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68 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 $f$ ...
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0answers
47 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 ...
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0answers
74 views

Chern classes of ideal sheaf of locally complete intersection

Let $C\varsubsetneq X$ be a reducible locally complete intersection closed pure $1$-dimensional subscheme in a smooth projective variety of dimension $3$ over a algebraically closed field $k$ of ...
3
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0answers
119 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 ...
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0answers
70 views

Probability two random intervals overlap

I'm working on an algorithm for orthogonal line intersection detection and am trying to analyze some things about it. For simplicity, we can consider the problem as follows: Given N randomly ...
7
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1answer
149 views

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 ...
1
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1answer
134 views

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 ...
4
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1answer
164 views

Bézout's theorem for arcs in the plane

Consider two polynomials $p,q \in {\mathbb R}[x,y]$, both of degree $d$. Let $\gamma_p$ and $\gamma_q$ be the two curves in ${\mathbb R}^2$ that are defined by these polynomials, and assume that these ...
0
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0answers
111 views

Transversal intersection in the moving lemma

Let $X$ be a smooth projective variety over an algebraically closed field and let $A,B$ be closed irreducible subvarieties of $X$. Chow's moving lemma which is proved in the book by Eisenbud and ...
0
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0answers
77 views

A question about Segre class

Suppose $C$ is a cone over $X$.(i.e.$C=\operatorname{Spec}S$, where $S$ is a sheaf of $O_X$ algebras.) The Segre class $s(C)$of $C$ is the class in $A_*(X)$ defined by ...
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0answers
42 views

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 ...
2
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1answer
186 views

What does the Chern-Schwartz-MacPherson class of a singular variety look like?

Let $A_\ast$ and $F_\ast$ be the functors $\textrm{Var}_\mathbb C\to \textrm{Ab}$ of Chow groups and constructible functions, respectively, with respect to proper maps. Then the ...
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0answers
97 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
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0answers
144 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) = ...
3
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0answers
88 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 ...
0
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1answer
104 views

Continuity of Intersection Multiplicities

I’m looking for a correct technical version (and in the best case a reference) for a statement of the following type: Consider a complex algebraic variety $X\subset\mathbb{P}^n$ and a sequence of ...
0
votes
1answer
109 views

Samuel multiplicity

Let $X$ be the hyper-surface defined by $$f:=\sum_{i=1}^k x_i^n=0$$ in $\mathbb{C}^k$. Let $Y$ be the non-reduced sub-scheme of $X$ defined by the ideal $$I=(x_1^{n-1},\dots , x_k^{n-1}) $$ What is ...
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0answers
122 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 $B$ ...
0
votes
1answer
89 views

Intersection multiplicty and global sections

Let $X$ be a smooth projective variety, $V, W$ closed subschemes in $X$ such that $V \cap W$ is finitely many points. Let $\mathcal{L}$ be a line bundle on $X$. Is there any relation between ...
1
vote
1answer
288 views

Non-proper intersection of surfaces

I'm interested in the first basic case of excess intersection in intersection theory: Let $X$ be a smooth projective 4-fold and let $S,T$ be two surfaces in $X$. Assume that the intersection $S\cap ...
2
votes
1answer
220 views

Self-intersection and generic point

The Wikipedia entry on intersection theory contains the following statement: [for C a curve, on a surface] "the self-intersection points of C is the generic point of C, taken with multiplicity C · ...
2
votes
1answer
324 views

A question about an intersection number

Let $\pi:Y\rightarrow \mathbb{P}^3$ be the blow-up of two points $p,q\in\mathbb{P}^3$, and then of the strict transform of the line $L$ spanned by them. Now, Let $E_p,E_q, E_{p,q}$ be respectively the ...
0
votes
1answer
121 views

Intersection Matrix of a resolution

Probably this is a very easy question. Let $f:X\rightarrow S$ be a resolution of a projective surface such that $$K_X = f^{*}K_S+\sum_ia_iE_i$$ with $a_i>0$. By Grauert-Mumford theorem the ...
2
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0answers
166 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
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0answers
135 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\} ...
1
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0answers
111 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 ...
0
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0answers
80 views

birational equivalence of linear sections of algebraic varieties

Let $X$ be an irreducible algebraic variety and suppose that $L$ is a linear space defined by the linear forms $l_1,l_2,\ldots,l_k$. I want to study $L\cap X$. I would like to know whether the ...
4
votes
0answers
110 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) ...
0
votes
1answer
104 views

curve through a point avoiding an hypersurface

Let $H$ be a closed hypersurface in $\mathbb{A}^{n}$, $n$ big enough over $\mathbb{C}$. Let $U$ be the complementary open subset. Let $x\in H$, Is it possible to find an curve ...
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0answers
143 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 $X$, ...
4
votes
1answer
170 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 ...
0
votes
0answers
114 views

Zero Dimension Intersection

Let $M$ be smooth and purely $r$-dimensional, $E$ be a vector bundle of rank $r$ over $M$, $s$ be a regular section of $E$ and $Z$ the zero scheme of $s$. Then $[Z]$ is dual to the top Chern class ...
1
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1answer
208 views

Analogy between variable substitution and the moving lemma

While reading the HoTT book, I came up with the following (very) vague analogy: Consider $f(x) :\equiv \lambda y.x+y$ of type, say, $f: \mathbb{N} \to \mathbb{N} \to \mathbb{N}$. Also assume you ...
1
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1answer
219 views

Rational normal curves as set-theoretic complete intersections

Let $C\subset\mathbb{P}^n$ be a rational normal curve of degree $n$. It is know that $C$ is a set-theoretic complete intersection and that, if $n\geq 3$, is a not a scheme-theoretic complete ...
2
votes
1answer
154 views

Chern and Segre classes

I've recently started to learn about Chern and Segre classes, and it seems to me that they are very similar, sharing the same important properties and having closely related definitions. Fulton's ...
3
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1answer
196 views

Intersection theory on M_{g,n}

Is there a paper\book that lists the top intersections of Hodge classes and tautological classes on $\overline{\mathcal{M}}_{g,n}$ for small $g$ and $k$, e.g. $g=2,3$ and $k=0,1,2$ ?
3
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1answer
243 views

Blow-up of $\mathbb{P}^4$ along a quadric surface

Let $Q\subset\mathbb{P}^3\subset\mathbb{P}^4$ be a smooth quadric surface, and let $X = Bl_Q\mathbb{P}^4$ the blow-up of $\mathbb{P}^4$ along $Q$. Let $H$ be the pull-back of the hyperplane section of ...
0
votes
1answer
168 views

Relation between intersection multiplicity and Hilbert-Samuel multiplicities

Suppose $X$, $Y$, $Z$ are projective varieties in $\mathbb{P}^n_K$ of dimension $n-1$, where $K$ is a field. $X$, $Y$, $Z$ intersect properly, and $P$ is one of their intersection irreducible ...
1
vote
1answer
173 views

Deformation space form the point of view of intersection theory

I'm interested in deformations of subvarieties of a toric variety $X$. Suppose we know a subvariety $V$ in the Chow group of $X$, for example, $V$ is a linear combination of powers of hypersurfaces. ...
5
votes
1answer
236 views

Positivity question on K3 surfaces

Let $X$ be a smooth projective complex K3 surface and $L, D$ two effective divisors, $L^2\geq0$ and $D^2\geq0$. (Q1). do we have $L\cdot D\geq0$ ? If either one has positive self-intersection, the ...
5
votes
1answer
222 views

Does $h^1(D)=0$ imply numerical connectedness on K3 surfaces?

Let $X$ be a complex K3 surface and $D$ an effective divisor on $X$. We shall say: $D$ is connected if its support is connected. $D$ is numerically connected if for any non-trivial effective ...
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0answers
269 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 ...
2
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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 ...
2
votes
1answer
112 views

Depth of Schubert cycles

For $a:a_1\geq \cdots\geq a_c$, let $\sigma_a$ be the corresponding Schubert cycle over $Gr(c,\infty)$. We say $a$ is of depth $k$ if $a_1-a_c=k$ ($c>1$). Let $a$ and $b$ be of depth $k_1$ and ...
2
votes
1answer
151 views

Deformation of transversal intersection

Fix a positive integer $n \ge 2$. Let $\pi:\mathcal{X} \to B$ be a family (flat, projective and surjective morphism) of projective subschemes of $\mathbb{P}^n$. Assume $B$ is reduced, irreducible. ...
4
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2answers
206 views

Upper bound for the product of Schubert cycles

Let $Gr(c,\infty)$ be the complex grassmannian of $c$-dimensional subspaces of the infinite dimensional complex space. Every finite dimensional grassmannian, $Gr(c,N)$, can be thought as a subspace of ...
4
votes
1answer
255 views

Intersections of complex submanifolds in $\mathbb{C}^N$

This is an exercise from Gromov's Partial differential relations. (page 5) Let $V$ and $V'$ be two closed complex submanifolds in $\mathbb{C}^N$ of complimentory dimension. Prove that $V$ and $V'$ ...
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1answer
617 views

What is the relation between Lefschetz fixed point theorem and Poincare-Hopf theorem on vector fields?

In Dubrovin/Fomenko/Novikov Modern geometry--Methods and applications, Part II, the (Poincare-)Hopf theorem is treated in section 15.2 (see theorem 15.2.7 on page 129), while the Lefschetz theorem on ...
2
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1answer
79 views

Which actions preserve non-complete intersections?

Let $X$ be a smooth projective variety and $Z$ is a closed subscheme in $X$ which is not a complete intersection in $X$. Assume the dimension of $X$ (resp. $Z$) is greater than $3$ (resp. $1$). Then, ...
2
votes
0answers
81 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 ...