Timeline for Intersection of curves in non-singular projective algebraic surfaces
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 20, 2021 at 8:46 | comment | added | Balazs | See Reid: Chapters on Algebraic Surfaces, arXiv:alg-geom/9602006, Chapter A for the basic statements (without proof) and lots of worked-out examples. | |
| Nov 20, 2021 at 6:36 | answer | added | Francesco Polizzi | timeline score: 1 | |
| Nov 19, 2021 at 19:06 | comment | added | Donu Arapura | @BMS Other people have caught my meaning. I second Daniel suggestion of reading that section of Hartshorne. Let me also observe the following which might help you. Lemma: if $C$ and $D$ are distinct irreducible curves on a variety $X$, then $C\cap D$ is finite. Proof: The intersection is a proper Zariski closed subset of $C$. | |
| Nov 19, 2021 at 17:55 | comment | added | Daniel Loughran | Some potential confusion of the OP may be that it is assumed in the question that $C$ and $D$ are irreducible, but not necessarily in the comments. But even in this case $C \cdot D$ is a well-defined integer, even when $C = D$. I would recommend that the OP read Chapter V.1 of Hartshorne (see in particular Theorem 1.1). | |
| Nov 19, 2021 at 17:37 | review | Close votes | |||
| Nov 27, 2021 at 3:02 | |||||
| Nov 19, 2021 at 17:33 | comment | added | Ben McKay | The integer $C\cdot D$ is never infinite. However, $C\cdot D \ge \#(C\cap D)$ only when $C$ and $D$ have no component in common. So your first sentence is not correct, as Donu Arapura says. | |
| Nov 19, 2021 at 17:19 | comment | added | Damian Rössler | I think Donu means that $C\cap C$ will be infinite. However, $C\cdot C$ wont be. | |
| Nov 19, 2021 at 17:15 | comment | added | BMS | @DonuArapura As far as I know $C \cdot D \geq \#(C \cap D)$ (since it is the sume of the intersection multiplicities at points of $C \cap D$), so I do not understand how $ \# (C \cap D)$ can be inifnite but not $C\cdot D$. On the other hand, when you say that "$C \cap D$ might be infinite, even in the plane", I guess you make reference to the projective plane, but here we have $\# (C \cap D) \leq nm$. You're answer doesn't convince me, but thank you anyway. | |
| Nov 19, 2021 at 17:06 | history | edited | BMS | CC BY-SA 4.0 |
added 20 characters in body
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| Nov 19, 2021 at 16:58 | comment | added | Donu Arapura | On any smooth projective surface, $C\cdot D$ is defined in such a way that it's an integer. So no, $C\cdot D$ won't be $\infty$. Of course $C\cap D$ might be an infinite set, even in the plane. | |
| S Nov 19, 2021 at 16:38 | review | First questions | |||
| Nov 19, 2021 at 16:51 | |||||
| S Nov 19, 2021 at 16:38 | history | asked | BMS | CC BY-SA 4.0 |