Return to Question

I am sorry, my question is very naive.

2nd Edit: Let us suppose that $V$ is a smooth complex projective variety, and $Y\subset V$ is a smooth divisor and has an ample conormal line bundle. We would like to contract $Y$ down to a point in $V$, such contractions do not always exist in the category of schemes (thanks to Donu Arapura and abx). However such contractions exist in the category of analytic spaces (Grauert) and algebraic spaces both under positivity conditions on the conormal line bundle (Artin), see for example: Conditions for the Existence of Contractions in the Category of Algebraic Spaces, Joseph Mazur, Transactions of the American Mathematical Society, Vol. 209, (Aug., 1975), pp. 259-265

$\textbf{Question 1:}$ if a contraction of $V$ is a projective variety $V'$ what can be said about the singular points of $V'$? What type of singularities can occur?

$\textbf{Question 2:}$ let us start with a singular projective variety $V'$ of dimension $n$ with at worst isolated singularities, and we suppose that the tangent cone of $V'$ at each of these singular points is an affine cone over a smooth projective hypersurface $Q$ of dimension $n-1$. Am I correct if I say that there exists a resolution $V$ of $V'$ whose exceptional divisors are isomorphic to the $Q$'s and that the conormal bundles are very ample?

I am sorry, my question is very naive.

2nd Edit: Let us suppose that $V$ is a smooth complex projective variety, and $Y\subset V$ is a smooth divisor and has an ample conormal line bundle. We would like to contract $Y$ down to a point in $V$, such contractions do not always exist in the category of schemes (thanks to Donu Arapura and abx, see V Example 5.7.3 in, Hartshorne's "algebraic geometry"). However such contractions exist in the category of analytic spaces (Grauert) and algebraic spaces both under positivity conditions on the conormal line bundle (Artin), see for example: Conditions for the Existence of Contractions in the Category of Algebraic Spaces, Joseph Mazur, Transactions of the American Mathematical Society, Vol. 209, (Aug., 1975), pp. 259-265.

$\textbf{Question 1:}$ if a contraction of $V$ is a projective variety $V'$ what can be said about the singular points of $V'$? Are these singularities classical do they have a special name?

$\textbf{Question 2:}$ let us start with a singular projective variety $V'$ of dimension $n$ with at worst isolated singularities, and we suppose that the tangent cone of $V'$ at each of these singular points is an affine cone over a smooth projective hypersurface $Q$ of dimension $n-1$. Am I correct if I say that there exists a resolution $V$ of $V'$ whose exceptional divisors are isomorphic to the $Q$'s and that the conormal bundles are very ample?

I am sorry, my question is very naive.

Edit: Let us suppose that $V$ is a smooth complex projective variety, and $Y\subset V$ is a smooth divisor and has an ample conormal line bundle.

$\textbf{Question 1:}$ if a contraction of $V$ is a projective variety $V'$ what can be said about the singular points of $V'$? What type of singularities can occur?

$\textbf{Question 2:}$ let us start with a singular projective variety $V'$ of dimension $n$ with at worst isolated singularities, and we suppose that the tangent cone of $V'$ at each of these singular points is an affine cone over a smooth projective hypersurface $Q$ of dimension $n-1$. Am I correct if I say that there exists a resolution $V$ of $V'$ whose exceptional divisors are isomorphic to the $Q$'s and that the conormal bundles are very ample?

I am sorry, my question is very naive.

2nd Edit: Let us suppose that $V$ is a smooth complex projective variety, and $Y\subset V$ is a smooth divisor and has an ample conormal line bundle. We would like to contract $Y$ down to a point in $V$, such contractions do not always exist in the category of schemes (thanks to Donu Arapura and abx). However such contractions exist in the category of analytic spaces (Grauert) and algebraic spaces both under positivity conditions on the conormal line bundle (Artin), see for example: Conditions for the Existence of Contractions in the Category of Algebraic Spaces, Joseph Mazur, Transactions of the American Mathematical Society, Vol. 209, (Aug., 1975), pp. 259-265

$\textbf{Question 1:}$ if a contraction of $V$ is a projective variety $V'$ what can be said about the singular points of $V'$? What type of singularities can occur?

$\textbf{Question 2:}$ let us start with a singular projective variety $V'$ of dimension $n$ with at worst isolated singularities, and we suppose that the tangent cone of $V'$ at each of these singular points is an affine cone over a smooth projective hypersurface $Q$ of dimension $n-1$. Am I correct if I say that there exists a resolution $V$ of $V'$ whose exceptional divisors are isomorphic to the $Q$'s and that the conormal bundles are very ample?

I am sorry, my question is very naive.

Let us suppose that $V$ is a smooth complex projective variety, and that $Y\subset V$ is a very ample smooth divisor.

A contraction of $Y$ in $V$ exists in the category of complex spaces (cf Grauert) and in the category of algebraic spaces (cf Artin) because $Y$ is ample.

$\textbf{Question 1:}$ do a contraction is projective when $Y$ is very ample? Any counter-example or reference?

$\textbf{Question 2:}$ if $Y$ is very ample and a contraction is a projective variety $V'$ what can be said about the singular points of $V'$? What type of singularities can occur?

$\textbf{Question 3:}$ let us start with a singular projective variety $V'$ of dimension $n$ with at worst isolated singularities, and we suppose that the tangent cone of $V'$ at each of these singular points is an affine cone over a smooth projective hypersurface $Q$ of dimension $n-1$. Am I correct if I say that there exists a resolution $V$ of $V'$ whose exceptional divisors are isomorphic to the $Q$'s and are very ample?

I am sorry, my question is very naive.

Edit: Let us suppose that $V$ is a smooth complex projective variety, and $Y\subset V$ is a smooth divisor and has an ample conormal line bundle.

$\textbf{Question 1:}$ if a contraction of $V$ is a projective variety $V'$ what can be said about the singular points of $V'$? What type of singularities can occur?

$\textbf{Question 2:}$ let us start with a singular projective variety $V'$ of dimension $n$ with at worst isolated singularities, and we suppose that the tangent cone of $V'$ at each of these singular points is an affine cone over a smooth projective hypersurface $Q$ of dimension $n-1$. Am I correct if I say that there exists a resolution $V$ of $V'$ whose exceptional divisors are isomorphic to the $Q$'s and that the conormal bundles are very ample?