All Questions
32 questions
10
votes
2
answers
3k
views
Kodaira-Spencer map in a concrete instance
Let $\pi:X_{\epsilon} \rightarrow \Delta$ be a family of (say smooth) projective plane curves parametrized by $\Delta:=\operatorname{Spec}(k[\epsilon])$, and let $X=X_0$ be the closed fiber.
Suppose ...
- 23.3k
8
votes
1
answer
381
views
Projective embedding in families of curves
Let $\pi:\mathcal{X} \to B$ be a family (flat, projective, surjective morphism) of projective curves (not necessarily reduced) where $B$ is smooth, irreducible. Suppose that for some closed point $b_0 ...
- 2,323
6
votes
1
answer
480
views
Is the Quot-scheme over non-singular curve reduced
Let $k$ be an algebraically closed field, $C$ a non-singular projective curve over $k$ of genus at least $2$ and $\mathcal{F}$ a locally free sheaf on $C$. Let $r,d$ be two integers satisfying $\...
- 1,981
5
votes
2
answers
984
views
Injectivity under flat base change of the Picard group on smooth projective curves
Let $K$ be a field of characteristic $0$, $X_K$ a smooth projective curve over $K$. Denote by $\bar{K}$ the algebraic closure of $K$. The base change morphism $X_{\bar{K}} \to X_K$, induces via the ...
- 503
5
votes
1
answer
299
views
First cohomology of tangent sheaf of rational curve
Let $C$ be a reduced, connected, projective and purely one-dimensional scheme of finite type over a field $k$.
Suppose that $C$ is rational, i.e. that the normalisation of $C$ is a disjoint union of ...
- 984
5
votes
1
answer
570
views
Degeneration of smooth curves and Picard-Lefschetz formula
Let $\pi:\mathcal{C} \to \Delta$ be a family of projective curves of genus $g \ge 2$ over the unit disc $\Delta$, smooth over the punctured disc $\Delta\backslash \{0\}$ and central fiber $\pi^{-1}(0)$...
- 2,022
5
votes
1
answer
367
views
Do singularities of plane curves deform independently?
If $C\subset\mathbb{P}^2$ is an integral curve of degree $d$, do its singularities deform independently as we vary $C$ over degree $d$ curves? If not, what about in the case $C$ is a nodal curve or $C$...
- 1,537
5
votes
1
answer
589
views
Tangent Space of the Hodge bundle on the moduli space of curves
Let $k$ be an algebraically closed field and $\mathcal M_g$ denote the moduli space (stack) of smooth curves of genus $g$ over $k$. Using the universal curve $\pi \colon \mathcal C_g \to \mathcal M_g$,...
- 445
4
votes
2
answers
335
views
Deformation of "Hecke modification"
Let $X$ be a smooth curve over $\mathbb{C}$. I wish to compute the deformation of the following data $(E,F,x)$. $E$ and $F$ are locally free sheaves over $X$ and $x$ is a point on $X$. They satisfy:
...
- 125
4
votes
1
answer
377
views
Deformation theoretic argument on dimension counting of naive Hurwitz scheme
I'm reading the Atanas Atanasov's course notes of Joe Harris' course Geometry of Algebraic Curves
and have a question about a suggested modification of an dimension
countinging argument applying ...
- 6,018
3
votes
1
answer
326
views
Segre embedding and intersections by hyperplanes
Consider the Segre embedding $$ \mathbb{P}^2 \times \mathbb{P}^2 \to \mathbb{P}^8.$$ Denote by $V$ the image of the Segre embedding and by $B$ the locus of triples $(H_1, H_2, H_3)$ with $H_i \in H^0(\...
- 2,323
3
votes
1
answer
702
views
Family of elliptic curves in $\mathbb P^3$
For points $p_1=[1,0,0,0], p_2=[0,1,0,0], p_3=[0,0,1,0]$, $p_4=[0,0,0,1]$ and $p_5=[1,1,1,1]$
in the projective space $\mathbb P^3$, Let $l_{ij}$ be the line through $p_i, p_j$.
Let
$$C=l_{12} \cup ...
- 1,841
3
votes
1
answer
254
views
Degeneration of curves in smooth families
Heuristically, I want to know, given a smooth, projective morphism from a scheme to a discrete valuation ring, if the generic fiber can be 'covered' by a family of geometrically integral curves, is it ...
- 2,323
3
votes
1
answer
190
views
Connected sum of algebraic curves, handlebody decomposition, and induction on genus
Are there any nice methods of taking algebraic curves $C_1, C_2$ of genera $g_1,g_2$ and producing a curve $C_3 = C_1 \# C_2$ of genus $g_3 = g_1+g_2$? I'm imagining doing this over any field, but ...
- 1,338
3
votes
0
answers
255
views
Intuition behind results in Mumford's “Lectures on curves on an algebraic surface”, II
NOTE: This is a followup to my question here.
These are some questions concerning Mumford's "Lectures on curves on an algebraic surface".
We concern ourselves with questions of the Picard variety $P$...
user97565
2
votes
0
answers
169
views
Exponential map of moduli space
Let $\mathcal{M}$ be a projective smooth moduli space over $\mathbb{C}$ (the specific example I have in mind is the moduli of curves $\mathcal{M}_g)$. Consider a point $[X]\in \mathcal{M}(\mathbb{C})$....
- 4,408
2
votes
0
answers
180
views
Linear projection from a point preserves flatness
Let $\pi:\mathcal{X} \to S$ be a flat family of affine curves contained in $\mathbb{C}^n$ for $n \ge 3$ i.e., $\mathcal{X} \hookrightarrow \mathbb{C}^n_S$ and the inclusion commutes with the natural ...
- 1,593
2
votes
0
answers
153
views
An analogue of Brill-Noether for hypersurfaces?
Let $d,g,r$ be natural numbers such that $d \geq 1$, $g \geq 2$, $r \geq 3$. Denote by $\mathcal{H}_{d,g,r}$ the Hilbert scheme classifying subschemes of $\mathbb{P}^r$ with Hilbert polynomial $P(x) = ...
- 1,981
2
votes
0
answers
345
views
Examples of semi-stable models of curves
Let $R$ be a discrete valuation ring with fraction field $K$ of characteristic zero and residue field $k$ of characteristic $p>0$. Assume $k$ is algebraically closed. I want to produce examples of ...
- 2,323
2
votes
0
answers
507
views
Fiber of the specialization map of Picard groups
Let $R$ be a Henselian discrete valuation ring with residue field $k$ of positive characteristic and fraction field $K$ of characteristic zero. Let $\pi:X_R \to \mathrm{Spec}(R)$ be flat, projective ...
- 2,323
2
votes
0
answers
357
views
Deformation of complete intersection curves
Let $\mathcal{X} \to B$ be a family of smooth surfaces in $\mathbb{P}^3$. Let $\mathcal{C} \to B$ be a family of curves in $\mathbb{P}^3$. Assume that for all $b \in B$, the fiber $\mathcal{C}_b$ is ...
- 833
1
vote
2
answers
325
views
Jacobian of a curve and field extension
Let $K$ be a field of characteristic zero and $X_K$ a smooth projective curve on $K$. Denote by $\bar{K}$ the algebraic closure of $K$ and $X_{\bar{K}}$ the base change of $X_K$ to $\bar{K}$. Under ...
- 503
1
vote
1
answer
203
views
Assumption of genus at least $2$ for stable curves
In the article "The irreducibility of the space of curves of given genus" by Deligne, Mumford, the definition of stable curves start with the assumption that the genus is at least $2$. Why is this ...
- 1,981
1
vote
1
answer
223
views
Does $\delta$-invariant give a sufficient condition for flatness for plane curve singularities?
Let $\pi:\mathcal{C}\to S$ be a morphism of schemes such that $\mathcal{C} \subset \mathbb{C}^2 \times S$ with the inclusion map commuting with the natural projection to $S$ and for all $s \in S$, $\...
- 2,126
1
vote
1
answer
315
views
Variation of global sections of line bundles
The underlying field is $\mathbb{C}$.
Let $\pi:\mathcal{C} \to \mathbb{A}^n$ be a flat family of projective curves (not necessarily smooth) of genus $g \ge 2$. Assume $\mathcal{C}$ is regular. Let $\...
- 1,981
1
vote
1
answer
374
views
Model over DVR for smooth projective curves
Let $C$ be a smooth, projective, geometrically irreducible curve of genus at least $2$ over a complete discrete valued field $F$ of characteristic zero (not necessarily algebraically closed). Let $R$ ...
- 503
1
vote
0
answers
88
views
Infinitesimal neighbourhoods and simultaneous normalization
Let $B$ be a local, complete, integral $\mathbb{C}$-algebra of Krull dimension $1$ and $n:B \to \mathbb{C}[[t]]$ the normalization map. Given any local artinian $\mathbb{C}$-algebra $A$, we say that ...
- 2,126
1
vote
0
answers
149
views
Smoothability of stable curves in mixed characteristic
Let $R$ be a complete DVR with residue field $k$ algebraically closed of characteristic $p$ and fraction field $K$ of characteristic zero. Let $C$ be a stable curve (in the sense of Mumford-Deligne) ...
- 2,323
1
vote
0
answers
106
views
Smoothing a map between curves
In the paper Families of Rationally Connected Varieties, the authors consider a map $X\rightarrow B$, where $X$ is obtained by taking disjoint copies of $B$ mapping isomorphically to $B$ and ...
- 1,537
1
vote
0
answers
125
views
Obstruction to Gorenstein Liaisons of space curves
Let $P_1,P$ be Hilbert polynomials of curves in $\mathbb{P}^3$. Denote by $H_{P_1,P}$ the flag Hilbert scheme parametrizing pairs $(C_1 \subset C)$ where $C_1, C$ are of Hilbert polynomials $P_1$ and $...
- 503
1
vote
0
answers
190
views
dimension of spaces of rational curves in a variety!
Do you know calculate the dimension of the space of rational curves of degree
m, through d given points, contained in some projective variety?
- 63
0
votes
1
answer
360
views
Deformations of the Fermat curve
We work over an algebraically closed field $k$, say of characteristic $0$, just in case, and we let $C$ be a smooth curve over $k$. First-order deformations of $C$ (or of any smooth variety for that ...
- 119