795 reputation
16
bio website mathematik.uni-mainz.de/…
location
age
visits member for 1 year, 11 months
seen 2 days ago

2d
comment Number of elements in fiber
@PeterWalker What might bother you is the definition of $\deg f$ is $f$ is not finite but only quasi-finite. In general, by $\deg f$ one denotes the degree of the corresponding extension of function fields. This makes sense for etale morphisms of integral schemes $X\to Y$ as etale morphisms are dominant and quasi-finite.
2d
comment Families of Elliptic Curves
See the books Modular functions of One Variable for some Tables, especially part IV and V.
2d
comment Scheme over S and morphisms
@Grobber Consider for all complex numbers $t$ the conic $x^2 + y^2 = tz^2$. You have probably a good idea of how these conics look like for all small enough real numbers $t$. For $t=0$, you get the union of two lines (namely, $x+iy =0$ and $x-iy = 0$). It was an important innovation of Grothendieck to realize this intuitive picture in the category of schemes as the relative morphism $X =$ Proj$(R[x,y,z]/(x^2 + y^2 - tz^2])\to$ Spec $R = S$, where $R$ is a ring and $t$ is an element of $R$. (In the example: $R$ would correspond to $\mathbb C[t]$.)
Jan
21
comment Cubic hypersurface of singular conics
Dear @user198182 , I can't write all the details now, but could this hypersurface be defined by the Jacobi matrix of the "universal" conic? For a conic given by $a= (a_0:\ldots:a_5)$ to be singular it is necessary and suffices for the Jacobi matrix of the associated quadric to be of rank zero. If $f_{a}(x,y,z)$ is the associated quadric, then this translates into the vanishing of the discriminant of $f_{a}$ probably. I wrote this in a hurry, so probably someone else could correct/complement this comment.
Jan
20
comment Question on Algebraic Hartogs Lemma for locally Noetherian normal schemes
But now use that $\cap_{\mathfrak p, ht = 1} A_{\mathfrak p} = A$, to see that it is the identity. Or am I missing something?
Jan
20
comment Question on Algebraic Hartogs Lemma for locally Noetherian normal schemes
@enoughsaid05 The (injective) morphism $A\to A_{\mathfrak p}$ corresponds to the morphism Spec $A_{\mathfrak p}\to X =$ Spec $A$. For all $\mathfrak p$, it factorizes through $\mathcal O_X(U)$, as Spec $A_{\mathfrak p} \to X$ factorizes through $U$ (by assumption). Therefore, we have $$ A \to \mathcal O_X(U) \to A_{\mathfrak p}$$ for all $\mathfrak p$ of height $1$. But, then we get $$A\to \mathcal O_X(U)\to \cap_{\mathfrak p, ht =1} A_{\mathfrak p}$$ where the map $A\to \cap_{\mathfrak p, ht = 1} A_{\mathfrak p}$ is injective (clearly).
Jan
20
comment Morphism $f\colon X\to S$ is proper iff $f^{-1}(V_j)\to V_j$ is proper for some open cover $\{V_j\}$ of $S$? (Lemma 28.42.3 of Stacks Project)
Dear @GeorgesElencwajg , thank you very much for your kind words. I have to say that I truly enjoy your many answers and questions on MSE and MO. Best, Ariyan
Jan
20
answered Morphism $f\colon X\to S$ is proper iff $f^{-1}(V_j)\to V_j$ is proper for some open cover $\{V_j\}$ of $S$? (Lemma 28.42.3 of Stacks Project)
Jan
19
comment Reference request: Zero set of global section
What if you take $(f,\ldots,f)\in \oplus_{i=1}^n\Gamma(\mathbb P^n,\mathcal O(1)) \subset \Gamma(\mathbb P^n,\mathcal O(1)^{\oplus n})$?
Jan
19
answered Jacobian of a Riemann surface and double unramified covers
Jan
19
comment Reference request: Zero set of global section
What happens if you take zero sets of global sections of $\mathcal O(1)$?
Jan
19
comment Properties of fibers of a morphism of varieties
@Dubious It seems like I miswrote many things in my first comment. I meant to say the following: If $X$ is an integral algebraic variety, then the locus of smoothness is dense (and open). On the other, if $X$ is irreducible and non-reduced, then the locus of smoothness is empty (and open), i.e., the locus of singularities is the entire variety. See tracing's answer for a proof of these facts. You can also take a look at Liu's book Prop 4.2.24 .
Jan
19
comment Properties of fibers of a morphism of varieties
Dear @tracing thank you for the correction. Also, your answer is certainly more clear than what I wrote.
Jan
18
comment Properties of fibers of a morphism of varieties
Even if you leave out the last condition, the reducedness follows from the fact that the image of your section (assumed to exist) lies in the smooth locus of $f$. So every fibre has a smooth point. Since they are all assumed to be irreducible, they are generically reduced (meaning reduced on a dense open), and thus reduced. (Probably this is the same answer as tracing gave.)
Jan
18
comment Properties of fibers of a morphism of varieties
If your fibers are reduced, then they are singular, and the locus of singularities is dense (so definitely not just a node). But your last condition is that every singularity on the fibre is isolated. So all fibres are reduced.
Jan
18
comment Properties of fibers of a morphism of varieties
@Dubious tracing is using Rn and Sn as defined (for instance) in math.nagoya-u.ac.jp/~takahashi/ncr.pdf. See also Serre's criterion for normality.
Jan
18
answered Question on Algebraic Hartogs Lemma for locally Noetherian normal schemes
Jan
14
revised Fibered surfaces with sections and generic fibers iso to projective space.
added 85 characters in body
Jan
14
comment Fibered surfaces with sections and generic fibers iso to projective space.
Dear @user101036 Please see the edited answer. The rank of $E$ is three. That's why you get $\mathbb P^2$. OW I see what I miswrote. I took the wrong vector bundle $E$. Of course, $f_\ast \mathcal O_X = \mathcal O_S$ (because the fibres are connected). See edited answer.
Jan
14
revised Fibered surfaces with sections and generic fibers iso to projective space.
added 85 characters in body