# Questions tagged [complete-intersection]

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### Surface of type $(2,2)$ on the Segre cubic scroll $\mathbb{P}^1 \times \mathbb{P}^2 \subset \mathbb{P}^5$

Let $S=\mathbb{P}^1 \times \mathbb{P}^2 \subset \mathbb{P}^5$ embedded with the Segre embedding given by $\mathcal{O}_S(1,1)$. If we intersect $S$ with a general smooth quadric $Q \subset \mathbb{P}^5$...
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### Isomorphisms of complete intersections

Let $X, Y\in \mathbb{P}^n$ be two singular Fano complete intersections of the same multidegree $(d_1,…,d_r)$. If we assume there is an isomorphism $f\colon X\rightarrow Y$ are there any assumptions so ...
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### Properties preserved in addition of ideals

If I and J are prime (radical) ideals then what are the conditions under which we can define a prime (radical) ideal from I+J?
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### Do height $h$ prime ideals in regular local rings contain regular sequences of length $h$?

Let $R$ be a regular local ring and let $P$ be a prime ideal of height $h$ in $R$. Is it always the case that $P$ contains a regular sequence of lenght $h$? This is clear if $h$ is $0,1$ or $\dim R$. ...
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### Rational connectedness of certain subvarieties of the linear series

Let $X$ be a smooth projective hypersurface in $\mathbb{P}^3$, $|\mathcal{O}_X(a)|$ be the complete linear system for some integer $a>0$. Ofcourse, a general element of the linear system is a ...
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### When every module is a scalar extension?

Let $A \subseteq B$ be commutative noetherian domains. Of course, if $M$ is an $A$-module, then $M \otimes_A B$ is a $B$-module. I am curious to know if there exist additional conditions on $A$ and $B$...
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### Complete Intersection

Let $I$ be an ideal of the polynomial ring $P=K[x_{1},...,x_{n}]$ that is generated by degree two polynomials ${f_1,...,f_k}$. The zero set $\mathcal{Z}(I)$ is isomorphic to an affine space of ...
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### A criterion for complete intersection in terms of the Hilbert series?

Let $(R, \mathfrak{m})$ be a complete local ring (of dimension $2$ if that makes a difference). I would like to be able to decide whether or not $R$ is a complete intersection (meaning, a quotient of ...
Let $H_1,H_2\cdots,H_{d-1}$ be hypersurfaces in $\mathbb{P}^d$, if the intersection $B:=H_1\cap H_2\cap \cdots \cap H_{d-1}$ is $1$-dimensional then it is called a complete intersection curve. What ...