5 added 38 characters in body; added 4 characters in body

The following question feels to me like a standard sort of 'fact' in birational geometry, but I can't seem to write down a correct set of details. Hopefully someone can point me to a reference and not a counter example!

Suppose $X$ is a variety (reduced and irreducible over an algebraically closed field, perhaps of characteristic zero) and suppose that there exist a very ample line bundle $L$ and a linear system $V \subset H^0(X,L)$ such that $Y = Bs(V)$ is the singular set of $X$ scheme theoretically, that $Y$ is smooth of codimension at at least 2, and that $\tilde X$, the blow up of $X$ along $Y$ is smooth. Further assume that $\phi_{|V|} X--> S$ birationally maps $X$ onto a smooth variety $S$. Let $\tilde \phi$ be the map from $\tilde X \to S$ induced by $V$. Further assume that, denoting by $f$ the map $\tilde X \to X$, that $f^{-1}(Y) = T$ surjects onto $S$. Let $v_1, \dots v_s$ be $s = \dim(S)$ general sections of $V$ so that the intersection $Z(v_1) \cap \dots Z(v_s) \cap S$ consists of finitely many smooth points say $p_1, \dots p_m$.

Denoting by $f$ the map $\tilde X \to X$, further

Also assume the $P = f( \tilde \phi^{-1}(\cup_{i=1:m} p_i))$ is a proper subset of $Y$. Then can one say that away from $P$, the sections $v_1 \dots v_s$ generate the ideal of $Y$ in $X$ ?

The case I have in mind is where $Y$ is a smooth curve embedded in a sufficiently ample manner so that 1) $Y$ is defined by quadrics and 2) $X = Sec(X)$ is singular only along $Y$. Then $V$ would be the quadrics through $Y$. The point would be to use this sort of an argument to establish a minimum depth of $Sec(Y)$ along $Y$.

This is my first question, so please feel free to correct etiquette with this question as well as the mathematics.

4 added 42 characters in body; added 21 characters in body; added 9 characters in body

The following question feels to me like a standard sort of 'fact' in birational geometry, but I can't seem to write down a correct set of details. Hopefully someone can point me to a reference and not a counter example!

Suppose $X$ is a variety (reduced and irreducible over an algebraically closed field, perhaps of characteristic zero) and suppose that there exist a very ample line bundle $L$ and a linear system $V \subset H^0(X,L)$ such that $Y = Bs(V)$ is the singular set of $X$ scheme theoretically, that $Y$ is smooth of codimension at at least 2, and that $\tilde X$, the blow up of $X$ along $Y$ is smooth. Further assume that $\phi_{|V|} X--> S$ birationally maps $X$ onto a smooth variety $S$. Let $\tilde \phi$ be the map from $\tilde X \to S$ induced by $V$. Further assume that Let $v_1, \dots v_s$ be $s = \dim(S)$ general sections of $V$ so that the intersection $Z(v_1) \cap \dots Z(v_s) \cap S$ consists of finitely many smooth points say $p_1, \dots p_m$.

Denoting by $f$ the map $\tilde X \to X$, further assume the $P = f( \tilde \phi^{-1}(\cup_{i=1:m} p_i))$ is a proper subset of $Y$. Then can one say that away from $P$, the sections $v_1 \dots v_s$ generate the ideal of $Y$ in $X$ ? The case I have in mind is where $Y$ is a smooth curve embedded in a sufficiently ample manner so that 1) $Y$ is defined by quadrics and 2) $X = Sec(X)$ is singular only along $Y$. Then $V$ would be the quadrics through $Y$. The point would be to use this sort of an argument to establish a minimum depth of $Sec(Y)$ along $Y$.

This is my first question, so please feel free to correct etiquette with this question as well as the mathematics.

3 edited body

The following question feels to me like a standard sort of 'fact' in birational geometry, but I can't seem to write down a correct set of details. Hopefully someone can point me to a reference and not a counter example!

Suppose $X$ is a variety (reduced and irreducible over an algebraically closed field, perhaps of characteristic zero) and suppose that there exist a very ample line bundle $L$ and a linear system $V \subset H^0(X,L)$ such that $Y = Bs(L)$ Bs(V)$is the singular set of$X$scheme theoretically and that$\tilde X$, the blow up of$X$along$Y$is smooth. Further assume that$\phi_{|V|} X--> S$birationally maps$X$onto a smooth variety$S$. Let$\tilde \phi$be the map from$\tilde X \to S$induced by$V$. Let$v_1, \dots v_s$be$s = \dim(S)$general sections of$V$so that the intersection$Z(v_1) \cap \dots Z(v_s) \cap S$consists of finitely many smooth points say$p_1, \dots p_m $. Denoting by$f$the map$\tilde X \to X$, further assume the$P = f( \tilde \phi^{-1}(\cup_{i=1:m} p_i))$is a proper subset of$Y$. Then can one say that away from$P$, the sections$ v_1 \dots v_s$generate the ideal of$Y$in$X$? The case I have in mind is where$Y$is a smooth curve embedded in a sufficiently ample manner so that 1)$Y$is defined by quadrics and 2)$X = Sec(X)$is singular only along$Y$. Then$V$would be the quadrics through$Y$. The point would be to use this sort of an argument to establish a minimum depth of$Sec(Y)$along$Y\$.

This is my first question, so please feel free to correct etiquette with this question as well as the mathematics.

2 edited body
1