User naga venkata - MathOverflow

Functorial properties of blow-up

2013-01-16T13:43:57Z

Let $X, Y$ be projective algebraic surfaces with isolated singularities. Suppose they are diffeomorphic to each other. Denote by $\phi$ the diffeomorphism from $X$ to $Y$. Then does there exists a blow up $X', Y'$ of $X, Y$, respectively such that there exists a diffeomorphism $\phi':X' \to Y'$ which commutes with $\phi$ (via the natural maps from $X'$, $Y'$ to $X$, $Y$ respectively arising from blow up)?

What happens if $dim X=dim Y >2$? What happens if we assume that $X, Y$ lie as fibers over closed points of a family parametrized by a quasi-projective variety $B$ which is simply connected under the analytic topology (the underlying field is always $\mathbb{C}$)?

Geometry of curves with $A_n$ singularity

2013-03-17T16:20:46Z

I am a beginner in the study of curves with $A_n$ singularities. I had the following questions: Fix $n \ge 1$ and $N \ge 3$

1) Fix $2$ integers $d, g$. Is it true that the set of curves of degree $d$ and arithmetic genus $g$ in $\mathbb{P}^N$ with an $A_n$ singularity can be seen as a subscheme of a Hilbert scheme?

2) If the answer to ($1$) is true, can the subscheme be an open subset of an irreducible component?

3) Is it very difficult to compute the dimension of such subschemes? Examples/ideas/references for such computations will be very interesting to see.

4) Can we say something similar for $D_n$ singularities?

Any suggestions for references on this topic is most welcome.

Family with a fixed special fiber over finite fields

2013-03-15T01:40:58Z

Let $X$ be a smooth projective variety over a finite field $\mathbb{F}_p$. What are the conditions for the existence of a projective variety $X'$ over $\mathbb{Q}_p$ such that $X$ is a special fiber of $X'$?

Properties of Gorenstein ideal

2013-03-14T13:44:37Z

Fix an integer $k>4$. For any integer $r>0$, denote by $S_{r}:=\mathbb{C}[X_0,X_1,X_2,X_3]_{r}$ the vector space of degree $r$ polynomials in $X_i$ with coefficients in $\mathbb{C}$. Let $W$ be a vector subspace of $S_{2k}$ of codimension $1$ such that $W_{2k+1}=S_{2k+1}$ where $W_{2k+1}$ is the degree $2k+1$ part of the ideal (of $\mathbb{C}[X_0,X_1,X_2,X_3]$) generated by $W$. Define an ideal $I$ by $I_r$ is the biggest vector space in $S_r$ such that $I_r \otimes S_{2k-r} \subset W$ for $0< r<2k$. This ideal is Gorenstein of socle degree $k$.

The question is when is this ideal generated in degree less than or equal to $k$ i.e., we can find a set of generators of $I$ such that the degrees of the generators is less than or equal to $k$? Is it true that the codimension of $I_r$ in $S_r$ is a strictly increasing function for $r \le k$?

A good reference for this topic will be great help as well.

Edit: In the first para we add an extra assumption that the induced pairing $S_j/W_j \times S_{2k-j}/W_{2k-j} \to S_{2k}/W_{2k} \cong \mathbb{C}$ is a perfect pairing for $j<2k$.

Non-uniqueness of smooth compactification

2013-03-11T21:11:24Z

Let $U$ be a smooth quasi-projective variety. Does there always exist a smooth compactification of $U$? If not always when can we have smooth compactification? In particular, suppose $X$ is a singular projective variety and $U$ is the smooth locus. The question is does there always exist a smooth projective variety $Y$ containing $U$? If we add the condition that $U$ is open in $Y$ is $Y$ uniquely determined upto isomorphism?

On the equation defining a surface

2013-03-08T15:23:44Z

Fix a curve $C$ and a line both in $\mathbb{P}^3$ not intersecting one another. Suppose that the ideal of the curve is generated by $P_i$ and the line is definied by linear polynomials $l_1, l_2$. Is it true that a general degree $d$ smooth surface in $\mathbb{P}^3$ containing $C \cup l$ is defined by an equation of the form $$ l_1\left(\sum_iP_iQ_i\right)+l_2\left(\sum_iP_iQ'_i\right) $$ for $Q_i, Q'_i$ homogeneous polynomials of degree equal to $d-1-\deg(P_i)$?

Pull-back of algebraic cycles under holomorphic maps

2013-03-09T22:42:45Z

Let $f:X \to Y$ be a holomorphic map between two smooth complex projective manifolds. Is there a good notion of pull-back of algebraic cycles by $f$ which preserves degree in the following sense: Suppose $V_1, V_2$ be two cycles in $Y$ with the same Hilbert polynomial and suppose $f^{-1}(V_i)$ are algebraic varieties. Then will the pull-backs of $V_i$ by $f$ have the same dimension and degree (as algebraic cycles)? A small comparison with the divisor case is as follows: As far as I understand that pull-back of line bundles preserves degree in the sense as above. By identifying Picard group with the divisor class group we have a similar statement as above.

Upper bound on the dimension of the Hilbert scheme of space cuves

2012-11-01T15:56:46Z

Denote by $H_{P,Q}$ the flag Hilbert scheme parametrizing a pair $(C,X)$ such that $X$ is a degree $d$ surface in $\mathbb{P}^3$ with Hilbert polynomial $Q$ and $C \subset X$ is a curve with Hilbert polynomial $P$. Let $p_1$ be the natural projection map from $H_{P,Q}$ to $H_P$.

Then the questions are:

1) Is there an upper bound on the dimension of $Im(P_1)$ in terms of the degree $d$?

2) If the dimension of $H_{P,Q}$ is large (say greater than $d^2$) then if we replace $Q$ by the Hilbert polynomial of degree $d-1$ surfaces in $\mathbb{P}^3$ then is the corresponding dimension of $Im(P_1)$ is equal to the one before? This is equivalent to saying that if dimension of $H_{P,Q}$ is large then for a generic curve in $Im(P_1)$, do we have that $I_{d-1}(C) \not= \emptyset$?

3) Can we also say that the degrees of the defining equations of a generic curve in $Im(P_1)$ is the same? That is to say is there a fixed $r$-tuple of integers $(a_i)$ such that a generic curve in $Im(P_1)$ is defined by $r$ equations $Q_i$ of degree $a_i$ respectively? (of course, I do not expect $Q_i$ to be fixed)

Partial/known results and ideas of approaching this problem are most welcome.

Blowing up a projective surface

2013-03-05T00:42:33Z

Let $X$ be a smooth degree $d$ ($d>5$) surface in $\mathbb{P}^3$. We now blow up $X$ at a point and embedd it in $\mathbb{P}^3$. The question is does this resulting surface have only ADE singularities? If not when is it the case? What is the degree of the final surface?

Can any local complete intersection subvariety be an intersection of smooth hypersurfaces

2012-11-15T11:31:25Z

Let $Z$ be a local complete intersection subscheme of dimension $m$ in $\mathbb{P}^{2m+1}$. Let $P$ be the Hilbert polynomial of $Z$. Denote by $Hilb_P$ the Hilbert scheme of local complete intersection subschemes with Hilbert polynomial $P$. Is it true that for a generic subscheme $Z'$ in $Hilb_P$, $I(Z')$ can be generated by polynomials that define smooth hypersurfaces in $\mathbb{P}^{2m+1}$ i.e., does there exists polynomials $P_1, ..., P_n$ such that $I(Z')=(P_1,...,P_n)$ and the zero locus of $P_i$ is smooth for all $i$?

A slightly weaker condition would be to ask if a generic hypersurface in $pr_2 Hilb_{P,Q}$ is smooth where $Hilb_{P,Q}$ is the flag Hilbert scheme of pairs $(Z \subset X)$ where $Z$ is a local complete intersection subscheme of dimension $m$ with Hilbert polynomial $P$ contained is a hypersurface $X$ in $\mathbb{P}^{2m+1}$ with Hilbert polynomial $Q$ and $pr_2$ denotes the natural projection map.

Uniqueness of decomposition of completely reducible representations

2013-02-11T14:09:32Z

Let $X$ be a smooth, separated scheme of finite type over $\mathbb{F}_q$ where $q=p^r$ for some $r>0$. Let $gcd(l,p)=1, \rho:W(X) \to GL_r(\mathbb{Q}_l)$ be a Weil representation which is semi-simple. Is $\rho$ decomposable uniquely into irreducible representations? Is a similar statement true for the semi-simplification of the weil representation (in the case it is not semi-simple)?

Section of a projective morphism

2013-02-07T15:37:28Z

Let $X \to Y$ be a projective morphism. So, this map factors through a closed immersion $i$ of $X$ into $\mathbb{P}^n \times Y$ for some $n$ followed by the projection map to $Y$. When is it possible to define a map $\phi$ from $Y$ to $\mathbb{P}^n \times Y$ such that the image of $\phi$ is contained in the image of the closed immersion $i$ such that $\phi$ composed with the projection map is identity? When can we say that $\phi$ is itself a closed immersion.

Partial dehomogenization and smoothness

2013-01-17T17:12:14Z

Let $P_1l_1+P_2l_2$ be a homogeneous degree $d$ polynomial in $\mathbb{C}[X_0,X_1,X_2,X_3]$ which defines a smooth surface in $\mathbb{P}^3$. Here $l_i$ are linear polynomials and $l_1 \not=\lambda l_2$ for any $\lambda \in \mathbb{C}$. Does there exist $\lambda' \in \mathbb{C}$ such that $P_1+\lambda' P_2$ defines a smooth surface in $\mathbb{P}^3$? If so can we further say that this is true for general $\lambda'$?

Normal sheaf of non-reduced space curves

2013-01-29T12:39:17Z

Let $C$ be a local complete intersection curve in $\mathbb{P}^3$ that is non-reduced. The natural map $C_{red} \to C$ induces the sujective map on the normal bundles, $\phi:N_{C|\mathbb{P}^3} \to N_{C|\mathbb{P}^3} \otimes \mathcal{O}_{C_{red}}$ .

The question is when is the induced map, $$ H^0(N_{C|\mathbb{P}^3}) \to N_{C|\mathbb{P}^3}\otimes \mathcal{O}_{C_{red}} $$ surjective? In other words, when does $H^1(\ker \phi)$ vanish?

Singular locus of a Hilbert scheme

2013-01-28T13:28:37Z

Consider the Hilbert scheme $H$ of conics in $\mathbb{P}^3$. It is easy to see that there exists a closed subscheme $H'$ of $H$ parametrizing $2$ lines intersecting at a point. This can be seen as the image of the Hilbert flag scheme $Hilb_{P_1,P_2}$ where $P_1$ (resp. $P_2$) is the Hilbert polynomial of a line (resp. a conic) under the second projection map. Is this subscheme singular at every point i.e., is $H$ singular at every point of $H'$?

Is there a general condition when a subscheme of a Hilbert scheme (for example the image under the projection from a Flag Hilbert scheme as above) singular at every point? We know from definition that this is equivalent to saying the dimension of the global sections of the normal sheaf of the curve is higher than the dimension of the scheme. What would be interesting would be to get a condition in terms of intersection of curves.

When is a smooth projective variety a fibration

2013-01-17T17:30:13Z

Let $X$ be a smooth projective variety. Is there a criterion (apart from the definition) for the existence of a projective curve $C$ and a proper surjective morphism $\pi:X \to C$?

Embedding of curves in surfaces

2013-01-11T22:47:09Z

Let $C_1 \cup C_2$ be a curve in $\mathbb{P}^3$ and $X$ be a smooth degree $d$ surface in $\mathbb{P}^3$ containing them and $d \ge 6$. Further, assume that the minimum degree polynomial in $I(C_1 \cup C_2)$ is of degree less than $d/2$. Is it true that there exists a smooth degree $d$ surface in $\mathbb{P}^3$ containing $C_1$ and a line? (The underlying field is always $\mathbb{C}$)

Linear equivalence and Hilbert function

2012-04-15T12:39:41Z

Let $X \subset \mathbb{P}^3$ be a smooth degree $d$ surface containing two irreducible curves $C_1, C_2$ linearly equivalent to each other. If we assume that $X$ is general (among all degree $d$ smooth surfaces in $\mathbb{P}^3$) then is it true that $I_d(C_1)=I_d(C_2)$?

Embedding a surface in a projective space

2012-12-31T12:13:19Z

Let $X$ be a smooth degree $d$ ($d >5$) surface in $\mathbb{P}^3$. Let $\pi:\tilde{X} \to X$ be a blow-up of $X$ at a point. When is it possible to embed $\tilde{X}$ into $\mathbb{P}^3$?

In general, when can we embed a projective surface in $\mathbb{P}^3$? When is the resulting surface smooth?

Upper bound on the dimension of linear series on a smooth hypersurface

2012-12-17T11:33:28Z

Let $X$ be a smooth degree $d$ hypersurface in $\mathbb{P}^3$ and $d \ge 5$. Let $C \subset X$ be a reduced curve such that $C$ is not a complete intersection curve in $\mathbb{P}^3$. Is it true that the dimension of the linear series $H^0(\mathcal{O}_X(C))$ is at most $2$? The motivation of this question is as follows:

As far as I understand $h^0(N_{C|X})$ computes the dimension of the Hilbert scheme of curves in $X$. Since $C$ is not a complete intersection and $X$ is a surface, there can exist at most one dimensional family of curves deforming $C$ in $X$ which would implying that $h^0(N_{C|X}) \le 1$. We now use the long exact sequence associated to the short exact sequence, $$ 0 \to \mathcal O_X \to \mathcal O_X(C) \to N_{C|X} \to 0 $$ along with the fact that $H^1(\mathcal{O}_X)=0$ since it is a smooth hypersurface in $\mathbb{P}^3$ to conclude the above result. Is there some mistake in this logic?

$d$ points on a curve which are in the base locus of a pencil of planes

2012-12-11T15:15:14Z

Let $C$ be a reduced curve in $\mathbb{P}^3$ of degree $d$. Does there exist $d$ points on $C$ such that there exists a $1-$dimensional family of hyperplanes in $\mathbb{P}^3$ passing through these points?

Hilbert function of a set of points

2012-12-11T09:59:46Z

Let $C$ and $C'$ be two reduced curves in $\mathbb{P}^3$, $C'$ is irreducible as well. For a general hyperplane $H$ in $\mathbb{P}^3$, does the following hold true:

1) There exist a line $l$ such that $codim (I_d(C \cup l) + I_d(H))\le codim (I_d(C \cup C') + I_d(H))$?

2) The ideal $I_d(C \cup l) + I_d(H)$ is a radical ideal (of the set of points $C \cup l \cap Z(H)$)?

3) Does there exists a line such that the natural map:

$I_d(C \cup l)+I_d(H) \to (I_d(C)+I_d(H))\cap(I_d(l)+I_d(H))$ is surjective?

cohomology of torsion sheaves and nilpotent sheaves

2012-12-03T15:11:26Z

Let $X$ be a scheme and $\mathcal{F}$ be a sheaf on $X$ which is torsion $\mathcal{O}_X-$module (i.e., every local section is annihilated by an element of the ring $\mathcal{O}_X(U)$) or nilpotent (i.e., a power of any local section is zero). When can we say that $H^i(\mathcal{F})$ or $H^i(Hom_X(\mathcal{F},\mathcal{O}_X))$ vanishes for $i>0$. For example if $X$ is a curve then does the first cohomology groups given above vanish?

Let us take for example $X$ is a non-reduced curve in $\mathbb{P}^3$ ($X_{red}$ not smooth). Can we say that $H^i(Hom_X(I_X/I_X^2,I_{X_{red}}/I_X))$ equal to zero for $i$ equal to $0$ or $1$? (Here $I_X, I_{X_{red}}$ denotes respectively the ideal sheaves of $X, X_{red}$ in $\mathbb{P}^3$.)

Upper bound on the number of generators of a local complete intersection curve in $\mathbb{P}^3$

2012-11-13T17:02:45Z

Let $C$ be a local complete intersection curve in $\mathbb{P}^3$ (not irreducible or smooth) of degree $e$. Suppose $f_1, f_2$ (and $e_i=\deg(f_i)$) are two of the lowest degree generators of $I(C)$. Then:

1) Is it true that the minimal number of generators of $I(C)$ is less than or equal to $2+(e_1e_2-e)$? (Intuitively I would think this to be true by using the degree formula from intersection theory which says that $e_1e_2=\sum_im_i\deg(C_i)$ where $C_i$ ranges over the irreducible components of the curve defined by $f_1$ and $f_2$ and $m_i$ are their multiplicities)

2) Is there any known result/approach to compute the upper bound on the minimal number of generators of $I(C)$ in terms of its degree?

Normal sheaf of non-reduced locally complete intersection space curves

2012-11-27T16:02:22Z

Let $C$ be a non-reduced locally complete intersection curve on a smooth degree $d$ surface in $\mathbb{P}^3$ (for example a non-reduced Cartier divisor). For simplicity we can assume that $d>deg(C)$. Denote by $C_r$ the reduced scheme associated to $C$. It is clear the normal sheaves, $N_{C|\mathbb{P}^3}$ and $N_{C_r|\mathbb{P}^3}$ are locally free. Also, recall under the closed immersion, $i:C_r \hookrightarrow C$ we have the canonical maps, $i^*:H^0(N_{C|\mathbb{P}^3}) \to H^0(i^*N_{C|\mathbb{P}^3})$ and $H^0(N_{C_r|\mathbb{P}^3}) \to H^0(i^*N_{C|\mathbb{P}^3})$.

The question is whether it is true that the image of the latter map is contained in the image of $i^*$.

Vanishing of Ext group

2012-10-23T08:33:41Z

Let $C$ be a cartier divisor on a smooth projective surface in $\mathbb{P}^3$. Then we get the short exact sequence $$0 \to \mathcal{O}_X(-C) \to \mathcal{O}_X(-C_{red}) \to F \to 0$$ for some sheaf $F$. We see that $F$ is supported on $C$. Assuming $C \not= C_{red}$ when is it possible to say that $Ext^2_X(F,\mathcal{O}_X)=0$?

linear system of non-reduced divisor and associated reduced divisors

2012-10-08T12:27:39Z

Let $X$ be a smooth degree $d$ $(d \ge 5)$ surface in $\mathbb{P}^3$. Let $D$ be an effective Cartier divisor (hence locally of complete intersection) on $X$ and $D_{red}$ the associated reduced scheme which is also a Cartier divisor on $X$. Then, we have a natural inclusion of ideal sheaves, $$0 \to \mathcal{O}_X(-D) \to \mathcal{O}_X(-D_{red}).$$

Taking the dual we have

$$\mathcal{O}_X(D_{red}) \to \mathcal{O}_X(D) \to 0.$$

Since, $\mathcal{O}_X(D_{red})$ and $\mathcal{O}_X(D)$ are locally free sheaves of rank $1$, this would imply that the kernel of the latter map is zero. This would imply that $\mathcal{O}_X(D_{red}) \cong \mathcal{O}_X(D)$ . This result is very surprising. Is there a mistake in the proof or is there an explaination for this behaviour?

Hilbert function of a Hilbert scheme

2012-10-01T11:04:21Z

Given a Hilbert scheme $H$ of curves in $\mathbb{P}^3$ satisfying certain Hilbert polynomial, is there any way of understanding the degree or arithmetic genus of an irreducible component of the reduced scheme $H_{red}$? If so can this be generalised to Hilbert scheme parametrizing higher dimensional projective varieties or to Hilbert flag schemes?

base-point free linear system

2012-09-22T07:43:30Z

Let $C$ be a reduced curve in a smooth degree $d$ ($d \ge 5$) surface in $\mathbb{P}^3$. Suppose $C=C_1 \cup ... \cup C_r$ with $C_i$ irreducible and $C_i^2<0$ for all $i$. Then is the linear system $|C|$ base-point free?

Hilbert scheme of curves in a degree $d$ surface in $\mathbb{P}^3$

2012-09-18T17:00:44Z

Fix a Hilbert polynomial $P$ of a non-plane curve in $\mathbb{P}^3$. By a curve we mean a reduced scheme of pure dimension $1$ i.e., it can be reducible but is reduced. Suppose that the degree of these curves is $e$. For a general curve $C$ in $Hilb_P$ (the Hilbert scheme corresponding to $P$) and a general smooth degree $d$ ($d \ge 5$) surface $X$ in $\mathbb{P}^3$ for $d \ge e+2$ containing $C$, should we expect that the dimension of the linear series $|C|$ which is equal to $h^0(\mathcal{O}_X(C))-1$ to be equal to zero? (Intuitively I would expect this because in this case if $C' \subset C$ irreducible then $C'^2<0$ implying $\dim |C'|=0$)

Comment by Naga Venkata

2013-03-18T17:13:00Z

Why the downvote?

Comment by Naga Venkata

2013-03-15T19:40:57Z

@Matt: I am interested in curves and surfaces. We can assume that $\dim X'= \dim X+1$.

Comment by Naga Venkata

2013-03-14T15:38:07Z

@Smith: I thought I found the solution which turned out to be incorrect.

Comment by Naga Venkata

2013-03-09T23:32:29Z

@Sawin: I understand this fact. I meant that the ratio of the degrees of the pull-back of two line bundles with the same degree is $1$. That is what I try to explain in the beginning.

Comment by Naga Venkata

2013-03-08T21:58:47Z

@Sawin: It will be very helpful if you could elaborate a bit more. Are you saying that the equation defining the surface has to be multiplied by certain power of $x_i$ for some $i$? Can we say that if there exists a smooth surface of degree $d$ containing $C$ and $l$ then there exists a smooth surface of degree $d$ defined by an equation as above?

Comment by Naga Venkata

2013-03-05T06:59:33Z

@Mustopa: Thank you. You are completely correct.

Comment by Naga Venkata

2013-02-23T11:32:36Z

There is a problem with the display

Comment by Naga Venkata

2013-02-11T16:24:35Z

@Humphreys: Weil representations are representations where $W(X)$ is a weil group, a subgroup of the etale fundamental group of $X$. It is not a finite group. A good reference is Deligne's paper on the proof of the weil conjectures.

Comment by Naga Venkata

2013-02-09T16:46:34Z

@Mohan, Smith: We can assume that both $X$ and $Y$ are smooth. The example I have in mind is applying Chow's lemma to a proper morphism $Y' to Y$ which gives rise to $X$ as described in the question.

Comment by Naga Venkata

2013-02-07T17:29:25Z

@IMeasy: This seems to a very general condition. Could you please elaborate or suggest some reference for your answer.

Comment by Naga Venkata

2013-01-29T12:41:55Z

There is some problem with the layout

Comment by Naga Venkata

2013-01-24T06:43:37Z

@Mohan: Thanks a lot for the answer. I understand most of it. However, could you please elaborate on the natural map from $O_C(−d−1)\to I_C/I^2_C$ induced by the degree d smooth surface $X$? Is it simply multiplication by the defining equation of $X$?

Comment by Naga Venkata

2013-01-13T13:25:02Z

@Sawin: $C_i$ are distinct curves. Answer to the last question is any line.

Comment by Naga Venkata

2012-12-17T11:34:27Z

There a problem with the display of the short exact sequence

Comment by Naga Venkata

2012-12-02T12:40:52Z

@Sasha: As far as I understand it arises from $i^∗(I_C/I_C^2) \to I_{C_r}/I_{C_r}^2$ and then taking the $Hom_{C_r}(−,\mathcal{O}_{C_r})$. This is a standard map while defining the tangent space to a flag Hilbert scheme.