All Questions
21 questions
1
vote
0
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210
views
Ex 1.1c Hartshorne Deformation Theory: Is this family flat?
This comes from my attempt to solve Exercise 1.1c in Hartshorne's Deformation Theory book, which says that a family of conics in $\mathbb{P}^2$ parameterised by some finitely generated $k$-algebra $A$ ...
6
votes
1
answer
491
views
Flatness of schemes
I am learning about flatness for the first time and I cannot wrap my head around why the definition with tensor products of a flat module implies geometrically that 1-parameter families of schemes ...
8
votes
3
answers
572
views
Lifting a complete intersection in $\mathbb{P}^n_{\mathbb{F}_p}$ to $\mathbb{Z}_p$
Suppose that you are given a (not necessarily smooth) projective variety $X \subseteq \mathbb{P}^n_{\mathbb{F}_p}$ of codimension $d$ that is a complete intersection. In other words, it can be defined ...
2
votes
1
answer
753
views
Push-forward of flat module under a finite, flat morphism
Let $f:X \to Y$ be a finite, faithfully flat morphism of noetherian, affine $\mathbb{C}$-schemes. One can assume $Y$ is non-singular. Let $A$ be a local artinian $\mathbb{C}$-algebra and $f_A:X_A \to ...
1
vote
1
answer
603
views
Isomorphism in fibers and flatness
Let $X$, $Y$ be (reduced) affine varieties and $f:X \to Y$ is a finite morphism which is an isomorphism over an open dense subset (for example a normalization map). Let $A$ be a local noetherian ring ...
3
votes
0
answers
214
views
Representability of Flattening stratification functor
Let $f:X \to Y$ be a projective morphism between noetherian scheme. Is there any know condition under which there exists a functorial stratification of $Y$ i.e., there exists a filtration by locally ...
1
vote
0
answers
249
views
On regularity of flat families over a DVR
Let $k$ be an algebraically closed field of characteristic zero and $R$ a discrete valuation ring over $k$. Let $\pi:X \to \mathrm{Spec}(R)$ be a flat, projective morphism such that the generic fiber ...
0
votes
1
answer
410
views
Can the specialization map be flat
Let $X$ be a projective variety over an algebraically closed field of characteristic zero. Let $\eta$ be a generic point of $X$ and $x$ be a closed point. By http://stacks.math.columbia.edu/tag/054F ...
1
vote
0
answers
191
views
Deformation of projective bundles
Let $\pi:\mathcal{X} \to \mathbb{P}^1$ be a flat, projective family of noetherian schemes with generic fiber a smooth, projective variety. Let $p:\mathcal{Y} \to \mathcal{X}$ be another flat, ...
0
votes
1
answer
349
views
Is the "addition" of flat morphisms flat?
Let $f:X \to Y$ be a flat, projective morphism between projective varieties. Let $F, G$ be coherent sheaves on $X$, flat over $Y$. Let $\phi_1, \phi_2$ be two morphisms from $F$ to $G$ such that:
1) ...
1
vote
0
answers
206
views
On simultaneous resolution of singularities in certain flat families
Let $X$ be a smooth projective variety (over the complex numbers) of dimension at least $2$, $B$ a finite set of closed points. Consider the closed subscheme $E:=B \times X + \Delta \subset X \times X$...
4
votes
1
answer
512
views
Being Cohen-Macaulay open in Hilbert scheme?
Let $H$ be the Hilbert scheme of closed subschemes of $\mathbb{P}^n$ with a given Hilbert polynomial. I would like to have a reference (preferably from a published paper or book, not stacks project) ...
3
votes
2
answers
410
views
Is the zero locus of a global section flat?
Let $f:X \to Y$ be a surjective, smooth projective morphism of noetherian schemes. Let $\mathcal{L}$ be an inverible sheaf on $X$ satisfying $f_*\mathcal{L}$ is locally free and $s \in H^0(\mathcal{L})...
3
votes
1
answer
506
views
Torsion free sheaves in flat families
Let $R$ be a dvr, $X$ a flat, projective, integral, normal $R$-scheme such every closed fiber is again integral, normal. Let $F$ be a torsion-free coherent sheaf on $X$, flat over $R$. Is it true that ...
0
votes
1
answer
918
views
Euler characteristic on flat families of quasi-projective schemes
Let $A$ be a noetherian integral domain (may be regular). Let $\pi:X \to \mathrm{Spec}(A)$ be a flat morphism. Suppose that each fiber of $\pi$ are quasi-projective. Let $\mathcal{F}$ be a coherent ...
1
vote
0
answers
139
views
A naive question on rational equivalence of varieties
Let $X$ be a projective scheme and $\pi:\mathcal{Z} \to \mathbb{P}^1$ a surjective morphism of finite type such that for any pair $t_0, t_1 \in \mathbb{P}^1$, the fibers $\mathcal{Z}_{t_0}$ and $\...
5
votes
1
answer
774
views
When is the flatness locus non-empty
Let $k$ be an algebraically closed field, $f:X \to Y$ be a surjective proper $k$-morphism locally of finite presentation between irreducible noetherian schemes. Assume that $Y$ is reduced. Under what ...
2
votes
0
answers
293
views
Flatness and intersection of fibers
Let $f:X \to Y$ be a flat, proper, surjective morphism between noetherian schemes. Assume $Y$ is irreducible and smooth over $\mathbb{C}$. Suppose that $X$ is the union of two schemes $X_1$ and $X_2$ ...
5
votes
1
answer
703
views
Simultaneous resolution of singularities in special cases of flat families of projective varieties
Let $\pi:\mathcal{X} \to B$ be a flat family of projective varieties. Assume that $B$ is irreducible. Suppose that $\mathcal{X}$ is smooth except for a closed subscheme, say $Y$ which is isomorphic to ...
3
votes
1
answer
1k
views
Morphism with non-reduced special fibre
Let $X, Y$ be irreducible projective varieties and $Y$ be smooth. Let $f:X \to Y$ be a flat projective morphism. Assume that a special fiber of $f$ is non-reduced i.e., there exists an irreducible ...
4
votes
1
answer
1k
views
Formal criterion of flatness
Let $k$ be a field, $S$ and $R$ be local $k$-algebras with residue field $k$ and $\phi:S\to R$ be a local homomorphism. Then $\phi$ induces (obviously) a natural transformation of "functors of points" ...