Timeline for Is the Segre embedding of two real varieties a real variety?
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Oct 11, 2022 at 13:51 | vote | accept | Ben | ||
| Oct 6, 2022 at 20:57 | comment | added | François Brunault | Each real algebraic subset of $\mathbb{R}^n$ has a canonical structure of affine algebraic variety over $\mathbb{R}$. Some authors use this implicitly. But for this question, to avoid ambiguity I would not say $X,Y$ are varieties, but rather algebraic sets. | |
| Oct 6, 2022 at 11:39 | answer | added | Johannes Huisman | timeline score: 4 | |
| Oct 6, 2022 at 11:26 | comment | added | YCor | @Ben because defining a real variety as zero locus in a real affine/projective space doesn't work well. There are nonempty projective real varieties with no real points, etc. | |
| Oct 6, 2022 at 10:17 | comment | added | Ben | @FrancescoPolizzi But that just cuts out the full image of the Segre. I want only those points that come from $X\times Y$. | |
| Oct 6, 2022 at 10:14 | comment | added | Francesco Polizzi | Yes, it is the zero locus of the $2 \times 2$ minors of the matrix $Z_{ij}$, where $Z_{ij}$ are the natural coordinates on the image of the Segre map. | |
| Oct 6, 2022 at 10:13 | comment | added | Ben | @YCor Why doesn’t it make sense? Perhaps I should rephrase everything in terms of real varieties in $R^n$ that form cones. Or maybe this is true for arbitrary real varieties in $R^n$. | |
| Oct 6, 2022 at 10:08 | comment | added | Ben | @FrancescoPolizzi You claim that the equations for $Seg(X\times Y)$ are easy to write down in terms of the equations for $X$ and $Y$ and the $2\times 2$ determinants? | |
| Oct 6, 2022 at 10:06 | comment | added | YCor | Is the question equivalent to whether the homomorphism (to the Segre variety) is surjective on real points? (In principle it makes no sense to say that a subset of the real projective space is a variety. However, it makes sense to ask whether it is Zariski closed.) | |
| Oct 6, 2022 at 10:05 | history | edited | YCor | CC BY-SA 4.0 |
formatting, added tag
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| Oct 6, 2022 at 10:02 | comment | added | Francesco Polizzi | Segre varieties are determinantal, and their defining equations have integer coefficients. | |
| Oct 6, 2022 at 9:50 | history | asked | Ben | CC BY-SA 4.0 |