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2d
revised If $\dim(X)=\dim(Y)>0$, and $X\to Y$ is onto, does affineness of $X$ imply affineness of $Y$?
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2d
revised If $\dim(X)=\dim(Y)>0$, and $X\to Y$ is onto, does affineness of $X$ imply affineness of $Y$?
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2d
comment If $\dim(X)=\dim(Y)>0$, and $X\to Y$ is onto, does affineness of $X$ imply affineness of $Y$?
@BenLim: you're right, smoothness is necessary. Thanks for the comment.
2d
answered If $\dim(X)=\dim(Y)>0$, and $X\to Y$ is onto, does affineness of $X$ imply affineness of $Y$?
May
18
comment Geometry of cubic 3-fold
In general you're right, but actually you don't need to worry about that here. If you have two distinct effective divisors on a surface and at least one is irreducible (as your curve $E$ is), then they intersect in finitely many points. The intersection number will then count those points (with multiplicity), so the only way it can be 0 is if the two divisors are indeed disjoint.
May
15
comment Geometry of cubic 3-fold
This is true for any smooth curve $E$ of genus 1 in a $K3$ surface $S$. See this answer: math.stackexchange.com/questions/717990/… to show that $E^2=0$ and $H^0(S,O_S(E))=2$. But that means the two sections must have disjoint vanishing sets, and therefore generate the bundle.
May
12
comment Divisors corresponding to hypersurfaces in Projective space
Answer: $k=\operatorname{deg} f$.
May
12
comment Dimension of linear system of divisor of two points on curve of genus greater than 2
@DanieleA: the hyperelliptic case is dimension 1. ("Linear system dimension" is vector space dimension minus one.)
May
12
answered Deforming line bundles on abelian varieties
May
9
comment Any quartic in $\mathbb P^3$ contains only finitely many lines.
You should make clear what the hypotheses on $X$ are: is it supposed to be smooth? For example, if $X$ is the cone over a quartic curve, it contains infinitely many lines.
May
8
comment Proving uniqueness of limits
I really thought the franchise ran out of steam after "highschool inequality 3: junior year". (Joking aside, your title could be improved to something more descriptive.)
May
8
comment Is exceptional divisor always a Projective bundle over the centre?
The exceptional divisor is a projective bundle if the closed subscheme is local complete intersection, i.e. everywhere cut out by a regular sequence. But it is not true in general: e.g. blowing up the threefold ordinary double point we get a quadric surface as exceptional divisor. See Section IV.2 of Eisenbud--Harris for details.
May
8
comment Serre's criterion and closure
If $X$ is R1, then it is normal iff it is S2.
May
8
comment Serre's criterion and closure
Certainly not. If $X$ is regular in codimension 1 then your hypothesis holds. But if $X$ is not normal then it is not $S_2$, by Serre's theorem (normal=R1+S2).
May
8
comment Ideal sheaf of intersection of two surfaces in $\mathbb P^3$
I assume $X$ is a complete intersection. Say the surfaces are zero sets of sections $f_1, f_2$. Consider the map $O(-d_1) \oplus O(-d_2) \rightarrow O_{\mathbf P^3}$ given by $(a,b) \mapsto af_1+bf_2$. What is the image and kernel?
May
6
reviewed Approve Why are degenerate conics not projectively equivalent to nondegenerate conics?
May
4
comment Automorphisms of del Pezzo surface
You didn't actually write a question, but I assume the question is "how do I prove it"? You should look in Chapter 8 of Dolgachev, Classical Algebraic Geometry (available online). Here is the idea for degree 2: sections of $-K_X$ define a 2:1 cover of $\mathbf P^2$ branched over a smooth quartic curve. Up to finite index, an automorphism of $X$ induces an automorphism of this quartic, so you are done. The story for degree 1 is similar, with a quadric cone now playing the role of the base surface.
May
3
comment Family of quartic surfaces in $\mathbb{P}^3$ that contain a fixed line or conic
OK. The motiviation to do this is still not so clear to me, but anyway" the sheaf you want is $\mathcal I_L \otimes \mathcal O_{\mathbf P^3}(4)$ where $\mathcal I_L$ is the ideal sheaf of the line. Then you can use the ideal sheaf sequence for $\mathcal I_L$ to get your answer. The same approach works for a conic too.
May
2
comment Family of quartic surfaces in $\mathbb{P}^3$ that contain a fixed line or conic
Why do you think this argument is not rigorous?
May
2
revised Resolving a node singularity on a plane curve
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