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1d
comment Can anyone comment on uniformizing parameters and uniformizing coordinates?
Dear Dmitrii, You are more likely to get an answer if you type out the precise definitions that you would like to compare.
Mar
21
comment Number of elements in a fiber
When you say "normal affine domain over an algebraically closed field", do you mean that the ring is finitely generated over the field? For general rings containing a field, I am not sure this is true. However, in the case of finitely generated $k$-algebras, this is true. The simplest argument involves blowing up the point and passing to the stalk at each generic point of the exceptional divisor: this reduces your problem to a problem about extensions of DVRs.
Mar
20
comment Twists of projective automorphisms
What precisely do you mean by "preserve the isomorphism class"? For instance, it seems to me that $\text{Aut}(\mathbb{P}^n_k,\mathcal{O}(1))$ is $\textbf{PGL}_{n+1}$. There is a slightly different notion that recovers $\textbf{SL}_{n+1}$. Some of this is discussed in my paper with de Jong, "Discriminant avoidance ..."
Mar
19
comment Cohen-Macaulay fibers
I do not understand your hypotheses. Is the domain $X$ of the morphism $\phi$ supposed to be finite? If so, then $\phi_*\mathcal{O}_X$ will not be a locally free sheaf. However, unless $\phi$ is finite, there is no reason that the induced morphism from $X$ to $\mathbb{P}(\phi_*\mathcal{O}_X)$ should be an immersion. Could you please clarify your hypotheses?
Mar
19
answered Relative tangent space to proper morphism and irreducibility of fibers
Mar
18
comment How to prove two manifolds are not birational?
Are your manifolds Moishezon? If not, what do you mean by "birational"? Did you mean to write "bimeromorphic"? For Kaehler manifolds, the weight 1 Hodge structures are birational invariants. Thus, if you know that the classifying map from $B$ to the period domains of weight 1 Hodge structures is injective, then distinct fibers are not bimeromorphic.
Mar
15
comment Covering of schemes and flatness
@user46578. You should read the rest of the section, or maybe just the rest of the exercise. The rank function is upper semicontinuous. So if you know the rank is constant at all closed points, then you know the rank is constant at all points.
Mar
14
answered Covering of schemes and flatness
Mar
14
revised Special linear sections of a hypersurface
added 480 characters in body
Mar
14
comment Presentation of the tautological bundle of the Grassmannian
Which tautological bundle: the quotient bundle of the trivial bundle or the subbundle of the trivial bundle?
Mar
14
revised Special linear sections of a hypersurface
added 47 characters in body
Mar
14
revised Special linear sections of a hypersurface
added 2607 characters in body
Mar
12
comment miniversality vs versality
Usually the phrase "is etale over the point $x$" means more than "is formally etale precisely at the point $x$". It means "there is an open neighborhood on which the morphism is etale". To pass from "formally etale at one point $x$" to "etale on an open neighborhood", Artin used the hypothesis that the functor / stack is limit preserving / finitely presented in an essential way. You can probably make counterexamples by considering functors / stacks that are not limit preserving.
Mar
12
comment Special linear sections of a hypersurface
That quotient is a stack, not a geometric invariant theory quotient. Also, the point of my answer, and of the article I linked, is that it is easier to prove the conclusion than to understand the morphism $\zeta$. This is just an incidence correspondence computation: form the incidence correspondence of pairs of a hypersurface in $\mathbb{P}^n$ together with a linear $\mathbb{P}^4$ section. If you do not (yet) know how to make incidence correspondence arguments, then you have more fundamental questions to answer than the one above.
Mar
11
revised Special linear sections of a hypersurface
edited body
Mar
11
answered Special linear sections of a hypersurface
Mar
5
comment Is the moduli space of curves arising from wild ramification smooth?
The branch morphism is discussed in a paper by Fantechi-Pandharipande. They probably only define it in characteristic 0 (to use for Gromov-Witten theory). However, the same construction works in all characteristics (or over Z).
Mar
5
comment Is the moduli space of curves arising from wild ramification smooth?
There is a Deligne-Mumford stack $\mathcal{N}$ of all finite, flat, generically etale morphisms $f:C\to \mathbb{P}^1$ such that $\Omega_f$ has length $b$. There is a branch morphism $\text{br}$ from $\mathcal{N}$ to the Hilbert scheme $\text{Hilb}^b_{\mathbb{P}^1_k/k} = \mathbb{P}^b_k$ that associates to $[f]$ the divisor (defined via "det-div" as in Knudsen-Mumford / Fogarty) of $f_*\Omega_f$. You ask about smoothness of an (open subset) of a fiber of $\text{br}$. This may hold. However, $\text{br}$ is not a smooth morphism. This is why the Oort conjecture is nontrivial.
Feb
26
comment How does one compute the first Chern class of a Line bundle defined as the Kernel of a linear map?
Off the top of my head, I doubt that there is a completely general formula. The divisor class $c_1(L)$ on $D$ may not be the pullback to $D$ of any divisor class on $M$, in which case, in what terms are you expecting to describe $c_1(L)$? In the specific example, I vaguely remember that Steven Kleiman's article on "The enumerative theory of singularities" has something about this.
Feb
23
comment reference for “curves over S are locally the base change of a curve over S' which is finite type over R”
"... what's your counterexample?" Let $A$ be $\mathbb{Q}[x_1,x_2,x_3,\dots]$, let $\mathfrak \subset A$ be $\langle x_1,x_3,\dots \rangle$. Let $S$ be $\text{Spec}(A)\setminus \{ \mathfrak{m} \}$. This is the union of the countably many open subsets $U_n = D(x_1)\cup \dots \cup D(x_n)$. Over each open subset $U_n$, $X\times_S U_n$ will be the blowing up of $\mathbb{P}^1\times U_n$ along a particular closed subscheme. For $U_2$, blowup $s(U_2) = \{[1,0]\}\times U_2$ over $Z(x_1)$. Then, over $U_3$, blowup further the strict transform of $s(U_3)$ over $Z(x_1,x_2)$, etc.