Questions tagged [intersection-theory]
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376 questions
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Intersection with very ample divisor and linear equivalence
Let $X$ be a smooth, projective variety and $D, E$ two effective divisors of $X$ which correspond to distinct elements on the cohomology group $H^2(X,\mathbb{Q})$. Denote by $H$ a very ample divisor ...
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Pushforward of curves
Let $Z$ be a subvariety of an irreducible projective variety $X$, and let $i:Z\rightarrow X$ be the inclusion.
Let $N_1(X),N_1(Z)$ be the $\mathbb{Q}$-vector spaces of curves in $X$ and $Z$ ...
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Higher Chow groups revisited
Let $X$ be an algebraic variety over a field $k$.
Bloch defines the "algebraic singular complex" using the algebraic simplices
$$\Delta^n = \text{Spec}(k[x_0,\dots,x_n]/(x_0+x_1+\dots+x_n=1) \subset ...
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Gysin map and blow up
Let $X$ be a smooth projective variety and $W \subset X$ a smooth, projective subvariety. Let $\pi:\tilde{X} \to X$ be the blow-up of $X$ along $W$. Let $E$ be the exceptional divisor of $\pi$ and $i:...
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Intersections of quadratic planes as elliptic curves
An elliptic curve defined over a field $k$ is a smooth projective curve of genus $1$, plus a $k$-rational point. Every elliptic curve can be written in a Weierstrass form, i.e. as a plane cubic curve ...
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First computations of intersection products – a formula in Fulton
The book is Fulton, Intersection Theory. My question pertains to Examples 6.1.4 and 6.1.5. In 6.1.4, we are looking at effective Cartier divisors $A,B$ and $D$ on a nonsingular surface $X$ with $A$ ...
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About Riemann-Roch without denominators
The Riemann-Roch without denominators can be expressed as follows:
Let $f: X\rightarrow Y$ be a closed embedding of quasi-projective smooth $k$-varieties of codimension $d$ for some field $k$. Let $E$ ...
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Is algebraic geometry related to conical intersection in potential energy surface of molecules?
I post this question here because it seems that the equations describing conical intersections in molecular potential energy surface are similar to what algebraic geometry research concerns.
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Why is flatness needed for the Segre classes of a family of cones to be equal in the Chow ring of the base
Let $X$ be an algebraic scheme and $\mathscr C$ a cone on $X\times\mathbf A^1$ and $C_t$ denote the restriction of $\mathscr C$ to $X\times\{t\}$ ($t=0,1$, or whatever). The claim in Fulton's ...
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Graphs of maps between manifolds as cycles and intersection theory
I'm guessing that the answer to this question is well-known, but I'm struggling to find anything to help me.
Let $X,Y$ be compact manifolds of dimension $n,m$ respectively. Let $f:X \to Y$ be a ...
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Irreducible components of normal cone $C_{X/Y}$ dominates X?
Assume $X$ is a subscheme of $Y$ and $X,Y$ are irreducible.
Then every irreducible component of the normal cone $C_{X/Y}$ dominates $X$?
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Blow-up and the Chow group of zero cycles
Let $\tilde{X}\to X$ be a blow-up of a variety $X$ (over an algebraically closed field).
Is it true that the Chow group of zero cycles of $\tilde{X}$ is isomorphic to that of $X$? What if $X$ is a ...
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Do A-infinity algebra(in Floer theory)have some kind of intersection theory and Poincare duality?
In Lagrangian Floer theory, we can define an A-infinity algebra. It is by first choosing a subset $X_L$ of chains in the Lagrangian submanifold $L$, and then defining boundary maps on(Actually, sum of ...
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intersections between closed curves on surfaces
I would like to find a result telling me that two simple closed curves $\alpha$ and $\beta$ (on a non-orientable surface $S$) are in minimal position if and only if there is not a disk in $S$ whose ...
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On a fiber square flat pullback commutes with proper pushforward
I'm working through Fulton's intersection theory book and I've been stuck on the end of Prop 1.7, i.e. that flat pullbacks commute with proper pushforwards for fibre squares. Specifically I ...
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What is the main failure in using Naive Chow group in Artin Stack
I'm reading Andrew Kresch's paper, Cycle groups in Artin Stacks.
The author defined Chow groups of Artin stacks by very technical way, instead of ordinary ways which he called 'naive chow group', ...
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Linear section of an algebraic variety
Let $\pi$ be a linear subspace of $\mathbb{P}^n$ and $X$ a reduced, irreducible variety of $\mathbb{P}^n$. Suppose that $\pi \cap X$ is reducible, hence $\pi \cap X=Y_1\cup Y_2 \cup \cdots Y_k$. When ...
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Reducible sections of algebraic varieties
Let $X$ be an irreducible variety. Is there some necessary condition on a hyperplane $H$ such $X\cap H$ is reducible? Also, suppose that $H\cap X$ is reducible, i.e., $H\cap X=Y_1\cup Y_2 \cup \cdots \...
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Zero dimensional components of an intersection
Let $X$ be a smooth projective algebraic variety over an algebraically closed field and let $A,B$ be closed irreducible subvarieties of complementary codimension in $X$. Let $n$ denote their ...
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Top intersections on the Hilbert scheme of points on a surface
The Picard group of $S^{[n]}$ is generated by the Picard group of $S$ (via a map $L \mapsto L_n$) and $E$, where $E = -\frac{B}{2}$, where $B$ is the exceptional divisor of the Hilbert Chow morphism.
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Degree of an isogeny in the endomorphism ring of the jacobian of a curve and self intersection index in its ring of correspondences
I hope this question is not too basic.
Let $C/\bar{k}$ be a nonsingular irreducible curve of genus $g$ and $\mathfrak{C}(C\times C)\cong \text{CH}^1(C\times C)$ be its ring of correspondences.
I am ...
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Is there a functor of points approach to algebraic cycles and intersection theory?
Motivation
Most of the algebraic geometry I have done so far was concerned with group schemes (e.g., abelian schemes, tori, unipotent groups). In that part of the field the "functor of points POV" is ...
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Intersections of divisors in blow-ups of $\mathbb{P}^n$
Let $p_1,p_2,p_3\in\mathbb{P}^n$ be three general points, $X$ the blow-up of $\mathbb{P}^n$ at $p_1,p_2,p_3$, then along the lines $\left\langle p_i,p_j\right\rangle$, and finally along the plane $\...
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How to think about the quotient field of an integral stack?
This is the definition given in Vistoli's paper.
Let $F$ be an integral stack. A rational function of $F$ is a morphism $G \rightarrow A^1_S$ defined on a nonempty open substack $G$ of $F$.
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Nonempty intersection in Grassmannian
Where can I find a proof of the following fact:
If $X_1$ and $X_2$ are subvarieties of $\mathbb{G}(k,n)$ of codimension $c_1$ and $c_2$ satisfying $c_1+c_2<n+1-2k$, then the intersection $X_1\cap ...
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Computing higher dimensional intersection numbers for complete intersections of $\mathbb P^n$
Let $X_1,X_2$ be two smooth hypersurfaces of degree $d$ in $\mathbb P^{n}$. Let $B=X_1\cap X_2$. Assume $B$ is smooth. Let $\mathcal N_{B/\mathbb P^n}$ be the normal bundle to $B$. Let $H$ be the ...
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Intersection multiplicity of limit linear spaces
Let $X\subset\mathbb{P}^N$ be a smooth projective variety. Let us fix a general point $q \in X$, and let $C\subseteq X$ be a smooth curve passing through $q$.
Now let $\Lambda_{\xi, q}$, with $\xi \...
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Class of the locus where two sections are proportional
Let $X$ be a smooth (complex) projective $n$-dimensional variety ($n\geq 3$) and $\mathcal E$ a vector bundle of rank $r<n$ generated by its global sections on $X$. Let $\sigma\in H^0(\mathcal E)$ ...
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Non-generic intersections of Schubert varieties?
Let $G$ be a linear algebraic group, $B$ a Borel subgroup, $P$ a parabolic subgroup containing $B$, and $W$ the Weyl group. For $w \in W$, the Schubert variety $X_w^P$ is the closure of the Schubert ...
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Tropical self intersection number of boundary divisor on toroidal embedding
Let $X_0 \subset X$ a toroidal embedding without self intersections and denote by $\overline{\Sigma}$ its corresponding (weakly embedded) extended conical simplicial complex. Let $D$ be a divisor on $...
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Finding two hypersurfaces of the same degree that intersect $X/\mathbb{F}_q$ smoothly
Let $X$ be a smooth projective variety over a finite field.
In [Poonen - Bertini theorems over finite fields] it is shown that one can find a smooth geometrically integral hypersurface $S$ of degree $...
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Self-intersection of sum of Eff cone generators on Picard rank 2 surfaces
Let $S$ be a smooth, projective, complex surface with Picard rank 2, whose effective cone is generated by two curves of negative self-intersection, $C_1$ and $C_2$ (i.e. $C_1^2<0$ and $C_2^2<0$)....
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Is sum $(E_i, E_j)$ non-positive, with $E_i$'s the exceptional components of a desingularization
Let $Y$ be an integral normal 2-dimensional scheme and let $X\longrightarrow S$ be a flat projective morphism, where $S$ is a Dedekind scheme.
Let $f:X\longrightarrow Y$ be a minimal resolution of ...
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Semi-continuity of intersection numbers
I always trusted the following quite vague statement:
If you have a family of effective divisors $D_1(t),\dots , D_k(t)$ on a $k$-dimensional projective variety $X_t$, where $t$ is a paramater say ...
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A strong form of Bezout theorem
Let $X$ be a smooth projective variety of dimension $n$, $U \subset X$, non-empty open set. For any integer $k>0$, does there exist $n$-hypersurface sections $Z_1,...,Z_n \in |\mathcal{O}_X(k)|$ ...
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The volume around a singular isolated root when equalities are loosened
Suppose I have a system of polynomial equations in $n$ real variables $f_i(x_1,\ldots,x_n)=0$, $i=1,\ldots,m$, such that $0$ is an isolated solution. Now I replace each of the equations with a double-...
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$8$-ary operation $(\mathbb{P}^2)^8 \text{ }-\to \mathbb{P}^2$, can we say anything about what this formula would look like?
My friend, who is currently taking an algebraic geometry course from an unnamed prolific poster on MO, told me about the following bonus question on one of his problem sets a few weeks ago.
...
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Common Point of Intersection of n-dimensional ellipsoids [closed]
Suppose we have two ellipses in 2-dimensions centered at the origin. It is easy to visualize that (unless one is contained in the other) they will have 4 points of intersection. Can we say that in ...
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Intersection of Subspaces with $O(3)$
Sorry for the confusion from earlier. I tried to fix the thread. The old version can be found below.
For $6$-dimensional subspaces $V$ of the space $\mathbb{R}^{3\times 3}$ of real three-times-three ...
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Chern classes of a family and Chern classes of a member
Let $X$ be a smooth projective variety over an algebraically closed field $k$ and $\mathcal E$ a family on torsion free coherent sheaves on $X$ parametrized by a smooth curve (over $k$) i.e. a ...
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Intersection product of pull back under projection
Let $X$ be a surface and $Y$ be a curve over $\mathbb{C}$. Let $L$ and $L'$ be ample line bundles on $X$ and $Y$ respectively. Consider the product $X\times Y$. Let $p$ and $q$ be the projection from $...
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Calculating the distinguished varieties of intersection product
In Fulton's Intersection theory Example 6.1.2，one considers two divisors on $\mathbf{P}^2$ given by $D_1=A+2B,D_2=2A+B$, where $A,B$ are lines meeting at a point.
Let $X=D_1\times D_2,Y=\mathbf{P}^2\...
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Can a curve intersect a given curve only at given points?
Clearly the question in the title has a positive answer for analytic (or smooth, or continuous ...) curves, but what about the algebraic category? More specifically, given an irreducible polynomial ...
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Showing that closure of all lines through a projective variety $Y$ has degree strictly less than $Y$
Let $Y$ be a variety of dimension $r$ and degree $d>1$ in $\mathbf{P}^n$. Let $P\in Y$ be a nonsingular point. Define $X$ to be the closure of the union of all lines $PQ$, where $Q\in Y$, $Q\ne P$. ...
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Is the Gysin pullback of an effective cycle effective?
Suppose $E$ is a rank $r$ vector bundle over a projective variety $X$, denote the zero section by $i\colon X\to E$. Given an effective cycle $a\in A_{k+r}(E)$, the Gysin pullback gives us a class $i^
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Smooth quadric hypersurface, Hilbert scheme is blowup of Grassmannian?
Let $Q \subset \mathbb{P}^n$ be a smooth quadric hypersurface. Where can I find a proof of/can anyone supply a proof of$$\text{Hilb}_{2m + 1}(Q) \cong \text{Bl}_{OG(3, n+1)}G(3, n+1)?$$Can we conclude ...
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Tor-amplitude [0, 1] in the setting of intersection theory on a regular surface?
The question is coming from Definition 1.5 in Deligne's Expose X in SGA 7 on intersection theory.
Let $X$ be a connected regular scheme of dimension $2$ and $Y \subset X$ a reduced divisor that ...
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