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Apr
21
answered What do Hecke eigensheaves actually look like?
Apr
1
comment Explicit equation for extension of a rational map?
If you like more coordinates: $\tilde X$ is covered by two charts: $w_1\ne 0$ and $w_2\ne 0$. On the former, the map is given by $\{w_1x,w_2y,w_1y,w_2z,w_1z\}$ (I just multiplied $\{x^2,y^2,xy,yz,xz\}$ by $w_1$, and then cancelled out $x$ using the identity $w_1y=w_2x$). On the other chart, it is a similar formula: multiply by $w_2$ and cancel $y$, getting $\{w_1x,w_2y,w_2x,w_2z,w_1z\}$. You now see the formula on the two charts is essentially the same, except that the middle term is written in two different ways.
Apr
1
comment Explicit equations for morphism determined by a linear system, Part II
Maybe it was better to edit your last question than create a new one? At any rate, I added more comments to your last question.
Mar
31
comment Explicit equation for extension of a rational map?
Geometrically, it is clear which curves appear in the system (pullbacks of conics minus the exceptional divisor). Do you want something more explicit? (More coordinates?)
Mar
31
comment The class of the diagonal in the symmetric product of a smooth curve
No, I don't know whether there is a nice description of $\mathrm{R\Gamma}$ (before taking $\det$) on $\mathrm{Sym}^{g-1}(C)$.
Mar
31
answered The class of the diagonal in the symmetric product of a smooth curve
Mar
6
awarded  Custodian
Mar
6
reviewed Leave Open Conceptual question about partitions in a given rectangular grid
Mar
3
comment About complete residues on curves
Yes, it is possible to extend it, for instance because $\Omega$ is the completion of $\Omega_{K|F}$ with respect to the topology defined by $v_P$, and $res_P$ is continuous in this topology (as follows immediately from the explicit formula).
Mar
2
comment Equi-dimensionality of special fibers in a flat family
I think Zariski's Main Theorem may be an overkill here, because the statement holds without the projectivity assumption. In fact, can't you just say that for every point $x\in X$, dimension of the fiber at x and dimension of X at x differ by 1 (Hartshorne, Proposition III.9.5). From this one easily sees that every component of X must dominate Y, therefore X has pure dimension d+1, and each fiber has pure dimension d.
Feb
29
comment Base change for quotient stack
No conditions are required. Basically, you have to check that $Y\to\mathcal{Y}$ is a presentation of the stack $\mathcal{Y}$, and that $Y\times_{\mathcal{Y}}Y$ is identified with $Y\times G$; both claims follow by base change from the corresponding claims about $\mathcal{X}$. (BTW, technically, if $\mathcal{Y}\to\mathcal{X}$ is representable, it may happen that $Y$ is an algebraic space, not a scheme, but this does not affect anything.)
Feb
15
awarded  Enlightened
Feb
15
awarded  Nice Answer
Feb
12
answered Counting isomorphism classes in open subsets of Bun_G
Dec
17
awarded  Yearling
Dec
7
comment Support of 0-dimensional sheaf and its dual
Note that the assumptions are too strong: the only relevant condition is that $E$ is Cohen-Macaulay (so that $Ext^i(E,\omega_R)=0$ for all but one value of $i$) and the same argument shows that the support of $E$ and the support of $E^D$ are equal as subschemes. (Here $E^D$ is the non-zero $Ext^i(E,\omega_R)$.)
Nov
23
comment Definition of étale (etc) for non-representable morphisms of algebraic stacks?
I would assume that an etale morphism of stacks $X\to Y$ is required to be `relative DM'; that is, that any base change from $Y$ to a scheme gives a DM stack. (Obviously, this is my opinion, but I don't see any other well-behaved definition.)
Nov
23
comment Is there a unique line bundle in the Kummer surface which pulls back to a totally symmetric line bundle?
First of all, do you want L' to pull back to L^2? If so then the answer to both of your questions (with L replaced by L^2 throughout) is yes, basically by a kind of descent (which works even though X->Y is not flat).
Nov
10
comment Vector bundles with symmetric perfect form
The `problem' with taking square roots is supposed to happen, because O(n) bundles are only locally trivial in etale topology, and taking square roots gives you an etale cover. It is too much to expect transition functions in the Zariski topology here.
Oct
21
comment Do all simple factors of jacobians of curves come from correspondences?
@Maarten Derickx: No, I don't, but it does not seem particularly hard. Fix base points $e\in E$, $c\in C$, we need to show that for any line bundle $L$ on $E\times C$, there exist $n_1$ and $n_2$ such that the line bundle $L(n_1(\{e\}\times C)+n_2(E\times \{c\}))$ is of the form $O(D)$ for smooth curve $D\subset E\times C$ (which then gives a correspondence between the two curves). However, $O((\{e\}\times C)+(E\times \{c\}))$ is ample, therefore, $L(n(\{e\}\times C)+n(E\times \{c\}))$ is very ample for $n\gg 0$, and the claim follows from Bertini's Theorem.