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My user name translates as Little Black Donkey. It is borrowed from a short story by Pádraic Ó Conaire, whose text you can find here.


4h
comment Hesse's Pencil: Base Points and Resolution of Singularities through Blow-ups
About the second part of your question: I'm not sure what exactly you are asking for. Do you want to check that after you blow up these 9 points, the curves in the pencil become disjoint? If so, following the strategy in the linked question should be a productive strategy.
4h
comment Hesse's Pencil: Base Points and Resolution of Singularities through Blow-ups
Dear Radz, your link does not work for me, but the solutions are easy to find by hand: as you write, one coordinate must be 0, another can be normalised to be 1, and then the third can be any cube root of -1. You can check this gives exactly 9 distinct points of $\mathbf P^2$.
5h
comment Hesse's Pencil: Base Points and Resolution of Singularities through Blow-ups
Well, your equation already describes the curves in the pencil as embedded in projective space (with homogeneous coords $[x,y,z]$). To find the basepoints (of which there are 9), just take two different values of the parameter $[\lambda,\mu]$ and intersect the two cubics you get. (Hint: there are two particularly simple choices for $[\lambda,\mu]$.)
1d
comment projective change of coordinates preserves smoothness
I think you misunderstood what the chain rule says here; there is no dot product. (For instance, the left-hand side of your equality is a vector, while the right-hand side is a scalar.)
2d
comment Quotients of varieties by polynomial relations
What you are describing is just the image of $V$ under the morphism $\mathbf C^n \rightarrow \mathbf C^m: \ x \mapsto (g_i(x))$. Note however that the image of $V$ need not be a subvariety of $\mathbf C^m$: the usual counterxample is the map $\mathbf C^2 \rightarrow \mathbf C^2: (x,y) \mapsto (x,xy)$.
2d
revised Dimensions of global sections of a divisor and its pullback
added 4 characters in body
2d
comment Dimensions of global sections of a divisor and its pullback
Dear @LiYutong, you're welcome.
2d
revised Conversion of angle from 360 degree to-90 degree
edited tags
2d
comment Dimensions of global sections of a divisor and its pullback
Dear Li, I wrote an answer to explain my claims. I don't have time to read through Iitaka's book, though.
2d
answered Dimensions of global sections of a divisor and its pullback
2d
comment Dimensions of global sections of a divisor and its pullback
You're correct: this is false as stated. But it is true if $h$ has connected fibres. Are you sure that hypothesis is not somewhere in the text, or implied by the context? (By the way, who are the authors?)
Jul
17
revised Is complex projective space simply connected?
edited title
Jul
17
comment Is complex projective space simply connected?
Yes, complex projective spaces are simply-connected. Indeed, $\mathbf{CP}^n$ has a cell structure with a single cell in each even degree, and the result follows from the <a href="en.wikipedia.org/wiki/… approximation theorem</a>. See Hatcher's book for the details.
Jul
11
comment what does projective line of degree one mean?
Where did you read this? It is difficult to answer without knowing the context.
Jul
10
answered Linear equivalence vs algebraic equivalence of divisors on smooth projective surfaces
Jul
10
comment Linear equivalence vs algebraic equivalence of divisors on smooth projective surfaces
I understood the question already. Do you know why the statement is false for curves? Once you do, it is not hard to turn this into an example for surfaces.
Jul
10
comment Linear equivalence vs algebraic equivalence of divisors on smooth projective surfaces
No. Hint: this isn't true for non-rational curves, e.g. elliptic curves; use that to come up with an example for surfaces.
Jul
9
comment Holomorphic line bundle with degree zero
$O(p)$ is the line bundle associated to the divisor $p$.
Jul
9
comment Holomorphic line bundle with degree zero
If $p$ and $q$ are distinct points on the torus, then $O(p) \otimes O(-q)$ is a nontrivial line bundle of degree zero.
Jul
8
comment Question about the restriction of a bimeromorphic map to divisors
+1. You dropped a minus sign at the start of the last paragraph.