Let $E$ be a vector bundle of rank $2$ on $\mathbb P^2$ and $Z$ be zero dimensional subscheme of length $k$ supported on a point $p \in \mathbb P^2$. What is the number $h^0(E(m) \times \mathscr O_Z)$ for an integer $m$? My intuition is that it should be $2k$. But it seems in some literature it...

Let $S=k[x_0,x_1;y_0,y_1]$ be the bihomogeneous coordinate ring of $\mathbb{P}^1\times\mathbb{P}^1$. Suppose that $F\in S_{(d_1,d_2)}$ is a generic bihomogeneous form of bidegree $(d_1,d_2)$, and let $F_1\in S_{(a_1,b_1)}$, $F_2\in S_{(a_2,b_2)}$ be generic forms of lower bidegrees. The question...

I'm reading CLS's Toric Varieties right now, and something is confusing me. By Theorem 1.1.17, an affine variety $V$ is toric iff $I(V)$ is toric, i.e. prime and generated by binomials. Now the variety $V = V(X+Y-Z) \subset \mathbb{C}^3$ seems to me to be toric simply because it's isomorphic to...

I am seeking guidance on how to properly classify the singular projective surface $S \subset \mathbb{P}^3$ defined by the degree 4 homogeneous polynomial $y^4 - 4x^3w + 8w^4 - z^2w^2 = 0$, which is known to contain some elliptic curve of rank 1, alongside a unique isolated singularity at $P_0 =...

I'm studying theorem III.7.5 from Hartshorne's book. There are a few things I don't understand in this proof. Theorem. Let $X$ be a projective scheme over a field $k$. Then $X$ has a dualizing sheaf $\omega_X^\circ$. Proof. Let $\dim(X) = n$. Embed $X$ as a closed subscheme of $P:=\mathbb{P}^N$...

Let $V$ be a finite dimensional complex vector space and $G$ be a finite subgroup of $G<\operatorname{GL}(V)$. My question is, essentially, when do the orbit space $V/G$ have finitely many symplectic leaves? My interest lies, specially, in the case when $V=h \oplus h^*$, and $G$ is given by a...

Let $E \subset \mathbb{R}^n$ be a Besicovitch (Kakeya) set, i.e., a compact set containing a unit line segment in every direction $e \in \mathbb{S}^{n-1}$. The Kakeya conjecture asserts that $\dim_{\text{H}}(E) = \dim_{\text{M}}(E) = n$ for all $n \ge 4$. Given the geometric obstructions in $n...

Let $S$ be a Dedekind scheme (using the less conventional definition: locally Noetherian, irreducible, all stalks normal and $\dim S \le 1$), $X$ an irreducible scheme, and $f: X \to S$ a dominant morphism of finite type. Consider a point $s \in S$, and let $T = \operatorname{Spec}...

I try to understand what the Grassmannian $\operatorname{Gr}(2, F^3)$ (over some field $F$) looks like geometrically (in as far this term makes sense when we do not specify the field), and in particular how two subsets (specified below) divide the whole thing among them. To my shame I do not...

Let $S$ be a K3 surface over some field $k$. That is, $S$ is a nice variety over $k$ such that the canonical bundle $\omega_S$ is trivial and $H^1(S, \mathcal{O}_S) = 0$. As an exercise for myself, I wanted to see if I can compute the Hodge numbers $h^{p,q} := \dim_k H^q(S,\Omega^p_S)$. I have...

(I'm sorry for my English.) Hello. I have been reading the book Advanced Topics in the Arithmetic of Elliptic Curves written by Joseph H. Silverman. In the course of the proof of Tate's algorithm (page 375, at the end of the proof of Step 8), there is an equality on the order as follows:...

It is a well-known fact that there exists a non‑principal ultrafilter $\mathcal{U}$ on the set of prime numbers such that the ultraproduct of the algebraic closures of the finite fields is isomorphic to the complex numbers: $$ \mathbb{C}\;\cong\; \prod_{\mathcal{U}}\ \overline{\mathbb{F}}_p . $$...

Let $\Phi=[F_1,\ldots,F_N]$ be a map, with $F_i$ homogenous polynomials of variables $X_1,\ldots,X_N$ of the same degree with integer coefficients. The map $\Phi$ mapping $Z^N$ into itself may share the following property: Let $P=(x_1,\ldots,x_N)\in Z^N$ be any point such that $\gcd...

I am self studying Algebraic geometry from Gortz and Wedhorn's Algebraic Geometry :1 Schemes. I have a question on Page $16$ of the textbook just after the definition of morphism of affine algebraic sets. Remark $1.29$: the definition (of morphisms between affine algebraic sets) shows that a...

I am unable to understand the proof of proposition $1.40$ given on Page $21$ of the textbook by Gortz and Wedhorn. Definition $1.30$ Let $X\subset \mathbb{A}^n{k}$ be the affine algebraic set The $k-$algebra $\Gamma(X)= k[T_1,...,T_n]\cong Hom (X, \mathbb{A}^1(k))$ is called the affine...

I am self studying algebraic geometry from the textbook of Daniel Perrin (Algebraic Geometry: An Introduction). On page 44 is the Proposition 3.5 which I am unable to prove and need help with. Proposition 3.5. Let $(X,O_X)$ and $(Y,O_Y)$ be two affine algebraic sets equipped with the affine...

I have a question in the proof ofCorollory $4.4$ of Daniel Perrin's Algebraic Geometry on Page $45$. Corollary $4.4$ A non empty algebraic variety can be uniquely written as a finite union of irreducible closed sets which do not contain each other. Proof:By quasi-compactness, we can write $X=...

I have 2 question in the proof of Proposition $4.6$ of algebraic geometry by Daniel Perrin.( Page $46$). Statement of Proposition $4.6$: Let $X$ be an algebraic variety and let $Y$ be a closed set in $X$. We define a sheaf of rings $O_Y$ of $Y$ by setting $O_Y(V)= ${$f:V\to k| \forall x\in V...

This question was asked in my assignment and I am stuck on it. Question:If $A$ is infinite integral domain , $F,G\in A[X_i|i\in I]$, $F\neq 0$ and if $A^I \setminus V_A(F)\subseteq V_A(G)$,then show that $G=0$. Attempt: Let $D(F)= A^I \setminus V_A(F)=${$a\in A^I : F(a)\neq0$} . Given...

Let $F_1,F_2$ be degree $1$ and $F_3$ be degree $2$ in $\mathbb Z[x_1,\dots,x_4]$. Let there be an unique common integer to $F_i$. Let them be algebraically independent of a fourth polynomial $G$ which also has the same common integer root. Is it possible for the system to not form a complete...