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

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75 views

Nonreducedness of schemes and projective morphisms

Let $X, Y$ be irreducible projective schemes over $\mathbb{C}$, $W \subset X \times Y$ a closed irreducible subscheme. Suppose that the natural projection map $pr_2:W \to Y$ is surjective on the ...
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67 views

The minimum genus of a family of degree $12$ algebraic curves which comes from the resultant of two quartic polynomials

Let $f(t)$ be a rational normal cubic curve in $\mathbb{P}^3$ (it is not contained in any plane) and also we assume that this cubic curve passes through two points $(0,0,0)$ and $(1,0,0)$. By an easy ...
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56 views

Chow group of zero-cycles generated by open dense subscheme [on hold]

Let $X$ be a smooth proper variety over a field $k$ (I do not think any assumption on $k$ is needed). Let $CH_0(X)$ be the Chow group of zero-cycles of $X$. Now let $U \subset X$ be any open dense ...
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29 views

Riemann bilinear identity and poles in the branching points

Suppose I have a hyperelliptic complex Riemann surface: $$ y^2=P_N(x) $$ If the genus is more than zero, $d\omega=dx/y$ is holomorphic. If I take some meromorphic form $dS$ then I can write down ...
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1answer
68 views

On Severi's definition of the complementary correspondence

In Weil's short note entitled "On the Riemann hypothesis in function-fields" he mentions the notion of the complementary correspondence associated to a given correspondence $T:C\rightarrow C$ where ...
4
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2answers
601 views

Salvaging Leibnizian formalism?

Can one justify Leibniz's formalism in a suitable algebraic or topological context? We have published some papers recently where we argue that Leibniz's formalism for the calculus wasn't ...
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100 views

Is there a smooth rationally connected, proper variety $M$ over $\mathbb{C}$ such that c_1(L)([A]) = 1 which is not projective space?

Is there a smooth, rationally connected, projective variety $M$, which is not a projective space and is equipped with an ample line bundle $L$, such that the homology class of the curves $[A]$ which ...
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0answers
64 views

support of embedded points in a curve

Let $C \subset \mathbb{P}^n$ be an one dimensional scheme. Suppose that $C$ decomposes as the union of a Cohen Macaulay reduced curve $\tilde{C}$ (in particular $\tilde{C}$ does not have embedded ...
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122 views

Is a semiabelian algebraic space a scheme?

Let $S$ be a scheme and let $A$ be a commutative separated smooth $S$-group algebraic space of finite presentation each of whose geometric fibers is an extension of an abelian variety by a torus. Is ...
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91 views

right adjoint functor for closed immersion of topoi

Let $i\colon (X,A)\rightarrow (Y,B)$ be a closed immersion of ringed topoi. Does functor $i_*\colon Mod(A)\rightarrow Mod(B)$ have a right adjoint?
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1answer
123 views

Divisors on an abelian surface

Let $A$ be an abelian surface given by the quotient of a product of two generic elliptic curves $E_1 \times E_2$ by the product $T_1 \times T_2$ of two translations by $2$-torsion points. Then $A$ ...
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1answer
141 views

Which singularities of log pairs do not depend on the resolution?

Let $(X,\Delta)$ be a log pair (we assume the coefficients of $\Delta\leq 1$, but could be negative rationals), and I use the definitions in the book of "Birational Geometry of Algbebraic varieties" ...
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1answer
145 views

Effective cone of C x C where C is a Fermat curve

A search of the literature reveals that for a curve $C$ of genus $\geq 2$, determining the effective cone of $C \times C$ is hard. My question is this: do we know a single example of a curve $C$ of ...
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0answers
69 views

The target of a finite morphism $f$ is a dense open in $S$, can you extend $f$ to have target $S$?

Let $U$ be a quasi-compact open dense subscheme of a scheme $S$ and let $X \rightarrow U$ be a finite surjective morphism. Does there exist a finite morphism $Y \rightarrow S$ such that the $U$-scheme ...
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0answers
99 views

G.W. invariants $<[pt],[pt]>_{0,[A]} \neq 0$ such that there exists ample L with $c_1(L)([A])=1$

Do there exist interesting examples of projective algebraic varieties such that the two-point genus 0 Gromov Witten invariants in homology class $[A]$, $GW<pt,pt>_{0,[A]}$, is non-zero, and ...
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75 views

Research of a reference about $G$-linearizations of line bundles on quasi-projective schemes

I am looking for some references for the following statement: Let $G$ be a linearly reductive algebraic group acting on a quasi-projective scheme $X$, over an algebraically closed field $K$. Let $L$ ...
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1answer
294 views

Quotients by the additive group $\mathbb G_a$

Geometric invariant theory doesn't work so well for non-reductive groups, since invariant rings are not generally finitely generated. However, in many cases the action of a non-reductive group has a ...
7
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2answers
368 views

modular forms, invertible sheaves, and quotients

I'm very confused about some contradicatory statements, and I hope someone can help me clarify this. Let $\Gamma$ be a congruence subgroup. It is well known that modular forms of weight $k$ for ...
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0answers
83 views

$\Gamma_Z(\widetilde M)\cong\widetilde{ \Gamma_Z(M)}$

Let $R$ be a Noetherian ring and let $M$ is an $R$-module. Consider the associated affine scheme $(\text{Spec R},\mathcal{O}_{\text{Spec R}})$ and Suppose $Z\subset X$ is a closed subset of ...
2
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0answers
87 views

$X$-points of reductive group schemes, if $X$ is a proper smooth curve over a finite field

Let $X$ be a connected proper smooth curve over a finite field (so the generic point of $X$ is the spectrum of a global field $K$), and let $G \rightarrow X$ be an affine $X$-group scheme of finite ...
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174 views

On Bott's Formula on projective Spaces

I'm following the book of Okonek, Schneider and Spindler, Vector Bundles on Complex Projective Spaces, and they say that is a useful exercise try to prove Bott's Formula that calculates the cohomology ...
3
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1answer
149 views

Deforming curves on Calabi-Yaus

What I'm looking for is some sort of 'Bertini-theorem' for curves. Let $X$ be a Calabi-Yau manifold and let $C$ be a union of rational curves on $X$. Are there any techniques or results that would ...
3
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0answers
67 views

Finding a divisor on a curve in a given linear equivalence class made with points over a “small” field

Suppose I have a curve $C$ defined over $\mathbb Q$ and a line bundle $\mathcal L$ on it, also defined over $\mathbb Q$. I am interested in trying to find a (non-effective) divisor $D=\sum_i a_ip_i$ ...
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1answer
95 views

Invertibility of random Vandermonde matrix

Let $\kappa, d \in\mathbb{N}$ and $f$ is a uniform probability measure on $\mathcal{D} = \left[-1,1\right]^{\kappa}$. In addition, let \begin{equation*} p = p\left(\kappa,d\right) := ...
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1answer
205 views

If $T$ is an $S$ scheme, and $X,Y$ are $T$-schemes, then do the $S$-valued points of $\mathcal{Hom}_T(X,Y)$ have an interpretation?

Let $X,Y$ be finite etale $T$ schemes for some scheme $T$ (assume the maps $X\rightarrow T,Y\rightarrow T$ are surjective). Then the sheaf $\mathcal{Hom}_T(X,Y)$ on $\text{Sch}/T$ (with the etale ...
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1answer
215 views

Surjective morphism from $X$ to itself is finite

Let $X$ be a projective variety. Why is any surjective morphism from $X$ to itself finite?
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2answers
192 views

Components of a Fiber Product

Suppose $X$, $Y$, and $Z$ are smooth irreducible schemes [EDIT: of finite type over an algebraically closed field of characteristic zero], and $X \to Z$ and $Y \to Z$ are dominant maps. I have a ...
4
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1answer
155 views

Explicit formula for the Poincare dual of a CM endomorphism of an elliptic curve

Let $E/\mathbf{C}$ be an elliptic curve with CM by the maximal order $\mathcal{O}_K$ of $K=\mathbf{Q}(\sqrt{-D})$ where $D$ is positive and square-free integer. To make it even more precise, let us ...
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0answers
145 views

Two functorial definitions of schemes

I have been reading a bit about the "functor of points" theory for schemes. There seem to be two ways of going about defining schemes this way: Equip the category $\textbf ...
2
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1answer
86 views

Deformation of Canonical singularities

Let $p: X \rightarrow T$ be a flat family of normal projective varieties over a variety $T$. Assume that $X_{t_{0}}=p^{-1}(t_{0})$, for a $t_{0}\in T$, has only canonical singularities of index $1$. ...
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0answers
132 views

Does GKZ's reflexivity theorem imply the Plucker formula?

Let $S\subset\mathbb{P}^n$, Gelfand-Kapranov-Zelevinsky defined its dual variety $S^\vee\subset\mathbb{P}^{n^\ast}$. In this paper (http://arxiv.org/pdf/math/0111179v1.pdf), the author obtained the ...
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1answer
158 views

Reference request: Deligne's conjecture (cycles)

In "The work of Tate" Milne says: "The relation between the two conjectures has been greatly clarified by the work of Deligne. He defines the notion of an absolute Hodge class on a (complete ...
2
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1answer
230 views

Reference request: Grothendieck´s period conjecture?

I would like to know if Grothendieck published something about this conjecture? Is there some book (or expository article) about this conjecture? Is there any connection between this conjecture and ...
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67 views

Hodge-Deligne Polinomial and flat morphism

I would greatly appreciate if you could address the following questions (please forgive any inaccuracies): Let $\phi:X\rightarrow T$ be a morphism of algebraic varieties or of complex manifolds , ...
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2answers
210 views

Finding an algebraic equation given divisors

I'm trying to find an algebraic curve that represents a specific Riemann surface and my question goes like this: Given divisors $(\omega_1) = P_1 + 5 P_2 + 2 P_3,$ $(\omega_2) = 5 P_1 + P_2 + 2 P_3,$ ...
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0answers
113 views

Rewrite sum of radicals equation as polynomial equation

My question is about a method described in Dr.Math forum for simplifying equations involving sums of radical functions. (The following is a transcription of the example given by Dr. Vogler): --- ...
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1answer
132 views

Examples of hyperelliptic curves with hyperelliptic quotients that have more automorphisms

Does there exist a hyperelliptic curve $X$ of genus $g\geq 2$ over the complex numbers such that $X$ has a hyperelliptic quotient $X\to Y$ (in the sense that $Y$ is hyperelliptic and the morphism ...
2
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0answers
76 views

Looking for concrete examples of the GW/DT correspondence

I would like to get my hands on some explicit manifestation of the Gromov-Witten / Donaldson-Thomas correspondence, explained for instance in MNOP I. This correspondence is an equality of power series ...
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1answer
170 views

About Weil's proof of “Weil conjectures for curves and abelian varieties”

I know that the Weil's proof of the Weil conjectures for curves and abelian varieties is made under the lenguage of his "Foundation of algebraic geometry", however in "Polarizations and Grothendieck's ...
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87 views

Differences and relationships between Motivic cohomology and Universal cohomology theory? [duplicate]

Differences and relationships between Motivic cohomology (Beilinson, Lichtenbaum and Voevodsky) and Universal cohomology theory (Grothendieck)?
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1answer
225 views

Comparison of etale and singular cohomology for varieties over number fields

Whilst reading Hartshorne's appendix C I came across the comparison theorem for etale cohomology and singular cohomology: Let $X$ be a smooth projective variety over a number field $K$ and $\ell$ a ...
2
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1answer
396 views

Base change through a field automorphism

Note:For a correct comprehension of the question see the "important edit" at the end. Consider a projective variety over $\mathbb C$, $X=\textrm{Proj}\frac{\mathbb ...
3
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2answers
217 views

Derived categories of curves equivalent then the curves are isomorphic

I am a beginner at derived categories and I'm looking for a proof of the following fact: If $X$ and $Y$ are smooth projective curves such that $D^b(Coh\,X)$ is equivalent to $D^b(Coh\,Y)$ then $X$ ...
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1answer
282 views

Easiest example where field of definition is not field of moduli

There are many examples of varieties over $\overline{\mathbb Q}$ whose field of moduli is $\mathbb Q$ but which can't be defined over $\mathbb Q$. What is the easiest such example? It should be a ...
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1answer
197 views

Hilbert function of points in $\mathrm{P}^2$

Let $\Gamma$ be a collection of $d$ points in $\mathrm{P}^2$, and $I$ the graded ideal of $\Gamma$.If $$ ...
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82 views

What is a “normal crossings divisor relative to S”?

I'm reading SGA 1 Expose XIII, as given here: http://arxiv.org/pdf/math/0206203v2.pdf and I'm trying to understand the given definition of a "diviseur a croisements normaux relativement a S" for some ...
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1answer
102 views

Empty real conic containing two pairs of conjugate points in the projective plane?

Given two conjugate pairs of points in general position in $\mathbb{CP}^2$, there is a pencil of real conics containing these four points. Is there a real empty conic in this pencil?
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1answer
188 views

Projection of a hypersurface from a point

Let $k$ be an algebraically closed field. We consider the projective space $\mathbb P_n$ over defined over $k$, the point $Q=(0:\dots:1)$, the hyperplane $H=\{X_n=0\}$ and a hypersurface $X$. We want ...
2
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1answer
191 views

Which polynomials define extensions of $k(t)$ unramified at the finite places

Let $k$ be an algebraically closed field of characteristic $p>0$. Let $L$ be the extension of $k(t)$ obtained by attaching a root of an irreducible polynomial $f\in k(t)[x]$. Is there a way to tell ...
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1answer
135 views

Linear systems on bielliptic surfaces

A bielliptic surface is a surface of type $S=E_1 \times E_2/G$ where $E_1, E_2$ are elliptic curves and $G$ is a finite group of translations of $E_1$ acting on $E_2$ such that $E_2/G=\mathbf{P}^1$. ...