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Components of an exceptional divisor

2011-08-26T07:12:06Z

Let $X$ be a projective variety and let $\tilde{X}$ be the blow-up of $X$ at a subscheme $Z$. Let $F$ be the exceptional divisor of $\tilde{X}$. I wonder:

What is the number of irreducible components of $F$?

Note that this number depends strongly on the scheme structure on $Z$. For example, when $Z$ is a line in $\mathbb{P}^2$ with an embedded point, $F$ has two components, whereas the blow-up of a line has only one. So is it true in general that the number of components of $F$ is at least the number of associated primes of $Z$? (I am mostly interested in a lower bound for this number.)

Generalized linear systems

2011-10-26T21:30:22Z

Let $X$ be an algebraic variety and let $Z\subset X$ be a subvariety. Let $[Z]$ be the class defined by $Z$ in the Chow group. Let $L(Z)$ be set of effective algebraic cycles on $X$ linearly equivalent to $[Z]$. When $Z$ is an effective divisor, this set is a linear system, and so has the structure of a projective space.

Is there any reasonable geometric structure on $L(Z)$ in general?

If so, has this problem been studied before? In particular, can one give estimates for the dimension of $L(Z)$?

Answer by unknown (google) for The concept of Duality

2011-08-26T11:07:24Z

Serre duality

Grothendieck duality

Verdier duality

Ample line bundle and Duality

2011-04-18T15:35:10Z

Let $X$ be a projective variety and let $L$ be a line bundle on $X$. Suppose for all locally free sheaves $M$ on $X$, $ H^i(X,{L^*}^{\otimes r} \otimes M)=0 $ for $i<\dim X$ and $r$ sufficiently big.

Does it follow that $L$ is an ample line bundle? Here $L^*$ denotes the dual of $L$.

This is of course clear if $X$ is smooth using Serre duality, but how is it in general?

After reading Laurent Moret-Bailly and Karl Schwede's comments, below I changed the condition '$M$ coherent' to '$M$ locally free'.

Comment by

2011-04-18T16:27:01Z

Yes, thanks, I edited.