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Jason Starr
Jason Starr
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For a graded algebra $\mathcal{A} = \oplus_d \mathcal{A}_d$ of $\mathcal{O}_X$-algebras whose associated graded pieces $\mathcal{A}_d$ are coherent and which is generated in degree $1$, the relative Proj, $P=\text{Proj}_X \mathcal{A}$ with its projection $p:P\to X$ comes with a natural invertible quotient $q:p^*\mathcal{A}_1 \to \mathcal{O}(1)$. Moreover, for every integer $d>0$, the induced surjection $\text{Sym}^d(q):p^*\text{Sym}^d(\mathcal{A}_1) \to \mathcal{O}(d)$ factors through the pullback of the natural surjection $\text{Sym}^d(\mathcal{A}_1) \to \mathcal{A}_d$. In fact this is the universal property of the pair $(p:P\to X,q:p^*\mathcal{A}_1 \to \mathcal{O}(1))$, i.e., for every $X$-scheme $f:T\to X$ and for every invertible quotient $r:f^*\mathcal{A}_1 \to \mathcal{L}$ such that every induced map $\text{Sym}^d(r):f^*\text{Sym}^d(\mathcal{A}_1) \to \mathcal{L}^{\otimes d}$ factors through $f^*\mathcal{A}_d$, there exists a unique $X$-morphism $h:T\to P$ such that $p\circ h$ equals $f$ and $h^*q$ equals $r$.

Since the blowing up is Proj of the blowup algebra $\mathcal{A}$, which is generated in degree 1, the functor represented is the functor of invertible quotients $\mathcal{L}$ of the pullback $f∗I$$f^∗I$ of $I$ such that for every integer $d>0$, the induced surjection $f^∗\text{Sym}^d(I)\to \mathcal{L}$ factors through $f^*(I_d)$.

For a graded algebra $\mathcal{A} = \oplus_d \mathcal{A}_d$ of $\mathcal{O}_X$-algebras whose associated graded pieces $\mathcal{A}_d$ are coherent and which is generated in degree $1$, the relative Proj, $P=\text{Proj}_X \mathcal{A}$ with its projection $p:P\to X$ comes with a natural invertible quotient $q:p^*\mathcal{A}_1 \to \mathcal{O}(1)$. Moreover, for every integer $d>0$, the induced surjection $\text{Sym}^d(q):p^*\text{Sym}^d(\mathcal{A}_1) \to \mathcal{O}(d)$ factors through the pullback of the natural surjection $\text{Sym}^d(\mathcal{A}_1) \to \mathcal{A}_d$. In fact this is the universal property of the pair $(p:P\to X,q:p^*\mathcal{A}_1 \to \mathcal{O}(1))$, i.e., for every $X$-scheme $f:T\to X$ and for every invertible quotient $r:f^*\mathcal{A}_1 \to \mathcal{L}$ such that every induced map $\text{Sym}^d(r):f^*\text{Sym}^d(\mathcal{A}_1) \to \mathcal{L}^{\otimes d}$ factors through $f^*\mathcal{A}_d$, there exists a unique $X$-morphism $h:T\to P$ such that $p\circ h$ equals $f$ and $h^*q$ equals $r$.

Since the blowing up is Proj of the blowup algebra $\mathcal{A}$, which is generated in degree 1, the functor represented is the functor of invertible quotients $\mathcal{L}$ of the pullback $f∗I$ of $I$ such that for every integer $d>0$, the induced surjection $f^∗\text{Sym}^d(I)\to \mathcal{L}$ factors through $f^*(I_d)$.

For a graded algebra $\mathcal{A} = \oplus_d \mathcal{A}_d$ of $\mathcal{O}_X$-algebras whose associated graded pieces $\mathcal{A}_d$ are coherent and which is generated in degree $1$, the relative Proj, $P=\text{Proj}_X \mathcal{A}$ with its projection $p:P\to X$ comes with a natural invertible quotient $q:p^*\mathcal{A}_1 \to \mathcal{O}(1)$. Moreover, for every integer $d>0$, the induced surjection $\text{Sym}^d(q):p^*\text{Sym}^d(\mathcal{A}_1) \to \mathcal{O}(d)$ factors through the pullback of the natural surjection $\text{Sym}^d(\mathcal{A}_1) \to \mathcal{A}_d$. In fact this is the universal property of the pair $(p:P\to X,q:p^*\mathcal{A}_1 \to \mathcal{O}(1))$, i.e., for every $X$-scheme $f:T\to X$ and for every invertible quotient $r:f^*\mathcal{A}_1 \to \mathcal{L}$ such that every induced map $\text{Sym}^d(r):f^*\text{Sym}^d(\mathcal{A}_1) \to \mathcal{L}^{\otimes d}$ factors through $f^*\mathcal{A}_d$, there exists a unique $X$-morphism $h:T\to P$ such that $p\circ h$ equals $f$ and $h^*q$ equals $r$.

Since the blowing up is Proj of the blowup algebra $\mathcal{A}$, which is generated in degree 1, the functor represented is the functor of invertible quotients $\mathcal{L}$ of the pullback $f^∗I$ of $I$ such that for every integer $d>0$, the induced surjection $f^∗\text{Sym}^d(I)\to \mathcal{L}$ factors through $f^*(I_d)$.

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Jason Starr
Jason Starr
  • 4.2k
  • 1
  • 94
  • 115

For a graded algebra $\mathcal{A} = \oplus_d \mathcal{A}_d$ of $\mathcal{O}_X$-algebras whose associated graded pieces $\mathcal{A}_d$ are coherent and which is generated in degree $1$, the relative Proj, $P=\text{Proj}_X \mathcal{A}$ with its projection $p:P\to X$ comes with a natural invertible quotient $q:p^*\mathcal{A}_1 \to \mathcal{O}(1)$. Moreover, for every integer $d>0$, the induced surjection $\text{Sym}^d(q):p^*\text{Sym}^d(\mathcal{A}_1) \to \mathcal{O}(d)$ factors through the pullback of the natural surjection $\text{Sym}^d(\mathcal{A}_1) \to \mathcal{A}_d$. In fact this is the universal property of the pair $(p:P\to X,q:p^*\mathcal{A}_1 \to \mathcal{O}(1))$, i.e., for every $X$-scheme $f:T\to X$ and for every invertible quotient $r:f^*\mathcal{A}_1 \to \mathcal{L}$ such that every induced map $\text{Sym}^d(r):f^*\text{Sym}^d(\mathcal{A}_1) \to \mathcal{L}^{\otimes d}$ factors through $f^*\mathcal{A}_d$, there exists a unique $X$-morphism $h:T\to P$ such that $p\circ h$ equals $f$ and $h^*q$ equals $r$.

Since the blowing up is Proj of the blowup algebra $\mathcal{A}$, which is generated in degree 1, the functor represented is the functor of invertible quotients $\mathcal{L}$ of the pullback $f∗I$ of $I$ such that for every integer $d>0$, the induced surjection $f^∗\text{Sym}^d(I)\to \mathcal{L}$ factors through $f^*(I_d)$.