing data $([L],[D_1],[D_2],[D_3],[C_1],[C_2],[C_3])$ of a line $L\subset \mathbb{P} V$, a triple of effective, degree $2$ Cartier divisors $D_i \subset L$, and a triple of effective, degree $2$ Cartier divisors $C_i\subset \mathbb{P} V$ (i.e., plane conics) such that $D_i \subset C_i$. Since $\text{pr}_A$ is a $\mathbb{P}^3$-bundle, also the projection $$\text{pr}_{A_3}:B_3 \to A_3, \ ([L],[D_1],[D_2],[D_3],[C_1],[C_2],[C_3]) \mapsto ([L],[D_1],[D_2],[D_3]),$$ is a $\mathbb{P}^3\times \mathbb{P}^3\times \mathbb{P}^3$-bundle. In particular, $\text{pr}_{A_3}$ is faithfully flat of fiber dimension $9$.

Now, consider the locally closed subset $I\subset A_3$ parameterizing data $([L],[D_1],[D_2],[D_3])$ such that the Cartier divisor $E=D_1+D_2+D_3 \subset L$ is invariant under a cyclic subgroup $G\subset \text{Aut}(L)$ of order $5$. Consider the dimension of $I$. First of all, $\mathbb{P}(V^\vee)$ has dimension $2$. For fixed $[L]$ in $\mathbb{P}(V^\vee)$, every cyclic subgroup $G$ of $\text{Aut}(L) \cong \textbf{PGL}_2$ lifts to $\text{Aut}(L,\mathcal{O}(1)) \cong \textbf{SL}_2$, i.e., $G$ is diagonalizable. The eigenspaces are two distinct points of $L$, say $p_0$ and $p_\infty$. Thus, the parameter space for such subgroups $G$ is just $L\times L \setminus \Delta$, which is $2$-dimensional. Finally, for fixed $L$ and fixed $G$, there is a $1$-dimensional parameter space for $G$-invariant, degree $6$ Cartier divisors $E\subset L$, namely the $7$ isolated points $m\underline{p}_0 + n\underline{p}_\infty$ for $m+n=6$ together with the two $\mathbb{A}^1$s parameterizing $\underline{p}_0 + \underline{G\cdot q}$, resp. $\underline{p}_\infty + \underline{G\cdot q}$ for $q\in L\setminus\{p_0, p_\infty\}$. Thus $I$ has dimension $2+2+1 = 5$. Since $\text{pr}_{A_3}$ is faithfully flat of fiber dimension $9$, $\text{pr}_{A_3}^{-1}(I)$ has dimension $5+9 = 14$.

Finally, consider the parameter space $$J =\mathbb{P}(\text{Sym}^2(V^\vee)) \times \mathbb{P}(\text{Sym}^2(V^\vee)) \times \mathbb{P}(\text{Sym}^2(V^\vee))$$ of triples $([C_1],[C_2],[C_3])$ of effective, degree $2$ Cartier divisors $C_i\subset \mathbb{P} V$, i.e., plane conics. Of course $J$ is isomorphic to $\mathbb{P}^5 \times \mathbb{P}^5 \times \mathbb{P}^5$, which has dimension $15$. Since $\text{pr}_{A_3}^{-1}(I)$ has dimension $14$, which is strictly less than $\text{dim}(J)$, the projection morphism, $$ \text{pr}_J : \text{pr}_{A_3}^{-1}(I) \to J, $$ cannot be dominant. Thus, the image is not Zariski dense. The Zariski closure of the image is a proper closed subvariety $K\subset J$. For every $([C_1],[C_2],[C_3])$ parameterized by a point of the dense, Zariski open subset $J\setminus K$ of $J$, there exists no $([L],[D_1],[D_2],[D_3])$ in $I$ such that $D_i\subset C_i$ for $i=1,2,3$.

I believe this is incorrect. Let $V$ be a $3$-dimensional vector space, so that $\mathbb{P}V$ is $\mathbb{P}^2$. Let $\mathbb{P}(V^\vee)$ be the dual projective space parameterizing lines $L\subset \mathbb{P}V$. Let $\mathbb{P}(S)\subset \mathbb{P} V \times \mathbb{P}(V^\vee)$ denote the universal line, and let $S$ denote the locally free sheaf on $\mathbb{P}(V^\vee)$ that is a subsheaf of $V\otimes\mathcal{O}_{\mathbb{P}(V^\vee)}$ and whose associated projective bundle is $\mathbb{P}(S)$. Let $A$ denote $\mathbb{P}(\text{Sym}^2(S^\vee))$, the parameter space for pairs $([L],[D])$ of a line $L\subset \mathbb{P} V$ and an effective, degree $2$ Cartier divisor $D\subset L$. Finally, denote by $$B \subset A \times \mathbb{P}(\text{Sym}^2(V^\vee)) $$ the closed subscheme parameterizing triples $([L],[D],[C])$ of a line $L\subset \mathbb{P} V$, an effective, degree $2$ Cartier divisor $D\subset L$, and an effective, degree $2$ Cartier divisor $C\subset \mathbb{P} V$ (i.e., a plane conic), such that $D$ is contained in $C$. Since the restriction linear transformation, $$ \text{Sym}^2(V^\vee) \to H^0(L,\mathcal{O}_L(2)),$$ is surjective, the projection morphism, $$\text{pr}_A : B \to A, $$ is a projective bundle of fiber dimension $3$, i.e., a $\mathbb{P}^3$-bundle.

Now denote by $A_3$ the self-fiber product $$ A_3 = A\times_{\mathbb{P}(V^\vee)} A \times_{\mathbb{P}(V^\vee)} A,$$ parameterizing data $([L],[D_1],[D_2],[D_3])$ of a line $L\subset \mathbb{P} V$, and a triple of effective, degree $2$ Cartier divisors $D_i \subset L$. Similarly, denote by $B_3$ the self-fiber product $$ B_3 = B\times_{\mathbb{P}(V^\vee)} B \times_{\mathbb{P}(V^\vee)} B, $$ parameterizing data $([L],[D_1],[D_2],[D_3],[C_1],[C_2],[C_3])$ of a line $L\subset \mathbb{P} V$, a triple of effective, degree $2$ Cartier divisors $D_i \subset L$, and a triple of effective, degree $2$ Cartier divisors $C_i\subset \mathbb{P} V$ (i.e., plane conics) such that $D_i \subset C_i$. Since $\text{pr}_A$ is a $\mathbb{P}^3$-bundle, also the projection $$\text{pr}_{A_3}:B_3 \to A_3, \ ([L],[D_1],[D_2],[D_3],[C_1],[C_2],[C_3]) \mapsto ([L],[D_1],[D_2],[D_3]),$$ is a $\mathbb{P}^3\times \mathbb{P}^3\times \mathbb{P}^3$-bundle. In particular, $\text{pr}_{A_3}$ is faithfully flat of fiber dimension $9$.

Now, consider the locally closed subset $I\subset A_3$ parameterizing data $([L],[D_1],[D_2],[D_3])$ such that the degree $6$ Cartier divisor $E=D_1+D_2+D_3 \subset L$ is invariant under a cyclic subgroup $G\subset \text{Aut}(L)$ of order $5$. Consider the dimension of $I$. First of all, $\mathbb{P}(V^\vee)$ has dimension $2$. For fixed $[L]$ in $\mathbb{P}(V^\vee)$, every cyclic subgroup $G$ of $\text{Aut}(L) \cong \textbf{PGL}_2$ lifts to $\text{Aut}(L,\mathcal{O}(1)) \cong \textbf{SL}_2$, i.e., $G$ is diagonalizable. The eigenspaces are two distinct points of $L$, say $p_0$ and $p_\infty$. Thus, the parameter space for such subgroups $G$ is just $L\times L \setminus \Delta$, which is $2$-dimensional. Finally, for fixed $L$ and fixed $G$, there is a $1$-dimensional parameter space for $G$-invariant, degree $6$ Cartier divisors $E\subset L$, namely the $7$ isolated points $m\underline{p}_0 + n\underline{p}_\infty$ for $(m,n)\in \mathbb{Z}_{\geq 0}\times \mathbb{Z}_{\geq 0}$ with $m+n=6$ together with the two $\mathbb{A}^1$s parameterizing $\underline{p}_0 + \underline{G\cdot q}$, resp. $\underline{p}_\infty + \underline{G\cdot q}$ for $q\in L\setminus\{p_0, p_\infty\}$. Thus $I$ has dimension $2+2+1 = 5$. Since $\text{pr}_{A_3}$ is faithfully flat of fiber dimension $9$, $\text{pr}_{A_3}^{-1}(I)$ has dimension $5+9 = 14$.

Finally, consider the parameter space $$J =\mathbb{P}(\text{Sym}^2(V^\vee)) \times \mathbb{P}(\text{Sym}^2(V^\vee)) \times \mathbb{P}(\text{Sym}^2(V^\vee))$$ of triples $([C_1],[C_2],[C_3])$ of effective, degree $2$ Cartier divisors $C_i\subset \mathbb{P} V$, i.e., plane conics. Of course $J$ is isomorphic to $\mathbb{P}^5 \times \mathbb{P}^5 \times \mathbb{P}^5$, which has dimension $15$. Since $\text{pr}_{A_3}^{-1}(I)$ has dimension $14$, which is strictly less than $\text{dim}(J)$, the projection morphism, $$ \text{pr}_J : \text{pr}_{A_3}^{-1}(I) \to J, $$ cannot be dominant. Thus, the image is not Zariski dense. The Zariski closure of the image is a proper closed subvariety $K\subset J$. For every $([C_1],[C_2],[C_3])$ parameterized by a point of the dense, Zariski open subset $J\setminus K$ of $J$, there exists no $([L],[D_1],[D_2],[D_3])$ in $I$ such that $D_i\subset C_i$ for $i=1,2,3$.

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Jason Starr
Jason Starr
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I believe this is incorrect. Let $V$ be a $3$-dimensional vector space, so that $\mathbb{P}V$ is $\mathbb{P}^2$. Let $\mathbb{P}(V^\vee)$ be the dual projective space parameterizing lines $L\subset \mathbb{P}V$. Let $\mathbb{P}(S)\subset \mathbb{P} V \times \mathbb{P}(V^\vee)$ denote the universal line, and let $S$ denote the locally free sheaf on $\mathbb{P}(V^\vee)$ that is a subsheaf of $V\otimes\mathcal{O}_{\mathbb{P}(V^\vee)}$ and whose associated projective bundle is $\mathbb{P}(S)$. Let $A$ denote $\mathbb{P}(\text{Sym}^2(S^\vee))$, the parameter space for pairs $([L],[D])$ of a line $L\subset \mathbb{P} V$ and an effective, degree $2$ Cartier divisor $D\subset L$. Finally, denote by $$B \subset A \times \mathbb{P}(\text{Sym}^2(V^\vee)) $$ the closed subscheme parameterizing triples $([L],[D],[C])$ of a line $L\subset \mathbb{P} V$, an effective, degree $2$ Cartier divisor $D\subset L$, and an effective, degree $2$ Cartier divisor $C\subset \mathbb{P} V$ (i.e., a plane conic), such that $D$ is contained in $C$. Since the restriction linear transformation, $$ \text{Sym}^2(V^\vee) \to H^0(L,\mathcal{O}_L(2)),$$ is surjective, the projection morphism, $$\text{pr}_A : B \to A, $$ is a projective bundle of fiber dimension $3$, i.e., a $\mathbb{P}^3$-bundle.

Now denote by $A_3$ the self-fiber product $$ A_3 = A\times_{\mathbb{P}(V^\vee)} A \times_{\mathbb{P}(V^\vee)} A,$$ parameterizing data $([L],[D_1],[D_2],[D_3])$ of a line $L\subset \mathbb{P} V$, and a triple of effective, degree $2$ Cartier divisors $D_i \subset L$. Similarly, denote by $B_3$ the self-fiber product $$ B_3 = B\times_{\mathbb{P}(V^\vee)} B \times_{\mathbb{P}(V^\vee)} B, $$ parameterizing data $([L],[D_1],[D_2],[D_3],[C_1],[C_2],[C_3])$ of a line $L\subset \mathbb{P} V$, a triple of effective, degree $2$ Cartier divisors $D_i \subset L$, and a triple of effective, degree $2$ Cartier divisors $C_i\subset \mathbb{P} V$ (i.e., plane conics) such that $D_i \subset C_i$. Since $\text{pr}_A$ is a $\mathbb{P}^3$-bundle, also the projection $$\text{pr}_{A_3}:B_3 \to A_3, \ ([L],[D_1],[D_2],[D_3],[C_1],[C_2],[C_3]) \mapsto ([L],[D_1],[D_2],[D_3]),$$ is a $\mathbb{P}^3\times \mathbb{P}^3\times \mathbb{P}^3$-bundle. In particular, $\text{pr}_{A_3}$ is faithfully flat of fiber dimension $9$.

Now, consider the locally closed subset $I\subset A_3$ parameterizing data $([L],[D_1],[D_2],[D_3])$ such that the Cartier divisor $E=D_1+D_2+D_3 \subset L$ is invariant under a cyclic subgroup $G\subset \text{Aut}(L)$ of order $5$. Consider the dimension of $I$. First of all, $\mathbb{P}(V^\vee)$ has dimension $2$. For fixed $[L]$ in $\mathbb{P}(V^\vee)$, every cyclic subgroup $G$ of $\text{Aut}(L) \cong \textbf{PGL}_2$ lifts to $\text{Aut}(L,\mathcal{O}(1)) \cong \textbf{SL}_2$, i.e., $G$ is diagonalizable. The eigenspaces are two distinct points of $L$, say $p_0$ and $p_\infty$. Thus, the parameter space for such subgroups $G$ is just $L\times L \setminus \Delta$, which is $2$-dimensional. Finally, for fixed $L$ and fixed $G$, there is a $1$-dimensional parameter space for $G$-invariant, degree $6$ Cartier divisors $E\subset L$, namely the $7$ isolated points $m\underline{p}_0 + n\underline{p}_\infty$ for $m+n=6$ together with the two $\mathbb{A}^1$s parameterizing $\underline{p}_0 + \underline{G\cdot q}$, resp. $\underline{p}_\infty + \underline{G\cdot q}$ for $q\in L\setminus\{p_0, p_\infty\}$. Thus $I$ has dimension $2+2+1 = 5$. Since $\text{pr}_{A_3}$ is faithfully flat of fiber dimension $9$, $\text{pr}_{A_3}^{-1}(I)$ has dimension $5+9 = 14$.

Finally, consider the parameter space $$J =\mathbb{P}(\text{Sym}^2(V^\vee)) \times \mathbb{P}(\text{Sym}^2(V^\vee)) \times \mathbb{P}(\text{Sym}^2(V^\vee))$$ of triples $([C_1],[C_2],[C_3])$ of effective, degree $2$ Cartier divisors $C_i\subset \mathbb{P} V$, i.e., plane conics. Of course $J$ is isomorphic to $\mathbb{P}^5 \times \mathbb{P}^5 \times \mathbb{P}^5$, which has dimension $15$. Since $\text{pr}_{A_3}^{-1}(I)$ has dimension $14$, which is strictly less than $\text{dim}(J)$, the projection morphism, $$ \text{pr}_J : \text{pr}_{A_3}^{-1}(I) \to J, $$ cannot be dominant. Thus, the image is not Zariski dense. The Zariski closure of the image is a proper closed subvariety $K\subset J$. For every $([C_1],[C_2],[C_3])$ parameterized by a point of the dense, Zariski open subset $J\setminus K$ of $J$, there exists no $([L],[D_1],[D_2],[D_3])$ in $I$ such that $D_i\subset C_i$ for $i=1,2,3$.

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