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Let $P_1,\dotsc,P_k$ be polynomials. Assume they are pairwise non-proportional (i.e., any two of them are linearly independent). Suppose $N$ is a power such that $P_1^N,\dotsc,P_k^N$ are linearly inde …

If $Z$ is a pure one-dimensional closed subscheme of degree $d$ in projective space, with $d$-dimensional tangent space at every closed point, then $Z$ is supported on a line. To see this, let $H$ be …

The answer to your questions is no. The ideals $\langle x, y(1-xy) \rangle$ and $\langle x, y \rangle$ are equal, and maximal; but $$ \langle x-\lambda, y(1-xy)\rangle \neq \langle x-\delta, y-\epsilo …

(Adapting earlier comment into an answer.) Assume for simplicity that $X$ and $Y$ are irreducible, or in general take an irreducible component of each. Let $Z$ be an irreducible component of the inter …

The answer to question 1 is affirmative. There are several lower bounds in various papers. I'll take the idea from https://arxiv.org/abs/1503.08253 (Buczyński and myself, "Some examples of forms of hi …

Here is an additional comment. Every homogeneous polynomial can be regarded as a symmetric tensor. Quadrics correspond to symmetric matrices; cubics correspond to order $3$ tensors; in general, a homo …

This is a bit of a folk theorem. Harris (Algebraic Geometry, Proposition 18.10 and Exercise 18.11) states it for general hyperplane sections, but actually proves it for all generically transverse hype …

I believe this would be a dual arrangement of a star arrangement. A star arrangement is a union of subspaces defined as follows. Let $H_1,\dotsc,H_d$ be a collection of hyperplanes and fix an integer …

No. The pre-closure of the join of two irreducible projective varieties is NOT necessarily quasi-projective. Let $X$ be a smooth plane conic and let $Y$ be a single point of $X$. The pre-closure of th …

The basepoint freeness part has been proved in dimension $5$ by Fei Ye and Zhixian Zhu, On Fujita's freeness conjecture in dimension $5$, Adv. Math., 2020 DOI:10.1016/j.aim.2020.107210, arXiv:1511.091 …

A set of points in the plane is called a star configuration of type $\ell$ if it is the set of pairwise intersections of some $\ell$ lines, no three concurrent. If the lines are defined by $L_1,\dotsc …

Here is an answer in terms of power sum decompositions of polynomials. A point $p \in \mathbb{P}^9$ corresponds to a homogeneous polynomial $P$ of degree $3$ in $3$ variable, defining a plane cubic. P …

That is the Noether-Lefschetz theorem. Searching online should find plenty of results in web pages and lecture notes. If you want a published source, how about: Mark Green, A new proof of the explicit …

Your $\vee$ is essentially multiplication of polynomials. The variety of tensors $x_1 \vee \dotsb \vee x_m$ corresponds to polynomials that factor as products of linear factors. Points of the (project …

Yes, it is possible for the degree to increase. Say $V \subset \mathbb{C}^3$ is reducible: a union of a curve of degree $d$ plus one more point that doesn't lie on the curve. Then $V$ has degree $d$. …