Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology.

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History of the connection between Riemann surfaces and complex algebraic curves

As noted in the question "Links between Riemann surfaces and algebraic geometry", there are strong connections between Riemann surfaces and algebraic geometry - for example, compact Riemann surfaces ...
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Borel-Serre compactification of $\mathbb{H}^3 / SL_2(\mathcal{O}_K)$

Let $\mathbb{H}^3$ be the three-dimensional hyperbolic space. Let $K$ be an imaginary quadratic number field and $\mathcal{O}_K$ its ring of integers. Then $SL_2(\mathcal{O}_K)$ acts on $\mathbb{H}^3$ ...
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85 views

Universal cover of elliptic curve without a point

This is something that I perhaps would be expected to know, but don't. Let $E_\tau$ be the elliptic curve ${\mathbb C}/({\mathbb Z +\mathbb Z} \tau)$. Consider the complement of a point in $E_\tau$, ...
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Degree of join of two varieties‏

Let $X\subset\mathbb{P}^n$ be an irreducible variety of degree $d$ and dimension $n-4$. Let $L\subset X$ be a line of mulitplicity $m$ in $X$. Assume that $m+1\leq d$ so that the general line spanned ...
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2answers
149 views

Example of linearization for GIT

Take a vector space $V$ (finite dimensional, over the complex numbers), let $G=SL(V)$. The group $G$ acts on $\mathbb{P}V$ and we can linearize its action to an action on the line bundle ...
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212 views

Non-vanishing of elements in cohomology of full Flag varieties

Consider the full flag variety $F_n$ consisting of full flags in $\mathbb C^n$. There is a collection of tautological bundles on $F_n$: $0=U_0\subset U_1\subset ...\subset U_{n-1}\subset U_n=\mathbb ...
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2answers
184 views

Basic questions on the Hilbert scheme/ Douady space

Let $X$ be a complex projective scheme (resp. complex analytic space). The Hilbert scheme (resp. Douady space) parameterizes closed subschemes (resp. complex analytic subspaces) of $X$. More ...
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Weil's Riemann Hypothesis for dummies?

Weil's Riemann Hypothesis is a deep result that I don't fully understand, but it has understandable corollaries which interest me. For example: (a) For any projective curve $X$ satisfying certain ...
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240 views

Canonical Sheaf of Projective Space

I am stuck on one step that occurs without explanation in several Algebraic geometry books. Starting from the exact sequence $$0\rightarrow \Omega_{\mathbb{P}^n}\rightarrow ...
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2answers
196 views

commutative algebra, diagonal morphism

can anyone help me with the following statement (it is part of a bigger proof where it is not explained). Let $B$ be a finite type $A$-algebra and consider the kernel $I$ of the diagonal ...
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calculating the Hilbert polynomial of a scheme given its primary decomposition

Given a scheme $X$ in $\mathbb{P}^n$, let $I_X$ be its, saturated, associated ideal. Suppose that a primary decomposition of this ideal is given by $$ I_X =I_1 \cap \ldots \cap I_2 $$ I was ...
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404 views

A letter from J. P. Serre

Which is the letter where J. P. Serre present "Analogues Kählériens de certaines conjectures de Weil" to Weil?
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259 views

Time-line until the publicaton of Weil of “Numbers of solutions of equations in finite fields”

In "On the history of the Weil Conjectures" Dieudonné says: "Appropriately enough, the story, as with so many problems in number theory, begins with Gauss...". C. F. Gauss, Disquisitiones ...
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163 views

Rational structures on the flag variety over a finite field [on hold]

Notations A variety over a field is defined to be a scheme of finite type over this field. An $\mathbb{F}_q$-rational structure of a variety $V$ over $\bar{\mathbb{F}}_q$ is a variety $V_0$ over ...
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1answer
96 views

Perfectness of the Jacobian of a curve

Let $C$ be a smooth projective curve over a field $K$ of characteristic $0$ (but not necessarily algebraically closed). Let $\mathcal{L}$ be a line bundle on $C$ of degree $0$. Fix an integer ...
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201 views

question about the induced homomorphism of etale fundamental groups

Background/Setup For any connected scheme $S$, let $\text{FEt}_S$ denote the category of finite etale $S$-schemes. Let $f : X\rightarrow Y$ be a morphism of connected schemes, then for any finite ...
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170 views

p-adic etale cohomology

Let $X$ be a smooth projective scheme over $\mathbb{Z}_p$, with special fiber $X_s$ over $\mathbb{F}_p$, generic fiber $X_{\eta}$ over $\mathbb{Q}_p$, and geometric generic fiber $\bar{X_{\eta}}$ over ...
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Homotopy type of complex algebraic varieties

In his 1974 ICM adress "Poids dans la cohomologie des variétés algébriques", Pierre Deligne explains that any finite polyhedron has the same homotopy type as a complex algebraic variety (section 6.). ...
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1answer
197 views

Second betti number of compact analytic spaces

Let $V$ be a proper singular complex algebraic variety, possibly nonprojective ($dim(V)=n>0$). I would like to know: 1) if its second Betti number is non zero, 2) same question but now $V$ is a ...
2
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117 views

What is the relation between Beilinson's conjectures and Standard conjectures of algebraic cycles?

Do Standard conjectures on the K-theory of varieties over finite field have implications in the motivic cohomology of Z where exist the correct formalism of Beilinson's conjectures? What is the ...
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1answer
157 views

Castelnuovo Mumford Regularity

Let $$ ‎‏0 ‎‎\rightarrow‎‎‏ ‎E_{n-1} ‎‎\rightarrow ‎... ‎‎‎\rightarrow ‎E_1 ‎‎‎\rightarrow ‎E_0 ‎‎‎\rightarrow I \rightarrow 0‎ ‎,‎ $$ be a minimal free resolution of ideal $I$, where $$ E_p = ...
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Are there any useful Grothendieck topologies for which the H1 of $GL_n$ is not the set of rank $n$ vector bundles

Let n be a positive integer and X a scheme. Then for all the Grothendieck topologies I know (Zariski, etale, fppf) the set $H^1(X,GL_n)$ is the set of (isomorphism classes of) rank $n$ vector bundles. ...
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354 views

Are there workable algebraic geometry approaches for the pentagon equation?

A pentagon equation is a system of polynomial equations of degree $3$ with several variables and integer coefficients, given by a fusion ring. A fusion ring is given by a finite set of integer ...
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78 views

How does one show that slope stability of a vector bundle is an open condition with respect to the polarisation?

I would like a source for the following result, which I expect to be true (probably well known): Let $X$ be a complex projective variety, $L$ an ample line bundle and $E$ a slope stable vector bundle ...
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607 views

GIT over integers

Let $G$ be a reductive algebraic group over ${\mathbb Z}$ (or a finite localization of a ring of integers $R$ in a number field) acting on an affine scheme of finite type $M=Spec(A)$ over $R$. We ...
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152 views

Small birational maps and singularities of the pair

Let $f:X\dashrightarrow Y$ be a small birational map, where $X,Y$ are normal $\mathbb{Q}$-factorial varieties. Let $\Delta_X\subset X$ be an effective $\mathbb{Q}$-divisor such that the pair ...
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47 views

Can Kuranishi families glue togheter to give a moduli space?

In his article "Moduli spaces of vector bundles on K3 surfaces and symplectic manifolds", Mukai constructs moduli of simple sheaves on a surface through a theorem that asserts the existence of a ...
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generalization Abhyankar's lemma

This question is related to a question I already asked on MO (smooth quotient out of a singular variety?), but I realized later that the hypotheses where not precise enough in my former question. Let ...
4
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1answer
191 views

Continued fraction expansion of an algebraic number and its conjugates

Let $w$ be an element of a Galois extension $L:\mathbb{Q}$ such that $\text{Gal}(L/\mathbb{Q})=\langle g\rangle$ is cyclic of order $n$ (here $\mathbb{Q}$ is rationals). Suppose we know the continued ...
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1answer
87 views

Boundedness of the number of curves negative on a varying big divisor

For a divisor $D$ on a smooth complex projective surface $X$, the stable fixed part is the maximal effective divisor $E$ which, for every $n \in \mathbb{N}$, is contained in every memeber of the ...
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Is every Noetherian *connected* ring a quotient of a Noetherian domain?

This question is a strengthening of this question (answered negatively), and arose due to David Speyer's answer here. Geometrically, this asks if every Noetherian connected affine scheme can be ...
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Global geometry of discriminant locus

Let $X$ be a smooth projective threefold, and $\pi : X \to S$ an elliptic fibration over a surface (i.e. flat, with general fiber an elliptic curve). I'm interested in constructing such fibrations ...
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1answer
77 views

A semi-ampleness criterion for homogeneous bundles on homogeneous spaces?

Let $X$ be a (compact) homogeneous space and $V$ be a homogeneous vector bundle on $X$ of rank $n$, and such that $\operatorname{dim}X\ge n$. Suppose $V$ has a section $s$, whose zeros $s=0$ form a ...
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Holomorphic vector fields tangential to a divisor [closed]

The work of Song-Wang http://arxiv.org/abs/1207.4839v1 pointed that while $D$ is a smooth simple divisor in $|-mK_X|$ for $m\in \mathbb{Z}^+$, then there is no any holomorphic vector field tangential ...
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1answer
102 views

Rational normal curves on Grassmanians

Consider the Grassmanian $G(k,n)$ ($k\le \frac{n}{2}$) and take its Plucker embedding. Consider now the space of all normal rational curves of degree $k$, contained in the Plucker embedding of the ...
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97 views

Family of exceptional divisors on a relative Hilbert scheme [closed]

I'm sorry to bother you again on the subject, but again I have trouble following a piece of an article of Beauville. In my previous question I asked how, from a family $\mathcal{X}\rightarrow B$ of ...
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131 views

Chern classes of tautological bundle over the Grassmannian G(2,4) [migrated]

I've the following problem: I know how to calculate chern classes of the tautological bundle over the Grassmannian $G=G(2,4)$ using the Shubert calculus. If I am right, the chern character should be ...
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1answer
96 views

Neron model: can number of components decrease after based change?

Suppose I have Neron model over some discrete valuation ring. Is there a result such that the number of components of the fiber over the closed point cannot decrease after some based change? In ...
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1answer
85 views

Generic deformation of Hilbert scheme of points on K3 surface

i was thinking about deformations of hyperkahler manifolds, in particular hilbert schemes of points on K3 surfaces and I think I realized something. I'm here to ask you if I'm right. Take $X^{[n]}$ ...
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History of the Proj construction in algebraic geometry

Projective geometry was introduced by fifteenth century Renaissance painters (like Alberti, da Vinci and Dürer) in the guise of perspective theory, although one could argue that Pappus was already ...
3
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1answer
131 views

Flat family with special fiber $\mathbb{C}\mathbb{P}^1$

Let $C=Spec \mathbb{C}[t]/(t^{n+1})$. Let $X$ be an algebraic (or complex analytic) scheme, flat over $C$ with the structure morphism $f\colon X\to C$. Assume that the special fiber is isomorphic to ...
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430 views

Has there been any progress on the standard conjectures on algebraic cycles?

What's the current state of these conjectures? Who is working on them? In "Standard conjectures on algebraic cycles" Grothendieck says: "They would form the basis of the so-called "theory of ...
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248 views

Existence of a ring with specified residue fields

Given a finite set of fields $k_1, \ldots, k_n$, is there a (commutative with $1$) ring $R$ with (maximal) ideals $m_i$ such that $R/m_i \cong k_i$? To prevent things from being too easy, I ...
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1answer
121 views

Pull-Back of an ample line bundle

here is my question: I work over the field of complex numbers, and my schemes are separated and of finite type. Let $X$ be a quasi projective scheme (but not projective), let $Y$ be a regular ...
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1answer
130 views

Holomorphic Foliations having transverse sections

In the introduction to the paper "On the Geometry of Holomorphic Flows and Foliations Having Transverse Sections" by Ito and Scardua, one reads the following "a holomorphic codimension one foliation ...
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1answer
274 views

When flatness of a morphism implies smoothness?

EDIT: Let $f\colon X\to C$ be a flat proper morphism of complex algebraic (or analytic) varieties. Assume the special fiber over a point $p\in C$ is smooth. Is it true that there exists a ...
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Calculating the Betti numbers of the Hilbert scheme of points on a surface

Let $S$ be a smooth (algebraic) surface over $\mathbb{C}$. Then the Hilbert schemes of points on $S$, denoted by $S^{[n]}$, are smooth. A formula for the Poincaré polynomials of $S^{[n]}$ was first ...
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1answer
59 views

Semicontinuity of degree of fibers for a proper map

Let $f:X \rightarrow T$ be a proper morphism from a projective scheme $X$ to a smooth projective curve $T$ over $\mathbb{C}$. I know that fiber dimension is upper-semicontinuous, but is the degree of ...
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2-torsion points on Frey-Hellegouarch curve. [closed]

Let us consider the so-called Frey-Hellegouarch curve $E: Y^2 = X(X-a^l)(X+b^l)$. Q: What are two independent ${\Bbb Q}$-rational points of order $2$ on $E$ ?
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Blow-up and tangent cone

Let $$ f\colon X \rightarrow Y$$ be a (possibly non-dominant) morphism of complex varieties which is birational with its image. Assume $f$ contracts a sub-variety $E$ of $X$ to a point $p$ of $Y$; ...