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It is well known that if you have a smooth quartic surface $X\subset \mathbb{P}^3$, it may or may not have lines in it. Indeed, $X$ has the following options, 64 (the maximal number), 32, 16etc. , or none.

Between the space of all such quartics in $\mathbb{P}^3$ which is $|\mathcal{O}_{\mathbb{P}^3}(4)|=\mathbb{P}^{34}$, those with at least one line in it form a divisor $D$. Therefore, intersections of such a divisor with curves in $\mathbb{P}^{34}$ may give enumerative information about points(quartics) in the intersection.

Is all the enumerative information about such quartics encoded in $D$?

I'd like to know more about such a divisor $D$. So, Could someone recommend free-references of this online?

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It is well known that if you have a quartic surface $X\subset \mathbb{P}^3$, it may or may not have lines in it. Indeed, $X$ has a the following options, 64 (the maximal number), 32, 16 etc. or none.

Between the space of all quartics in $\mathbb{P}^3$ which is $|\mathcal{O}_{\mathbb{P}^3}(4)|=\mathbb{P}^{34}$, the surfaces those with at least one line in it form a divisor $D$. Therefore, intersections of such a divisor with curves in $\mathbb{P}^{34}$ may give enumerative information about points(quartics) in the intersection.

Is all the enumerative information about such quartics in $\mathbb{P}^3$ encoded in $D$?

I'd like to know more about such a divisor $D$. So, Could someone recommend free-references of this online?

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# Noether-Lefschetz locus in enumerative geometry.

It is well known that if you have a quartic surface $X\subset \mathbb{P}^3$, it may or may not have lines in it. Indeed, $X$ has a following options, 64 (the maximal number), 32, 16 etc. or none.

Between the space of all quartics in $\mathbb{P}^3$ which is $|\mathcal{O}_{\mathbb{P}^3}(4)|=\mathbb{P}^{34}$, the surfaces with at least one line in it form a divisor $D$. Therefore, intersections of such a divisor with curves in $\mathbb{P}^{34}$ may give enumerative information about points(quartics) in the intersection.

Is all the enumerative information about quartics in $\mathbb{P}^3$ encoded in $D$?

I'd like to know more about such a divisor $D$. So, Could someone recommend free-references of this online?