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We use $\mathbb{G}(l,k+l)$ to denote the Grassmannian consisting of all $l$-vector in $\mathbb{C}^{k+l}$. Given a $k$-vector $v$ in $\mathbb{C}^{k+l}$, let $H_{v}$ be the set of all $l$-vector in $\...
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I am trying to understand several technical points in Alina Marian’s paper （arxiv：math/0601275；arXiv:math/0605097), particularly Section 2 concerning the degeneracy loci $D_i^0$ and $D_{i,p}^0$ inside ...
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I try to study a specific operation of pullback and pushforward related to flag varieties. Let $Y=Gr(r-1,2r-1)$ be the variety of $r-1$ subspaces in a vector space of dimension $2r-1$. I also have $Y'=...
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Thanks to work of many people (Madapusi, Charles, Maulik...) the Tate conjecture is now known for K3 surfaces over any finitely generated field. A smooth quartic surface is K3. Consider instead a ...
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I am working with a particular multivariate resultant and I suspect that behind there is a geometric interpretation that I cannot see, so I would be happy if some one could help me to understand that. ...
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Let $C,D\subset \mathbb{P}^n$ be irreducible and reduced curves, for some $n\ge 2$. For a point $p\in \mathbb{P}^n$ we can define the intersection multiplicity $$I_{p}(C,D)=\mathrm{dim}(\mathcal{O}_{p,...
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Let $L / k$ be a finite extension of fields, let $M$ be a pure motive over $k$ (with rational coefficients), and suppose that the base change $M_L$ of $M$ to $L$ is isomorphic to $\mathfrak{h}^1(A)$. ...
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Let $k$ be the function field of a $d$-dimensional regular integral finite-type scheme $Y$ over $\mathbb{Z}$. Conjecture 2 in Tate's paper in the Woods Hole proceedings predicts (among other things) ...
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Let $V_n^d(\mathbb{Q})$ denote the set of realizable volume polynomials of degree $d$ in $n$ variables over $\mathbb{Q}$ - these are polynomials of the form $\frac{1}{d!}\int_Y (\sum x_i D_i)^d$ where ...
DimensionalBeing's user avatar
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The background of my question is the intersection theories introduced in Chapters 42 and 43 of the Stacks Project. We work over an algebraically closed field. Let $Y$ be a smooth projective variety ...
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I see Kiritchenko's Intersection theory note claims that the Chow ring with rational coefficients of a smooth scheme $X$ is generated multiplicatively by Picard group (page 18). Is this claim true ? ...
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Assume $C_1$ and $C_2$ are two plane algebraic curves in $E = \mathbb{A}^2$, the affine plane, given by equations $f(x,y) = 0$ and $g(x,y) = 0$ of degrees $d$ and $e$ respectively. Now the bisector ...
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I have been trying to get into equivariant cohomology from an algebraic geometry perspective, and it seemed like Fulton's most recent book would be a halfway decent place to start. His approach is to ...
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Let $X$ be a (nice enough) variety over a field $k$, with a regular closed immersion $i:Z\rightarrow X$. All the cohomologies appeared are supposed to be étale cohomology. Fix the coefficient ring $\...
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This question regards example 4.1.1 from Fulton's Intersection theory. Namely we have the following statement: Example 4.1.1. For any cone $C$, $s(C\oplus 1) = s(C)$. Here we define a cone $C$ over ...
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Assume $X:=X_d$ is an hypersurface of degree $d$ in $\mathbb{P}^n$. Assume in addition that $d<n+1$, hence $X$ is a Fano variety. The hyperplane section theorem of Lefschetz ensures that the only ...
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The starting point of my question is the following exercise from Liu's book in the framework of arithmetic surfaces (mixed characteristic). In particular in the following image $S$ is a Dedekind ...
manifold's user avatar
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The following is the adjunction formula for regular arithmetic surfaces (taken from Liu's book): where $S$ is a Dedekind scheme of dimension $1$ and $X$ is an integral, projective scheme of dimension ...
manifold's user avatar
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After over one week and quite a lot of views on this question, I would like to ask a refined version here. Let X be a minimal Calabi-Yau threefold in the sense of [1] and let $D$ be a Weil divisor on $...
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I would like to have examples of smooth projective varieties $X$ and $Y$ that have the same rational Chow ring but non-isomorphic Chow motives $h(X)$ and $h(Y)$.
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Given $X$ a normal projective surface over $\mathbb{C}$ and $p$ be a singular point of $X$ . Let $\pi: Y \rightarrow X$ be the blowup of $X$ at the point $p$. Is it true that $\pi^{\star}(K_{X}) \...
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I refer to the article of van Geemen https://arxiv.org/pdf/math/9903146. What van Geemen calls the Kuga-Satake-Hodge conjecture suggests that for a K3 surface $X$ over $\mathbb{C}$, the summand $h^2(X)...
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In his book “Mixed motives and algebraic K-theory”, Jannsen generalizes the Tate conjecture to a potentially singular projective variety $X$ over a finitely generated field. The statement is the same ...
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The vanishing part of the Beilinson-Bloch conjecture asserts that for a smooth projective variety $X$ over a number field $K$, $\dim_{\mathbb{Q}} \operatorname{CH}^i(X) \otimes_{\mathbb{Z}} \mathbb{Q} ...
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What are some examples of smooth projective varieties $X$ over a finite field for which the Tate conjecture for divisors is known, and which admit a smooth morphism to a smooth projective curve? I am ...
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Let $k$ be an algebraically closed field. Let $X$ be an $n$-dimensional affine, simplicial toric variety over $k$. There exists an $n$-dimensional simplicial cone $\sigma$ in $\mathbb{R}^n$ such that $...
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I can't quite follow Proposition $2.1$ of "UNIVERSAL UNRAMIFIED COHOMOLOGY OF CUBIC FOURFOLDS CONTAINING A PLANE". I posted this on Math stackexchange but got no answer. Let $X$ be a smooth ...
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Let $\Delta$ be a polytope and consider the projective toric variety $P_{\Delta}.$ Given a curve $C \subset \mathbb{P}_{\Delta},$ which is not toric, is it true that in the Chow group we have $$ C = \...
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We are working on algebraic closed field $k$. Let $\mathbb{F}_1$ be the Hirzebruch surface $\mathbb{P}(\mathcal{O}\oplus\mathcal{O}(-1))$, $C_0$ and $C_{\infty}$ are its zero and infinity sections ...
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Consider the complex projective plane $P^2$. A rational curve in $P^2$ of degree $\leq 3$ is either a line, a smooth conic, a nodal cubic, or a cuspidal cubic. I am looking for some "nontrivial&...
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Let's borrow the quadric intersection $I$ from another question. More precisely, let $k$ be an algebraically closed field of characteristic $\neq 2$ and $a_1, a_2, \cdots, a_n \in k^*$ be some ...
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Let $X$ be a smooth projective variety over $\mathbb{F}_q$. Is any cycle in the Chow group $CH^i(X)$ which is trivial in $\ell$-adic cohomology automatically torsion? For abelian varieties I believe ...
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In a toric variety $T$ of dimension $11$ I have a subvariety $W$ of which I would like to compute the dimension. On $T$ there is a nef but not ample divisor $D$ whose space of sections has dimension $...
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Let $X\to\operatorname{Spec} \mathbb Z$ be an arithmetic surface which is projective, regular and integral. Let $D$ and $E$ two divisors intersecting at a point $x\in X$ that lies over the prime $p$. ...
manifold's user avatar
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The Hodge index theorem states that the intersection matrix associated to curves on a smooth algebraic surface has a specified signature---namely, if the intersection matrix has size $n \times n$ then ...
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Let $X\subset \mathbb{P}^3$ be the surface defined by the equation $xy-zw=0$, and consider the curve $E \subset X$ defined by the equation $x=z=0$. Question. Can $E$ be contracted to a point?
George's user avatar
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Is there a version of the Bloch–Beilinson conjecture for smooth projective varieties over global fields of positive characteristic? The conjecture I’m referring to is the “recurring fantasy” on page 1 ...
Bma's user avatar
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Let $p,q\geq 3$ be integers. Let $M$ be a compact oriented smooth $(p+q)$-manifold and $P$ and $Q$ compact submanifolds of dimensions $p$ and $q$ intersecting transversely. Assume that $M,P$ and $Q$ ...
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Is there a name for a normal, projective variety such that every effective divisor is ample? Examples of such varieties are projective space, weighted projective spaces, and simple Abelian varieties ...
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$\DeclareMathOperator\Tot{Tot}\DeclareMathOperator\ch{ch}\DeclareMathOperator\td{td}\DeclareMathOperator\ker{ker}\DeclareMathOperator\rk{rk}$I was wondering if the following is correct: Let $X=\Tot(L)$...
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Let $i:X\hookrightarrow Y$ be an embedding of two non-singular projective varieties over $\mathbb{C}$. Consider the blow-up $f:Y' = Bl_XY \to Y$, and the corresponding embedding $j:E\hookrightarrow Y'$...
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In Proposition 1.14, page 25 in the book "3264 and all that Intersection Theory in Algebraic Geometry" the authors define a right exact sequence: $$ Z(\mathbb{P}^1 \times X) \rightarrow Z(X) ...
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Sorry if this question is not appropriate for this site, but I haven't got an answer on stackexchange. It's well known that there are divisors (on a normal projective variety over the complex numbers) ...
Calculus101's user avatar
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Let $A, B \in \mathbb{R}^{d \times d}$ denote two symmetric positive definite matrices. I am interested in solutions $V_r \in \mathbb{R}^{d \times r}, 1 < r < d$ to the system of quartic ...
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Let X be a complex projective variety and E be an irreducible effective divisor on it. Then, I want to know whether the following set is finite: {C | C be an irreducible curve and C.E<0}. I know ...
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Let $K$ be a field. A polynomial $F \in \mathbb{Q}[X_1, \dots, X_r]$ which is weighted homogeneous of degree $n$ with respect to the grading $\deg(X_k) = k$ is called numerically non-negative for nef ...
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The short question is: Say $p:\bar{X}\rightarrow S$ is a proper and normal morphism with the following properties: S is integral and smooth over a certain base field $k$, $\bar{X}$ has a smooth and ...
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Let $\mathbf{F}_p$ be the field with $p$ elements, and $X = (\mathbf{P}^1_{\overline{\mathbf{F}}_p})^\text{perf}$ the inverse perfection of the projective line over $\mathbf{F}_p$. Let $\Gamma$ be the ...
Tim's user avatar
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Let $X$ be a smooth projective curve over the complex numbers, and take $\Delta$ the diagonal divisor on $X\times X$. Using the adjunction formula, one computes $\Delta\cdot\Delta =2-2g$ for $g$ the ...
Tim's user avatar
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Let $\varphi\colon X_1\to X_2$ be dominant proper morphism of finite degree (in particular $\dim X_1=\dim X_2$) between varieties. Let $D \subset X_2$ be a Cartier divisor. Is it true that $$\varphi_*...
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