User fei ye - MathOverflow

Enriques classification of algebraic surfaces

2013-04-29T02:58:31Z

Surfaces can be classified using Kodaira dimensions and birational invariants. Denote by $\kappa(S)$ the Kodaira dimension of a smooth projective surface $S$. For a smooth projective surface, an equivalent characterization of Kodaira dimension uses the pluricanonical genus $P_{12}=\dim H^2(S, 12K_S)$ and $K_S^2$. When $K_S^2=0$ and $K_S$ is nef, we see that $0\leq \kappa(S)\leq 1$.

Is there an easier way (which means not using the classification of surfaces of $\kappa=0$) to proof that that $P_{12}=1$ and $K_S$ is nef implies $\kappa(S)=0$ or that $P_{12}\geq 2$, $K_S^2=0$ if $K_S$ is nef and $\kappa(S)=1$.

Geometry meaning of higher cohomology of sheaves?

2010-01-10T05:01:13Z

Let $X$ be a projective algebraic variety over a algebraic closed field $k$. Let $\mathcal{F}$ be a coherent sheaf on $X$. We know that $H^0(X, \mathcal{F})$ is the vector space of global sections of $\mathcal{F}$. This gives us a geometric illustration of $H^0$. For example, let $I_D$ be the ideal sheaf of a hypersurface $D$ of degree > 1 in a projective space $\mathbb{P}^n$, then it is easy to see that $$H^0(\mathbb{P}^n,I_D\otimes\mathcal{O}_{\mathbb{P}^n}(1))=0.$$

In fact, there is no hyperplane containing $D$, which means that there is no global section of $\mathcal{O}_{\mathbb{P}^n}(1)$, which are hyperplanes, containing $D$. Hence $H^0(\mathbb{P}^n, I_D\otimes\mathcal{O}_{\mathbb{P}^n}(1))=0$. However, my first question is how to understand higher cohomologies of sheaves in geometric ways. The following questions then come out:

1) How to understand Serre's vanishing theorem, i.e., is there a geometric way to think about the vanishing of $H^q(X, \mathcal{F}\otimes A^n)$ for $n>>1$, where $\mathcal{F}$ is coherent and $A$ is ample.

2) How to understan Kodaira's vanishing theorem geometrically.

Maybe a concrete question will help, say $D$ a subvariety of $\mathbb{P}^n$, how to determine geometrically whether $H^0(\mathbb{P}^n, I_D\otimes\mathcal{O}_{\mathbb{P}^n}(1))$ vanishes or not.

Smooth projective varieties of Picard number one

2013-02-26T05:44:57Z

Is there a classification theory of smooth projective varieties with Picard number 1?

By Lefschetz theorem, if $X$ is a complete intersection variety of dimension at least $3$, then the Picard number $\rho(X)=1$.

Can some one give an example of smooth projective variety with picard number 1 which is not a complete intersection? What kind of additional conditions, especially cohomological conditions, will force such a variety to be a complete intersection?

multiplicities of rational singularities in higher dimension

2012-11-15T08:20:15Z

For a normal surface rational singularity, we know that the multiplicity of is bounded by $e-1$ where $e$ is the embedding dimension (See for example Miles Reid's book "Chapters on algebraic surfaces").

I am wondering if this inequality also holds in higher dimension. If not, what can we say about the multiplicities.

Dimension and singularities of the minimal log canonical center

2012-11-01T03:58:45Z

As explained in Singularities of pairs by Karl Schwede, log canonical center is quite useful for induction. For example, to study effective freeness by induction on dimension of log canonical centers. In general, log canonical centers are not smooth.

Question: How does the dimension and multiplicity of a log canonical center related to the pair itself?

Let $(X, D)$ be a log canonical pair and $Z$ the minimal log canonical center of the pair? What property of $(X, D)$ will control the dimension of $Z$? How singular could $Z$ be?

Cohen-Macaulay sheaves which are not locally free

2012-09-07T04:33:17Z

A coherent sheaf $\mathcal{F}$ over a Noetherian scheme $X$ is called (maximal) Cohen-Macaulay if $depth_{\mathcal{O}_x}(\mathcal{F}_x) = \dim\mathcal{O}_x$ for any $x\in X$, where $\mathcal{O}_x$ is the local ring of $X$ at $x$.

Is there a simple example of $(X, \mathcal{F})$ such that $\mathcal{F}$ is Cohen-Macaulay but not locally free?

For regular schemes, I think they are equivalent. What about singular schemes? Under what conditions, a Cohen-Macaulay sheaf is locally free?

Evidences on Hartshorne's conjecture? References?

2010-02-03T16:07:56Z

Hartshorne's famous conjecture on vector bundles say that any rank $2$ vector bundle over a projective space $\mathbb{P}^n$ with $n\geq 7$ splits into the direct sum of two line bundles.

So my questions are the following:

1) what is an evidence for this conjecture?

2)why is the condition on $n\geq 7$, but not other numbers?

3)any recent survey or reference on this conjecture?

Punctured spectrums of local rings

2012-03-26T14:20:44Z

Let $A$ be a local ring with the unique maximal ideal $\mathfrak{m}$. The punctured spectrum of $A$ is the open subset $\text{Spec}(A)\setminus {\mathfrak{m}}$. I have seen many papers (for instance Horrocks' papers) studying vector bundles over algebraic varieties (in particular, projective spaces) by putting them over a punctured spectrum of a local ring.

However, I am wondering why punctured spectrum is better than varieties in this satiation. I feel that geometric pictures of varieties are clearer than that of a punctured spectrum. More essential, are the categories of coherent sheaves over a variety and its punctured spectrum equivalent? How much information about coherent sheave (in particular, vector bundles) can be recovered from punctured spectrums? For second question, I am thinking examples that reflect some relations between those two gadgets.

Ext modules of coherent sheaves and associated modules

2012-05-29T14:25:56Z

Let $\mathcal{E}$ and $\mathcal{F}$ be two coherent sheaves on a polarized projective variety $(X,\mathcal{O}_X(1))$.

Denote by $E=\Gamma_*(\mathcal{E})=\oplus_{k\in\mathbb{Z}}\mathcal{E}(k)$, $F=\Gamma_*(\mathcal{F})$, and $R= \Gamma_\ast(\mathcal{O}_X)$.

Is it true that $$\oplus_{k\in\mathbb{Z}}Ext^i_{\mathcal{O}_X}(\mathcal{E}, \mathcal{F}(k))=Ext^i_R(E, F)?$$

If the equality does not hold in general, when and to what extent it will hold?

projective dimension of finitely generated modules over char 0

2012-05-29T07:50:28Z

In the paper "On modules of finite projective dimension over complete intersection" Dutta proved that a finitely generated module over a local complete intersection ring over char $p>0$ has finite projective dimension if it satisfies certain condition on Tor. I am wondering if there are any similar results over char $0$. I am mainly interested in finitely generated module over homogeneous coordinate rings of projective varieties.

Equivalent definitions of arithmetically Cohen-Macaulay varieties

2012-04-27T18:35:46Z

Let $X\subset \mathbb{P}^n$ be a projective algebraic variety with coordinate ring $R$. $X$ is said to be arithmetically Cohen-Macaulay if $R$ is a Cohen-Macauly ring. A equivalent definition is that the natural morphism $$R\to \oplus_{k\in\mathbb{Z}}\mathcal{O}_X(k)$$ is bijictive and $H^i(X,\mathcal{O}_X(k))=0$ for all $k\in\mathbb{Z}$ and $1\leq i\leq \text{dim} X-1$. Does anyone know a proof for the equivalence? Without assuming that $X$ is Cohen-Macaulay, I can only prove that $R$ is Cohen-Macauly at $0$, the vertex of the affine cone. How to prove $R$ or the section ring is Cohen-Macauly at other points of the affine cone? Did I miss something from the idea? Or is there an complete alternative proof?

Direct summands of direct sum of line bundles on projective varieties

2012-03-25T09:02:41Z

Serre-Swan's theorem (see the MO discussion) says that any locally free sheaf over an affine variety is a direct summand of a free sheaf. However, this is not true on projective varieties. It is not hard to check that a non-trivial line bundle with non-zero global sections can not be a direct summand of a free sheaf. The reason is that, being a direct summand of a free sheaf implies that the dual line bundle also has non-zero global sections. But that implies the line bundle is trivial.

I am wondering if the same holds for vector bundles, i.e. if a vector bundle and its dual over a projective variety both have non-zero global sections, then the vector bundle is trivial.

Another I think related question is the following:

Is a direct summand of a direct sum of line bundles on a projective variety also a direct sum of line bundles?

Fundamental group of the complement of a conic-line arrangement

2011-12-18T11:37:01Z

This is a problem concerning a lemma in Oka's paper "On the fundamental group of the complement of a reduced curve in $\mathbb{P}^2$". Let $C$ be a curve in $\mathbb{P}^2$ and $L$ be a general line to $C$. The lemma says that $\pi_1(\mathbb{P}^2-C\cup L)$ is abelian if and only $\pi_1(\mathbb{P}^2-C)$ is abelian. Now consider the arrangement of a conic $C$ and three lines $L_1$, $L_2$ and $L_3$ in $\mathbb{P}^2$. Assume that the conic passes through the three double points $L_1\cap L_2$, $L_1\cap L_3$ and $L_2\cap L_3$. We can show that $\pi_1(\mathbb{P}^2-C\cup L_1\cup L_2\cup L_3)$ is abelian. Let $L_\infty$ be a general line to the arrangement. According to Oka's lemma, the fundamental group $\pi_1(\mathbb{C}^2-C\cup L_1\cup L_2\cup L_3)=\pi_1(\mathbb{P}^2-C\cup L_1\cup L_2\cup L_3\cup L_\infty)$ should be abelian. However, without the projective relation, it is impossible to prove that the group is abelian. I don't know where I made mistakes. The only mistake that I suspect is that $\mathbb{P}^2-C\cup L_1\cup L_2\cup L_3\cup L_\infty$ does not equal $\mathbb{C}^2-C\cup L_1\cup L_2\cup L_3$. But why they do not equal.

$\textbf{Added:}$ Here are the fundamental groups. The computation uses braid monodromy method. The fundamental group of the affine complement is $A=<1, 2, 3, 4 \mid 431=314=143, 432=324=243, 132=321=213>$. The fundamental group of the projective complement has an extra relation. The group is $G=<1, 2, 3, 4 \mid 431=314=143, 432=324=243, 132=321=213, 43^221=e>$, where $e$ is the group identity.

Irreducibility of intersections of quadric hypersurfaces

2011-11-30T09:34:43Z

It is known (see the MO question " Varieties cut by quadrics") that every projective variety can be realized as a scheme-theoretic intersection of quadrics. Is there a way to determine if the intersection of irreducible quadric hypersurfaces is irreducible? For example, consider equations of the following form: $$ (x_2-x_1)(y_{2i}-y_{1j})-(x_3-x_1)(y_{3k}-y_{1j})=0, $$ where $i=1, 2, \dots, M$, $j=1,2,\dots, N$ and $k=1, 2, \dots, R$, and $M$, $N$ and $R$ are fixed natural numbers. Is there a way to determine if the intersection is irreducible? More generally, what about equations of the following form:

$$ (x_n-x_m)(y_{ni}-y_{mj})-(x_r-x_m)(y_{rk}-y_{mj})=0, $$ where $m:n:r\neq 1:1:1$.

When is an algebraic variety $\mathbb{Q}$-factorial?

2011-08-23T09:19:18Z

A variety is $\mathbb{Q}$-factorial if every global Weil divisor is $\mathbb{Q}$-Cartier. How bad singularities are allowed so that the algebraic variety is still $\mathbb{Q}$-factorial? Is a singular curve $\mathbb{Q}$-factorial? For example is a nodal-cuspidal plane curve $\mathbb{Q}$-factorial?

Answer by Fei YE for Stable base loci

2011-08-28T14:23:13Z

I don't know if the following is the elementary theory of multiplier ideals mentioned in the paper.

In the case of $B_+(D)$, I think the key point is that we can assume that $M$ is a nef and big line bundle. Since $M$ is big, the multiplier ideal $\mathcal{J}(|M|)$ is not trivial at $Bs(|M|)={{f^{-1}(x)}}$ (Theorem 11.2.21, Lazarsfeld). But for a nef and big line bundle $M$, the multiplier ideal $\mathcal{J}(|M|)$ is trivial (Proposition 11.2.18, Lazarsfeld).

Minimal number of nodes in a complex line arrangement.

2011-08-11T08:05:27Z

Let $\mathcal{A}$ be a collection of $n$ lines. Assume that $\mathcal{A}$ is not a pencil. It is known (see http://www.springerlink.com/content/320p742475v6q746/) that if all lines are in $\mathbb{RP}^2$, then there are at least $6n/13$ nodes.

What is the minimal number of nodes, if all lines are in $\mathbb{CP}^2$?

Multiplicity of a singular point

2011-04-27T21:07:43Z

Let $X$ be a smooth projective complex variety. Assume that $Z$ is a subvariety. Let $T$ be a generic complete intersection of codimension $\dim Z-1$. Assume that $p$ is a point in $Z_T:=T\cap Z$. Is there a formula relating $mult_p(Z)$ and $mult_p(Z_T)$?

Stability of multiplier ideals under small perturbations

2011-05-29T19:30:07Z

Let $G$ be an ample $\mathbb{Q}$-divisor on a smooth variety $X$. Let $D$ be a $\mathbb{Q}$-divisor linearly equivalent to $G$. Let $f: Y\to X$ be a common log resolution of $G$ and $D$. We define the multiplier ideal of a divisor $G$ as $I(G)=f_*O_Y(K_{Y/X}-[f^*G])$. Are the multiplier ideals $I(G)$ and $I((1-t)G+tD)$ the same for sufficiently small $t>0$?

snake lemma in category of groups

2011-01-24T21:03:01Z

In Wiki, under the item "category of groups", it states that the snake lemma fails in category of groups, however the nine lemma is valid. However, in the preface of the book " Mal'cev, protomodular, homological and semi-abelian categories ", it says "And category theory could not grasp either the conceptual foundations of the homological lemmas: the Nine Lemma, the Snake Lemma, which remain valid and strongly meaningful in the category Gp of groups, even if this category does not belong to the abelian setting in which these lemmas are generally proved in a significant categorical way."

Does the snake lemma fail or not in the category of groups? Any counter example or reference concerning that?

Homotopy type of complement of a plane algebraic curves.

2011-01-20T04:59:30Z

Assume that $X$ is the complement of a plane algebraic curve $C$ in $\mathbb{C}^2$ and Y is the complement of the union of $C$ and a line $L$ (not contained in $C$). Assume that $Y$ is $K(\pi, 1)$. Is it true that $X$ is $K(\pi, 1)$? Why or why not?

Why do automorphism groups of algebraic varieties have natural algebraic group structure?

2009-12-14T00:04:35Z

I am not sure that all automorphism groups of algebraic varieties have natrual algebraic group structure. But if the automorphism group of a variety has algebraic group structure, how do I know the automorphism group is an algebraic group. For example, the automorphism group of an elliptic curve $A$ is an extension of the group $G$ of automorphisms which preserve the structure of the elliptic curve, by the group $A(k)$ of translations in the points of $A$, i.e. the sequence of groups $0\to A(k)\to \text{Aut}(A)\to G \to 0$ is exact, see Springer online ref - automorphism group of algebraic variaties. In this example, how do I know $\text{Aut}(A)$ is an algebraic group.

How to resolve a wedge product of vector bundles

2010-10-13T03:58:49Z

Let $X$ be an algebraic variety. Consider an exact sequence $$0\to A\to B\to C\to 0$$ of vector bundles on $X$. I have seen in different papers the following type resolution of wedge product of $C$ (or $A$) $$0\to S^kA\to S^{k-1}A\otimes B\to S^{k-2}A\otimes \wedge^2B\to \cdots\to \wedge^kB\to \wedge^k C\to 0.$$

Question: does this resolution come from certain geometric context? Is there a proof which involves certain geometric aspects, for example, using projective bundles associated to the vector bundles?

Serre type vanishing theorem of coherent sheaves on quasi-projective variety?

2010-04-28T19:16:33Z

For a projective variety $X$, Serre's vanishing theorem says that $H^i(X, \mathcal{F}(n))=0$ for any coherent sheave, $i>1$ and sufficiently large $n$. I am wondering, is there a similar type of vanishing theorem on quasi-projective varieties, namely, let $Y$ be a quasi-projective variety, what can we say about the vanishing of $H^i(Y, \mathcal{F}(n))$ under the same setting as projective case.

Or is there a similar type of theorem for local cohomology, say, when is $H^i_{pt}(X, \mathcal{F}(n))$ vanishing?

Derived categories of symmetric products

2010-09-29T16:46:39Z

Let $X$ be a smooth projective algebraic variety and $D^b(X)$ be the derived category of coherent sheaves on $X$. Denote by $Sym^nX$ the $n$-th symmetric product of $X$. Can we describe the derived category $D^b(Sym^nX)$ in terms of $D^b(X)$. If so, how are they related? Is there any reference?

This question is intrigued by the question Hilbert schemes of points and exceptional collections asked by Cat.

Vanishing of Self-Ext groups of vector bundles

2010-08-13T02:47:54Z

Let $E$ be a rank two vector bundle on $\mathbb{P}^n$. Assume that $\text{Ext}^1(E, E)=0$. Will $\text{Ext}^2(E, E)$ be zero? Why? Any geometric explanation (in terms of deformation theory?)?

Edit: As pointing out by Angelo, in the case $n=2$, the answer is no. However, I really want to know when $n\geq 4$.

Can $\mathbb{P}^n$ be regarded as an algebraic vector bundle over some algebraic variety?

2010-01-27T14:59:11Z

Is it possible that $\mathbb{P}^n$ is an algebraic vector bundle over some algebraic variety? This is an interesting question that my friend asked in a student seminar. I believe that the answer is NOT. Because the only global sections of $\mathcal{O}_{\mathbb{P}^n}$ are constants. However as a total space of vector bundles $E$ over $X$, the global sections are all in $\Gamma(X, \oplus Sym^n E)=\oplus \Gamma(X,Sym^n E)$ which may not be $\mathbb{C}$ in general. The trouble case is when $\Gamma(X, E)=0$? Am I correct? Is there any other explanation?

Edit: Thanks to everyone for your comments and answer. It seems that compactness is the correct way to follow.

However, I still wondering that why the argument on sections doesn't work. In other words, is there an example of a non-proper algebraic variety whose structure sheaf only has constant global sections.

Proof of a Theorem in the paper "Construction of bundles on P^n" by Horrocks

2010-04-07T04:42:23Z

I am trying to understand Horrocks's construction of vector bundles. However I have been stuck on the proof the first theorem in the paper.

In the paper, a trivial bundle is a direct sum of Hopf bundles $\mathcal{O}(p)$.

Theorem: Let $E$ be a vector bundle without a trivial direct summand. Then there exist a trivial bundle $T$ such that $E\oplus T$ has a filtration $$E\oplus T=F^0 \supseteq F^1\supseteq F^2\supseteq\cdots\supseteq F^N=0$$ with $F^i/F^{i+1}$ a twisted exterior power of the tangent bundle $T_{\mathbb{P}^n}$.

Here is his proof:

Take a resolution $L$ of the dual of $E$ by trivial sheaves which is exact as a resolution of graded modules. The dual $L^*$ can be dismantled into Koszul complexes.

Here are my questions. How to break up $L^*$ into Koszul complexes? What are the Koszul complexes? Where does the $T$ come from?

Thanks in advance!

Answer by Fei YE for Nef divisors with few global sections

2010-04-13T15:02:08Z

How about the trivial divisor $\mathcal{O}_X$. It is nef because the intersection with any curve is 0. But $H^0(X, \mathcal{O}_X)=\mathbb{C}$.

Finite morphisms between algebraic varieties are flat?

2010-04-09T01:43:52Z

Let $f: X\to Y$ be a finite (surjective) morphism between two algebraic varieties. I know when $X$ and $Y$ are non-singular and $\dim Y =1$, $f$ is flat. But in general, is it true that $f$ is a flat morphism?

Comment by Fei YE

2013-05-23T15:42:47Z

I guess that you mean that $H^1(\mathbb{P}^n, E(t))=0$ for any $t\in \mathbb{Z}$.

Comment by Fei YE

2013-05-01T05:05:07Z

Dear Philip, thanks for the answer. Maybe my question is not clear. The classification method is exactly what I want to avoid. I am looking for a direct way of proof. Thanks again.

Comment by Fei YE

2013-04-30T09:30:47Z

Thanks, MP. You are right, the second one is easier. I should ask the other way around.

Comment by Fei YE

2013-02-26T08:29:40Z

Thanks a lot for the comment. Are there other types?

Comment by Fei YE

2012-11-02T02:09:19Z

Dear Professor Kovács, thanks a lot for the references.

Comment by Fei YE

2012-09-08T08:08:51Z

Dear Professor Liu, thank you so much for the answers.

Comment by Fei YE

2012-09-07T07:11:47Z

Wow, thanks a lot! You always provide nice examples.

Comment by Fei YE

2012-09-07T07:04:19Z

Thank you very much. What if we put good conditions on both the scheme and the sheaf. For instance, $X$ is a complete intersection and $\mathcal{F}$ is a reflexive sheaf?

Comment by Fei YE

2012-05-31T03:44:04Z

Thanks a lot. The last comment seems very useful.

Comment by Fei YE

2012-05-30T10:50:25Z

Sorry. What I really mean is that $H^i(X, \mathcal{F}{k})=0$ for all $k$. Thank you for your answer.

Comment by Fei YE

2012-05-30T06:00:47Z

What if we assume that $H^i(X, \mathcal{F})=0$ for all $i>0$?

Comment by Fei YE

2012-04-28T08:58:55Z

@Piotr: Thank you very much for your answer. I like the argument.

Comment by Fei YE

2012-03-29T08:20:14Z

Thanks for your answer. I don't understand why the punctured spectrum is of smaller dimension. Can you explain it? Thanks!

Comment by Fei YE

2012-03-27T08:23:08Z

Dear Mahdi, can you say more about the connection between problems over punctured spectrums of local rings and projective varieties, and expand your comments to an answer?

Comment by Fei YE

2012-03-26T15:37:11Z

Is there any relation (such as an isomorphism) between a projective variety and the completion of its punctured spectrum?