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Mar
16
comment Cohomology of structure sheaf of abelian variety
For an abelian variety $X$ of dimension $n$ the cotangent sheaf is trivial $\Omega^1_X \cong \mathcal{O}_X^{\oplus n}$. Hence, using the symmetries of Hodge numbers, you can compute $h^i(X,\mathcal{O}_X)=h^0(X,\Omega^i_X)=h^0(X,\mathcal{O}_X^{\oplus \binom{n}{i}})=\binom{n}{i}$
Mar
16
comment Flat schemes over artinian local ring with isomorphic special fibers
You're welcome!
Mar
15
comment Why are Unique Factorization Domains (UFD's) geometrically significant?
No, locally UFD means that the stalks are UFD. You're right that this does not imply that the affine open sets are UFD.
Mar
15
answered Flat schemes over artinian local ring with isomorphic special fibers
Mar
15
comment Why are Unique Factorization Domains (UFD's) geometrically significant?
If $X$ is an integral normal scheme, it is always true that the Picard group is isomorphic to the group of Cartier divisors modulo linear equivalence. Cartier divisors are by definition those that can be locally defined by a single equation and if $X$ is locally UFD then it is easy to see that every divisor is a Cartier divisor.
Feb
16
comment Comparison of spaces of global sections
Let $E=D'-D$. Then $E$ is effective by hypothesis and we have a short exact sequence $0 \to \mathcal{O}(-D'+D) \to \mathcal{O} \to \mathcal{O}_E \to 0$. Now tensor this by $\mathcal{O}(D')$ and take global sections.
Jan
25
comment Why do we need to take the closure in the definition of projective closure?
I think the closure needs to be taken in $\mathbb{P}^n$, not in $U_0$.
Jan
21
comment Computing the restriction $(\Lambda^2T_{\mathbb{P}^n})_{|L}$ for line in $\mathbb{P}^n$
Yes, if $n=4$. For any $n$ it is $\mathcal{O}_L(2)^{\oplus \binom{n-1}{2}}\oplus \mathcal{O}_L(3)^{\oplus (n-1)}$.
Jan
20
comment Computing the restriction $(\Lambda^2T_{\mathbb{P}^n})_{|L}$ for line in $\mathbb{P}^n$
You can use that exterior powers commute with pullbacks.
Jan
13
comment If $L$ is a line bundle on a scheme $X$, what is the ring $\oplus_{n \geq 0} \Gamma(X, L^{ \otimes n})$?
Just a remark about the case $\mathscr{L}$ very ample: if $X\subseteq \mathbb{P}^r$ is the embedding determined by $\mathscr{L}$ and if by homogeneous coordinate ring of $X$ one means $k[X_0,\dots,X_r]$ modulo the ideal of $X$, then this is not always the same as $\bigoplus_{n=0}^{\infty} H^0(X,\mathscr{L}^{\otimes n})$. For this to be true we need that the embedding is projectively normal, i.e. that all the maps $H^0(\mathbb{P}^r,\mathcal{O}(n)) \to H^0(X,\mathscr{L}^n)$ are surjective.
Dec
14
answered Determine the dimension of the set of surfaces of $\mathbb{P}^{3}$ that contain certain conic.
Dec
11
comment Index of speciality of general line bundles
Yes! That's exactly what it means.
Dec
11
revised Index of speciality of general line bundles
deleted 441 characters in body
Dec
11
answered Index of speciality of general line bundles
Dec
9
comment If $\varphi\colon X\to Y$ is a dominant morphism of affine varieties, $X$ irreducible, then $\varphi X$ contains nonempty open of closure.
Any morphism of $k$-algebras $k[X]\to k$ must be of the form $ev_x$ for some $x\in X$. Indeed, suppose that $k[X]=k[x_1,\dots,x_n]/I$ for some ideal $I$: then a map $g\colon k[X]\to k$ corresponds to choosing $a_i=g(x_i)$ and then this morphism becomes the evaluation in $(a_i,\dots,a_n)$
Dec
3
answered Cohomology of rational quartic in $\mathbb{P}^3$
Dec
2
comment Repost: Pushforward by double covering
You are welcome!
Dec
2
answered Repost: Pushforward by double covering
Dec
1
comment Computing cohomology over projective curve in $\mathbf{P}^3$
You're welcome! In the rational curve case, you have $d=3$ so that $\mathcal{O}_X(d-3)=\mathcal{O}_X$.
Dec
1
answered Computing cohomology over projective curve in $\mathbf{P}^3$