Questions tagged [intersection-theory]
The intersection-theory tag has no usage guidance.
175 questions with no upvoted or accepted answers
23
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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 ...
- 5,504
11
votes
0
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607
views
The virtual fundamental class as derived intersection
Say $X$ is a smooth projective variety and $\beta\in H_2(X)$ is a class. Then there is a finite-type proper scheme (or in general, stack) $SM : = \overline{\mathcal{M}}_{g,n}(X,\beta)$ of stable maps ...
- 7,972
10
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0
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327
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Integrality of primary genus $0$ Gromov-Witten invariants of a Fano manifold
Suppose $(X,\omega)$ is a positively monotone compact symplectic manifold, i.e., after a positive scaling of the symplectic form, we have $c_1(T_X) = [\omega]$ in de Rham cohomology ($T_X$ has well-...
- 1,761
10
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331
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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 ...
- 7,338
8
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569
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Bloch Ogus spectral sequence
Let $X$ be a smooth projective variety over $\mathbf{C}$, and $p : X_{\rm an}\to X_{\rm Zar}$ the obvious map of sites.
The Leray spectral sequence
$$H^r(X_{\rm Zar}, R^sp_*\mathbf{C})\Rightarrow H^{...
user92332
8
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0
answers
204
views
Chow ring of extended tropicalizations
In Allermann-Rau '09, the authors define the Chow groups of an arbitrary abstract tropical cycle. In particular, one may take the tropical cycle to be the tropicalization of a subvariety of a torus. ...
- 166
7
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0
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183
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Intersection numbers via residue formula
$\newcommand{\sslash}{\mathbin{/\mkern-7mu/}}$With a friend we are trying to understand residue formulas in the article "Cohomology pairings on singular quotients in geometric invariant theory&...
7
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Applications of E8 manifold
The $E_8$ Cartan matrix is given by,
$$
K_{E_8}=\begin{pmatrix}
2 & -1 & 0 & 0 & 0 & 0 & 0 & 0 \\
-1 & 2 & -1& 0 & 0 & 0 & 0 & 0 \\
...
- 10.5k
7
votes
0
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551
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 ...
- 2,384
7
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0
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188
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-...
- 6,162
7
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0
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570
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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 ...
- 4,265
6
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0
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250
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Singling out irreducible components
Let $V\subset \mathbb{A}^n$ be a variety defined by equations of degree $\leq D$, or, what is the same, an intersection of hypersurfaces of degree $\leq D$. Let $V^+_0$ be an irreducible component of $...
- 20.2k
6
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0
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254
views
If cohomology theory corresponds to intersection theory, valuation theory corresponds to -?
This is a meta question I asked myself. Cohomology theory is dual to an intersection theory. Is there anything valuation theory corresponds to in general?
For instance, McMullen's polytope algebra is ...
- 1,356
6
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0
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241
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Bezout theorem for germs of holomorphic functions
UPDATE.
It was pointed out by @Dmitri that two smooth curves given by $f=y$ and $g=y+x^k$ in $\mathbb C^2$ provide a simple counterexample.
Let $f_1, \ldots, f_p, g_1, \ldots, g_q$ be germs of ...
- 1,238
6
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0
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355
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Schubert cycles on Grassmannian bundles
Let $X$ be smooth variety and let $\mathcal{E}$ be a vector bundle on $X$ of rank $n$. On the total space of the Grassmannian bundle $\pi:G(k,\mathcal{E})\to X$ we have the tautological exact sequence ...
- 735
6
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313
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Intersection of curves in abelian varieties
Fix an abelian variety $A$ of dimension at least $3$ and a (smooth, projective, irreducible) curve $C$ inside $A$ that generates $A$ as an algebraic group, over an algebraically closed field. Now, fix ...
- 30.5k
6
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0
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591
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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 ...
- 69
6
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0
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298
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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 ...
- 1,534
6
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363
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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
$$
\operatorname{ch}(f_!\alpha).\operatorname{Td}(Y)
=
f_*\left(\operatorname{ch}(\alpha).\...
- 2,011
6
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0
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320
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A question on infinitesimal deformation (related to intersection theory)
Let $X$ be a connected projective scheme in $\mathbb{P}^n$. Assume, $2 \le \dim X \le n-2$. Let $H$ be a general hyperplane in $\mathbb{P}^n$. Denote by $Z:=X.H$ and $Z'=X.H^m$ for $m \gg 0$. Then ...
- 833
6
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426
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Relationship between virtual cohomological dimension and tautological rings for moduli spaces of curves
Here's the short version of the question. For $M_{g,n}$, $M_{g,n}^{rt}$, $M_{g,n}^{ct}$ and $\overline M_{g,n}$ it seems that the virtual cohomological dimension is given by the complex dimension plus ...
- 40.3k
6
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882
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Riemann-Roch and Grothendieck duality: general case of Fulton's example 18.3.19
Fulton's "Intersection theory" book contains the following fact (example 18.3.19):
Let $X$ be a Cohen-Macaulay scheme over a field. Assume $X$ can be imbedded in a smooth scheme (so it has a ...
- 30.6k
5
votes
0
answers
135
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Geometric interpretation of pairing between bordism and cobordism
In page 448 of these notes, a pairing between bordism and cobordism
$$\langle \ ,\ \rangle: U^m(X)\otimes U_n(X)\rightarrow \Omega^U_{n-m}$$
is defined as follows. Assume $x\in U^m$ is represented by $...
- 171
5
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0
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163
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How to compute the class defined by intersection with a square?
$\DeclareMathOperator\Gr{Gr}$Let $\Gr(k,n+k)$ (of course, one can do also for $\Gr(k,\infty)$) be the complex Grassmannian of $k$-planes in $n+k$-dimensional linear space.
It is well-known that ...
- 719
5
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0
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397
views
Vector bundles vs algebraic cycles
For integral schemes, the Picard group is isomorphic to the group of Cartier divisors modulo linear equivalence.
What is the correct analog of this isomorphism, for higher codimension Chow groups vs ...
user119470
5
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0
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486
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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 $\...
user21574
5
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0
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293
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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) ...
- 1,509
5
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0
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217
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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 ...
- 40.3k
5
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479
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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 ...
- 1,889
4
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110
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Euler factors from bad primes and the Beilinson-Bloch vanishing conjecture
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} ...
- 531
4
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0
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167
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Is the group of homologically trivial cycles in a variety over a finite field torsion?
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 ...
- 531
4
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0
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137
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Inverse direction of Hodge index theorem
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 ...
- 2,175
4
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0
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184
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Obstruction to finding a Whitney disk
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$ ...
4
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0
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226
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K-theoretic derivation of Bézout theorem
In the paper "$K$-Theory and Intersection Theory" by Henri Gillet in Handbook of K-theory, 2005 (link behind paywall at Springerlink), the author says:
"When the ground field $k = \mathbb C$, Bézout’...
4
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0
answers
261
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Motives up to homological equivalence
Let $X$ be a smooth projective variety over a field $k$ finitely generated over its prime field, and $M_{hom}(X)$ the category of motives modulo $\ell$-adic homological equivalence.
(1) Is $M_{hom}(...
user120812
4
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0
answers
245
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Hard Lefschetz for cycles
Let $X$ be a smooth projective variety over a field $k$. It is known by work of Deligne, that the Lefschetz operator:
$$
L^k:H^{2n-2k}\left(X_{\overline{k}},\mathbf{Q}_{\ell}\right)\to H^{2n+2k}\left(...
user92332
4
votes
0
answers
347
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Complete intersections in projective spaces
Let $X$ be an arbitrary smooth projective variety over a field $k$.
Do there exist:
a smooth complete intersection $X'$ in a projective space.
a surjective morphism of $k$-varieties $X'\to X$ ?
user119470
4
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265
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Is there an analogy of Sumihiro's equivariant Chow's lemma for DM stack?
There is an analogy of Chow's lemma for a DM stack $X$ written in the Laumon's book 'Champ algebrique'. There exists a generically finite, proper surjective morphism $Y \to X$ from a quasi-projective ...
- 421
4
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181
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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 ...
- 1,217
4
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0
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225
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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 ...
- 357
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221
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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 $...
- 573
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0
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168
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Are motives of K3 surfaces of abelian type?
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)...
- 658
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Tate conjecture for singular varieties in terms of intersection homology
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 ...
- 658
3
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148
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Simple Grothendieck-Riemann-Roch computation with relative Todd class
$\DeclareMathOperator\Tot{Tot}\DeclareMathOperator\ch{ch}\DeclareMathOperator\td{td}\DeclareMathOperator\ker{ker}\DeclareMathOperator\rk{rk}$I was wondering if the following is correct: Let $X=\Tot(L)$...
- 139
3
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0
answers
393
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Why is the "wrong" definition of intersection of varieties the "right" one for generalized Bézout?
For ease of notation, define the degree of a variety to be the sum of the degrees of its irreducible components. The generalized Bézout theorem (due to Fulton and Macpherson) states that, for $V_1$, $...
- 20.2k
3
votes
0
answers
225
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Intersection of two modules (and sub-modules) under tensors
I have a (hopefully quick) question regarding an intersection of two tensor modules. Let $K$ be a field and $A, B$ finitely-generated modules over the Laurent series $K((X))$. Let $\tilde{A}$ be a (...
- 31
3
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0
answers
174
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Intersection theory on schemes with Gorenstein singularities
Is there a good reference/book on Intersection theory on schemes with Gorenstein singularities? Does the construction of the intersection of cycles discussed in Fulton's book also hold for schemes ...
user497552
3
votes
0
answers
121
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Detecting non-principal Weil divisors on normal varieties using curves
Let $X$ be a normal projective variety over an algebraically closed field $k$. Given any morphism $f:Y\to X$, there is a pullback homomorphism $f^*:\text{Cl}(X)\to\text{Cl}(Y)$, where $\text{Cl}(X)$ ...
- 1,363
3
votes
0
answers
91
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A proper morphism restricts to a closure of a point on the generic fiber
Let $\pi:X^{e}\rightarrow Y^{d}(e\geq d)$ be a proper and dominant morphism of projective varieties over field $k$. Moreover, $Y$ is assumed to be smooth. Denote $\eta$ the generic point of $Y$, $X_\...
- 113
3
votes
0
answers
95
views
Singling out lower-dimensional components
Let $V\subset \mathbb{A}^n$ be defined by equations of degree $\leq D$. (That is, $V$ is an intersection of hypersurfaces of degree $\leq D$.) Assume $V$ is not pure-dimensional. Let $V^-$ be the ...
- 20.2k