## Quintic surface singular along lines

What is the maximum number of lines, along which a quintic surface in $$mathbb{P}^3$$ can be singular ?

## dimension of quintic hypersurfaces singular at given number of points

How many quintic hypersurfaces are there which are singular at given points (need not be general) of length at least 20?Is there any upper bound of the dimension of such quintics ?

## ag.algebraic geometry – On the intersection numbers of the generators of \$text{Pic}(X)\$ of a smooth quintic surface

There exists the following result in the literature: There exists a polarized $$K3$$ surface $$(X, H)$$ of genus $$3$$ and a smooth irreducible curve $$C$$ on $$X$$ satisfying $$C^2 =4$$, $$C.H=6$$ such that $$text{Pic}(X) cong mathbb Z(H) oplus Z(C)$$. The theorem follows from (https://arxiv.org/pdf/math/9805140.pdf) theorem $$1.1(iv)$$.

Now Let’s consider $$X$$ to be a smooth quintic hypersurface in $$mathbb P^3$$ with Picard number $$2$$. Then I think it can be shown that $$text{Pic}(X) cong mathbb Z(H) oplus Z(H’)$$, where $$H$$ is the hyperplane class and $$H’$$ is some divisor. Now in order to locate the ample line bundles in $$text{Pic}(X)$$ using the Nakai-Moishezon criterion, we must know the intersection numbers $$H.H’$$ and $$H’^2$$.

In this context my question is the following: Does there exist in the literature an analogous existence result as the first-mentioned theorem for smooth quintic hypersurface with Picard number $$2$$?

To be more precise: Does there exist a polarized smooth quintic hypersurface $$(X, H)$$ in $$mathbb P^3$$ and a smooth irreducible curve $$C$$ for which the intersection numbers $$C^2$$ and $$C.H$$ are known and $$text{Pic}(X) cong mathbb Z(H) oplus Z(C)$$?

Can someone give me any reference which could be even remotely useful in the context of finding out such $$(X,H)$$ and $$C$$

Any help from anyone is welcome.

## group theory gr. – resolution of auther cayley for the quintic equation

hello i want to get different numbers using cayley resolvent

and I put integers x1 = 1 x2 = 2 x3 = 3 x4 = 4 x5 = 5

when to swap the integers in

= (X1x2 + x2x3 + x3x4 + x4x5 + x5x1
-X1x3 -x3x5 -x5x2 -x2x4 -x4x1) ^ 2

I've had (1,25,81,121) four digits but I'm not sure
is this the right way to think such a resolute?

## Algebra geometry – Curves of degree 3 on the Calabi – Yau quintic

Robbert Dijkgraaf said:1
Espace Calabi – Yau, the quintique:

"A classic result of the 19th century indicates that the number of lines – curves of degree one – is equal to 2875. The number of degree two curves was calculated only around 1980 and proves much larger : 609 250. But the number of curves of degree three required the help of string theorists ".

I understand from OEIS sequence A076912 that
the number is 317,206,375.

Q. Can any one tell me (or describe) how degree-3 matters
was settled with "the help of string theorists"?

1Robbert Dijkgraaf. "Quantum questions inspire new mathematics."
quanta.
The best writing in mathematics 2018Ed M. Pitici. p.80. Princeton.