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15
votes
0answers
541 views

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 ...
10
votes
0answers
211 views

Smooth, complete varieties on which “zero is effective”

I will say zero is effective on a complete, smooth variety $X$ if some positive linear combination of irreducible varieties is rationally equivalent to zero. In other words, zero is effective if there ...
7
votes
0answers
153 views

Chow ring of extended tropicalizations

In Allermann-Rau '09, the authors define the Chow groups of an arbitrary abstract tropical cycle. In particular, one may take the tropical cycle to be the tropicalization of a subvariety of a torus. ...
6
votes
0answers
179 views

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 ...
6
votes
0answers
286 views

A question on infinitesimal deformation (related to intersection theory)

Let $X$ be a connected projective scheme in $\mathbb{P}^n$. Assume, $2 \le \dim X \le n-2$. Let $H$ be a general hyperplane in $\mathbb{P}^n$. Denote by $Z:=X.H$ and $Z'=X.H^m$ for $m \gg 0$. Then ...
6
votes
0answers
147 views

Divisibility of all entries in an intersection form

What are situations where one can conclude that all entries of an intersection form are divisible by a fixed integer? More precisely: $F \subset S$ is a proper connected (usually reducible) half-...
6
votes
0answers
279 views

Flat morphisms whose fibers are affine spaces

Let $f:X \to Y$ be a flat morphism, such that each fiber is isomorphic to the affine space $\mathbb{A}^n$. Then is is true that $f$ is a Zariski affine bundle? If not, is it at least an ètale affine ...
6
votes
0answers
369 views

intersection theory on proper algebraic spaces

I have a question about the second example in Hartshorne's Algebraic Geometry, Appendix B, section 3 (given by Hironaka?). It is an example of a compact complex Moishezon 3-fold $X$ which is not an ...
5
votes
0answers
201 views

Cycle classes that are killed by pushing forward from normalization

Let $X$ be a non-normal algebraic variety and $f \colon X' \to X$ its normalization. Is there a general description $\mathrm{ker}\left(\mathrm{CH}_k(X') \to \mathrm{CH}_k(X)\right)$? Are there ...
5
votes
0answers
282 views

Relationship between virtual cohomological dimension and tautological rings for moduli spaces of curves

Here's the short version of the question. For $M_{g,n}$, $M_{g,n}^{rt}$, $M_{g,n}^{ct}$ and $\overline M_{g,n}$ it seems that the virtual cohomological dimension is given by the complex dimension plus ...
5
votes
0answers
719 views

Riemann-Roch and Grothendieck duality: general case of Fulton's example 18.3.19

Fulton's "Intersection theory" book contains the following fact (example 18.3.19): Let $X$ be a Cohen-Macaulay scheme over a field. Assume $X$ can be imbedded in a smooth scheme (so it has a ...
4
votes
0answers
75 views

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 $...
4
votes
0answers
161 views

Pullback/pushforward of bivariant intersection classes

In chapter 17 of Fulton's Intersection Theory, he defines a bivariant intersection theory. I'm a bit puzzled by the pushforward/pullback he defines on page 322-323, though; they seem not analogous to ...
4
votes
0answers
311 views

Pull-push formula?

There are many contexts in which the push-pull formula $f_*(f^*(\alpha)\cdot \beta) = \alpha \cdot f_*(\beta)$ holds. I am interesting mostly in the case of algebraic K-theory and Chow rings (under ...
4
votes
0answers
174 views

Why write GRR with the relative tangent sheaf?

The first published version of the Grothendieck-Riemann-Roch theorem, GRR for short, was written in the form $$ \operatorname{ch}(f_!\alpha).\operatorname{Td}(Y) = f_*\left(\operatorname{ch}(\alpha).\...
4
votes
0answers
124 views

Strategy to prove formula for top chern class from knowlege of chern character

I am trying to prove a conjecture that involves an enumerative problem. In the course of doing so, the following situation came up. I have a sequence of (smooth, complex, rationally connected) ...
4
votes
0answers
274 views

where to learn K-group of coherent sheaves modulo numerical equivalence?

I am trying to emerge from my complete ignorance about intersection theory. I have a bias toward sheaves, so I like the idea of doing intersection theory with the K-group of coherent sheaves. From ...
3
votes
0answers
58 views

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 \...
3
votes
0answers
115 views

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 $C_2^2<0$)....
3
votes
0answers
168 views

Computing intersection number of two arithmetic line bundles

I have some questions in Arithmetic Arakelov geometry Let $\mathcal X\to Spec(\mathcal O_K)=C$ be an arithmetric projective variety over $C$ , where $\mathcal O_K$, ring of number filed $K$ and $\...
3
votes
0answers
149 views

On the local Euler obstruction for singular varieties

Let $X$ be a complex algebraic variety (not necessarily irreducible, nor reduced). Then the local Euler obstruction is a group isomorphism $$\textrm{Eu}: Z_\ast X\to F_\ast X,$$ where $Z_\ast X$ is ...
3
votes
0answers
101 views

Intersection numbers on blow ups of toric varieties

Suppose we have a smooth, complete toric varietiy $X_{\Sigma}$ of dimension $n$. Let $\sigma_k \in \Sigma(k)$ a smooth $k$-dimensional cone in $\Sigma$ and suppose we make the toric blow up at the ...
3
votes
0answers
174 views

Chow ring of a $\mu_2$-gerbe

Suppose that $X$ is a stack, and $Y \to X$ is a $\mu_2$-gerbe. Is there any relationship between the integral Chow rings (in the sense of Edidin and Graham) of $X$ and $Y$? (I assume they become ...
3
votes
0answers
292 views

Bezout's theorem for non-proper intersections?

Is there a version of Bézout's theorem for non-proper intersections? For my specific problem, the setup is as follows: Let $P_1,P_2,P_3,P_4\in\mathbb{C}[z_1,z_2,z_3,z_4]$, and suppose that (as a ...
3
votes
0answers
166 views

Reference request: Samuel's multiplicity and degree

I am looking for references for the following simple facts. Let $Y\subset \mathbb{P}^n$ be a variety (or pure-dimensional algebraic set). For $P\in Y$ denote by $e_p(Y)$ the (Samuel's) multiplicity ...
3
votes
0answers
140 views

Topology of K3 as a sum of two abelian fibrations.

Let $E$ be a blow-up of $\mathbb{P}^2$ at 9-points in the bases locus of a pencil of elliptic curves (A $T^2$ fibration over $S^2$). K3 surfaces is obtained by removing a fiber from two copies of $E$ ...
2
votes
0answers
95 views

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^![...
2
votes
0answers
183 views

The self intersection class of exceptional divisor of 3-fold blown up along a curve

Suppose $X$ is a smooth complete variety of dimension $3$, let $\sigma\colon\widetilde{X}\to X$ the blow-up along smooth curve $C\subset X$, let $\sigma^{-1}(C)=E$ be the exceptional divisor, let $f$ ...
2
votes
0answers
186 views

When is the intersection number well-defined?

According to 1.34 of Birational Geometry of Algebraic Varieties(J.Kollár, S.Mori), there are at least 4 ways(classical approach, cohomological approach, general intersection theory and topological ...
2
votes
0answers
155 views

Is complex surface in CP(3) a two handlebody?

Consider a complex surface given by homogeneous equation in $\mathbb{C}P^3$. Without loss of generality, take \begin{equation} S = \{[x:y:z:w] \in \mathbb{C}P^3~ |~ x^d + y^d + z^d + w^d = 0\} \end{...
2
votes
0answers
174 views

bijection of moduli space of equivariant holomorphic embeddings

Consider the moduli space $\mathcal{M}$ of equivariant holomorphic embeddings of closed oriented Riemann surfaces into a generic quintic three-fold $X$ in $\mathbb{P}^4,$ of given degree $d \in H_2(X,...
2
votes
0answers
127 views

Computing intersection of cycles on the product of Grassmannians/Deligne-Lusztig varieties

My collaborators and I are preparing an interesting manuscript where the computation leads to something related to what we believe to be in the area of Schubert calculus; but none of us knows much ...
2
votes
0answers
176 views

Negative self intersection and section of the conormal sheaf for a singular complex curve

Let $M$ be a complex $2$-dimensional manifold and $C$ be a compact complex curve in $M$ (possibly singular). Let us suppose that there exists a holomorphic function $f\in\mathcal{O}(M)$ such that $f$...
2
votes
0answers
201 views

Segre class of cones and Base change of projective cones

I'm trying to work out a result in Fulton's intersection theory and I think I need the following basic result about base change of projective cones (whose support may not be the entire base scheme). ...
2
votes
0answers
147 views

What is known about the structure of $\mathbb Z[c_1(\mathcal O_V(1))]$ for a projective $\Bbbk$-variety $V$?

Motivation: Following Fulton's Intersection Theory, the Chern class of an arbitrary algebraic $\Bbbk$-scheme $X$ can be constructed as follows. First, define the graded by codimension abelian group $...
2
votes
0answers
322 views

intersection theory

Assume that $X$ is a smooth 3-fold (over $\mathbb C$). Let $V$ be a smooth divisor on $X$ and let $S_1,S_2$ be prime divisors on $X$. Assume that given a curve $C$ on $X$ not contained in $V$, then $...
1
vote
0answers
31 views

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 $...
1
vote
0answers
107 views

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 $Z_1,...,Z_n \in |\mathcal{O}_X(k)|$ ...
1
vote
0answers
70 views

When is a collection of toric subvarieties in $\mathbb P^r$ generic?

Suppose $Z_1,\dots,Z_n \subset \mathbb P^r$ are toric sub-varieties. By this I mean, there is a fixed co-ordinate system $\mathbb P^r=\{[z_0:\dots:z_r]\}$ and each subvariety is defined by the ...
1
vote
0answers
73 views

Endomorphism of Chow goup induced by a birational map

Let $\phi:X\dashrightarrow Y$ be a birational map between smooth projective $k$-varieties ($k=\bar k$) and $\Gamma$ be the closure of the graph of $\phi$. In Fulton's intersection theory example 16.1....
1
vote
0answers
75 views

Intersection of affine subspaces with specific Veronese variety

Consider the mapping $$ \Psi: \mathbb R^2 \times \mathbb R^2 \to \mathbb R^5, \\ \Psi(x^{(1)},x^{(2)}) = \begin{pmatrix} x_1^{(1)} + x_1^{(2)} \\ x_2^{(1)} + x_2^{(2)} \\ (x_1^{(1)})^2 + (x_1^{(2)})^...
1
vote
0answers
77 views

What is the Segre class of a generating line of a cone

Suppose $U=\textrm{Proj}\ k[X,Y,Z,W]/(XY-Z^2)$ is the projective closure of an affine cone, let $V$ be a generating line of the cone $V=V(Y,Z)$, how do we calculate the Segre class $s(V,U)$? (We can ...
1
vote
0answers
148 views

Intersection Multiplicity

Let $X$ be an hyper-surface in an affine space defined by an equation $F$. We can assume that the ground field is $\mathbb{C}$ and $X$ is normal. Take functions $f_1,\dots, f_n$ on $X$ and let $B$ ...
1
vote
0answers
133 views

Sufficient conditions to get complete intersection curves

Let $H_1,H_2\cdots,H_{d-1}$ be hypersurfaces in $\mathbb{P}^d$, if the intersection $B:=H_1\cap H_2\cap \cdots \cap H_{d-1}$ is $1$-dimensional then it is called a complete intersection curve. What ...
1
vote
0answers
151 views

excess intersection theory

Can the excess intersection theory be applied to the following problem: I have a non-singular irreducible variety $X$ of dimension $k$ and degree $d$ and $k+1$ hyperplane sections of $X$, $H_1,H_2,\...
1
vote
0answers
115 views

Bounds for intersection multiplicity

Let's for simplicity work in $\mathbb{C}^n$. Suppose that $f_1,\dots, f_n$ are polynomials and $0$ is an isolated solution of the system $f_1(z)=\dots=f_n(z)=0$. I want to bound from below the ...
1
vote
0answers
127 views

Non-proper intersection of projective schemes

Let $X, Y$ be projective varieties in $\mathbb{P}^n$ for $n>10$. Assume that dimensions of $X,Y$ are greater than $n/2$. My first question is as follows: Is there any criterion (...
1
vote
0answers
153 views

Is -(E,E) greater or equal to 2 for a minimal resolution

I'm quite confused by the terminology minimal resolution and minimal model. Let $f:X\longrightarrow Y$ be a minimal resolution of singularities, where $Y$ is a normal surface. Let $E$ be an ...
0
votes
0answers
101 views

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 ...
0
votes
0answers
95 views

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 $Q\in Y$, $Q\ne P$. ...