Locus of tangent lines to a smooth curve of degree $d$ and genus $g$
$begingroup$
Suppose $Csubseteqmathbb{P}^3$ is a smooth curve of degree $d$ and genus $g$ (let's say we are working over $mathbb{C}$). Let $T(C)$ be the locus of tangent lines to $C$. In other words,
$$
T(C) = { L inmathbb{G}(1, 3) | L text{ is tangent to C}}
$$
Here $mathbb{G}(1, 3)$ is the Grassmannian of lines in $mathbb{P}^3$. How can I compute the class of this locus $[T(C)]$ in the Chow ring? More precisely, what does $[T(C)]$ look like in $A^{3}(mathbb{G}(1,3))$?
Attempt: We know that $[T(C)]=c sigma_{1, 2}$ where $sigma_{1, 2}$ is the Schubert cycle corresponding to lines in $mathbb{P}^3$ that pass through a point $p$ and contained in a plane $H$ (where $p$ and $H$ are general but fixed). So we just need to figure out the constant $c$. To do this, can fix a general line $L_{0}$, and intersect this class $[T(C)]$ with the Schubert cycle $sigma_{1}$ (which consists of all lines incident to $L_0$). Thus, $c$ is equal to the number of lines $L$ that is tangent to $C$ (at some point) such that $Lcap L_{0}neqemptyset$. How can we determine this number in terms of the degree $d$ and genus $g$ of the smooth curve $C$?
algebraic-geometry algebraic-curves tangent-line intersection-theory schubert-calculus
edited Jan 5 at 12:38
Matt Samuel
38.7k63769
asked Nov 4 '16 at 11:57
PrismPrism
5,05731981
$endgroup$
add a comment |
$begingroup$
Suppose $Csubseteqmathbb{P}^3$ is a smooth curve of degree $d$ and genus $g$ (let's say we are working over $mathbb{C}$). Let $T(C)$ be the locus of tangent lines to $C$. In other words,
$$
T(C) = { L inmathbb{G}(1, 3) | L text{ is tangent to C}}
$$
Here $mathbb{G}(1, 3)$ is the Grassmannian of lines in $mathbb{P}^3$. How can I compute the class of this locus $[T(C)]$ in the Chow ring? More precisely, what does $[T(C)]$ look like in $A^{3}(mathbb{G}(1,3))$?
Attempt: We know that $[T(C)]=c sigma_{1, 2}$ where $sigma_{1, 2}$ is the Schubert cycle corresponding to lines in $mathbb{P}^3$ that pass through a point $p$ and contained in a plane $H$ (where $p$ and $H$ are general but fixed). So we just need to figure out the constant $c$. To do this, can fix a general line $L_{0}$, and intersect this class $[T(C)]$ with the Schubert cycle $sigma_{1}$ (which consists of all lines incident to $L_0$). Thus, $c$ is equal to the number of lines $L$ that is tangent to $C$ (at some point) such that $Lcap L_{0}neqemptyset$. How can we determine this number in terms of the degree $d$ and genus $g$ of the smooth curve $C$?
algebraic-geometry algebraic-curves tangent-line intersection-theory schubert-calculus
edited Jan 5 at 12:38
Matt Samuel
38.7k63769
asked Nov 4 '16 at 11:57
PrismPrism
5,05731981
$endgroup$
add a comment |
$begingroup$
Suppose $Csubseteqmathbb{P}^3$ is a smooth curve of degree $d$ and genus $g$ (let's say we are working over $mathbb{C}$). Let $T(C)$ be the locus of tangent lines to $C$. In other words,
$$
T(C) = { L inmathbb{G}(1, 3) | L text{ is tangent to C}}
$$
Here $mathbb{G}(1, 3)$ is the Grassmannian of lines in $mathbb{P}^3$. How can I compute the class of this locus $[T(C)]$ in the Chow ring? More precisely, what does $[T(C)]$ look like in $A^{3}(mathbb{G}(1,3))$?
Attempt: We know that $[T(C)]=c sigma_{1, 2}$ where $sigma_{1, 2}$ is the Schubert cycle corresponding to lines in $mathbb{P}^3$ that pass through a point $p$ and contained in a plane $H$ (where $p$ and $H$ are general but fixed). So we just need to figure out the constant $c$. To do this, can fix a general line $L_{0}$, and intersect this class $[T(C)]$ with the Schubert cycle $sigma_{1}$ (which consists of all lines incident to $L_0$). Thus, $c$ is equal to the number of lines $L$ that is tangent to $C$ (at some point) such that $Lcap L_{0}neqemptyset$. How can we determine this number in terms of the degree $d$ and genus $g$ of the smooth curve $C$?
algebraic-geometry algebraic-curves tangent-line intersection-theory schubert-calculus
edited Jan 5 at 12:38
Matt Samuel
38.7k63769
asked Nov 4 '16 at 11:57
PrismPrism
5,05731981
$endgroup$
Suppose $Csubseteqmathbb{P}^3$ is a smooth curve of degree $d$ and genus $g$ (let's say we are working over $mathbb{C}$). Let $T(C)$ be the locus of tangent lines to $C$. In other words,
$$
T(C) = { L inmathbb{G}(1, 3) | L text{ is tangent to C}}
$$
Here $mathbb{G}(1, 3)$ is the Grassmannian of lines in $mathbb{P}^3$. How can I compute the class of this locus $[T(C)]$ in the Chow ring? More precisely, what does $[T(C)]$ look like in $A^{3}(mathbb{G}(1,3))$?
Attempt: We know that $[T(C)]=c sigma_{1, 2}$ where $sigma_{1, 2}$ is the Schubert cycle corresponding to lines in $mathbb{P}^3$ that pass through a point $p$ and contained in a plane $H$ (where $p$ and $H$ are general but fixed). So we just need to figure out the constant $c$. To do this, can fix a general line $L_{0}$, and intersect this class $[T(C)]$ with the Schubert cycle $sigma_{1}$ (which consists of all lines incident to $L_0$). Thus, $c$ is equal to the number of lines $L$ that is tangent to $C$ (at some point) such that $Lcap L_{0}neqemptyset$. How can we determine this number in terms of the degree $d$ and genus $g$ of the smooth curve $C$?
algebraic-geometry algebraic-curves tangent-line intersection-theory schubert-calculus
algebraic-geometry algebraic-curves tangent-line intersection-theory schubert-calculus
edited Jan 5 at 12:38
Matt Samuel
38.7k63769
asked Nov 4 '16 at 11:57
PrismPrism
5,05731981
edited Jan 5 at 12:38
Matt Samuel
38.7k63769
asked Nov 4 '16 at 11:57
PrismPrism
5,05731981
edited Jan 5 at 12:38
Matt Samuel
38.7k63769
edited Jan 5 at 12:38
Matt Samuel
38.7k63769
edited Jan 5 at 12:38
Matt Samuel
38.7k63769
38.7k63769
asked Nov 4 '16 at 11:57
PrismPrism
5,05731981
asked Nov 4 '16 at 11:57
PrismPrism
5,05731981
asked Nov 4 '16 at 11:57
PrismPrism
5,05731981
5,05731981
add a comment |
add a comment |
1 Answer
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votes
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I came across to the following beautiful argument in Notes on Grassmannians and Schubert Varieties by Chipalkatti (see page 27).
Given a general line $Lsubseteqmathbb{P}^3$ we want to compute the number of tangent lines to $C$ that intersect $L$. Pick a general line $M$ (which will be skew to $L$) and consider the projection away from $L$, i.e. consider the map
$$
f: C longrightarrow mathbb{P}^1 cong M
$$
given by sending $xin C$ to the intersection of $M$ and the plane $overline{Lx}$ (generated by $L$ and $x$). Now we will apply the Riemann-Hurwitz theorem!
What is the degree of $f$? Given a general point $yin M$, it is easy to see that
$$
f^{-1}(y) = {xin X: x in overline{Ly}cap C}
$$
which consists of $d$ distinct points (where $d$ is the degree of $C$), because $yin M$ was general. Moreover, since $L$ and $M$ are general, the ramification points $f$ will have degree $1$ (i.e. simple ramification points). The ramification points of $f$ are exactly those points $xin C$ such that $L$ meets the tangent line $ell_{x}$ to $C$ at $x$. So Riemann-Hurwitz tells us
$$
2g - 2 = d(2cdot 0 - 2)+text{# ramification points}
$$
Thus, the number of tangent lines that meet $L$ is precisely:
$$
2g + 2d - 2 = 2(g+d-1)
$$
We conclude that
$$
[T(C)] = 2(g+d-1) sigma_{1,2}
$$
in the Chow ring of $mathbb{G}(1, 3)$.
answered Nov 8 '16 at 1:21
PrismPrism
5,05731981
$endgroup$
add a comment |
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1 Answer
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1 Answer
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$begingroup$
I came across to the following beautiful argument in Notes on Grassmannians and Schubert Varieties by Chipalkatti (see page 27).
Given a general line $Lsubseteqmathbb{P}^3$ we want to compute the number of tangent lines to $C$ that intersect $L$. Pick a general line $M$ (which will be skew to $L$) and consider the projection away from $L$, i.e. consider the map
$$
f: C longrightarrow mathbb{P}^1 cong M
$$
given by sending $xin C$ to the intersection of $M$ and the plane $overline{Lx}$ (generated by $L$ and $x$). Now we will apply the Riemann-Hurwitz theorem!
What is the degree of $f$? Given a general point $yin M$, it is easy to see that
$$
f^{-1}(y) = {xin X: x in overline{Ly}cap C}
$$
which consists of $d$ distinct points (where $d$ is the degree of $C$), because $yin M$ was general. Moreover, since $L$ and $M$ are general, the ramification points $f$ will have degree $1$ (i.e. simple ramification points). The ramification points of $f$ are exactly those points $xin C$ such that $L$ meets the tangent line $ell_{x}$ to $C$ at $x$. So Riemann-Hurwitz tells us
$$
2g - 2 = d(2cdot 0 - 2)+text{# ramification points}
$$
Thus, the number of tangent lines that meet $L$ is precisely:
$$
2g + 2d - 2 = 2(g+d-1)
$$
We conclude that
$$
[T(C)] = 2(g+d-1) sigma_{1,2}
$$
in the Chow ring of $mathbb{G}(1, 3)$.
answered Nov 8 '16 at 1:21
PrismPrism
5,05731981
$endgroup$
add a comment |
$begingroup$
I came across to the following beautiful argument in Notes on Grassmannians and Schubert Varieties by Chipalkatti (see page 27).
Given a general line $Lsubseteqmathbb{P}^3$ we want to compute the number of tangent lines to $C$ that intersect $L$. Pick a general line $M$ (which will be skew to $L$) and consider the projection away from $L$, i.e. consider the map
$$
f: C longrightarrow mathbb{P}^1 cong M
$$
given by sending $xin C$ to the intersection of $M$ and the plane $overline{Lx}$ (generated by $L$ and $x$). Now we will apply the Riemann-Hurwitz theorem!
What is the degree of $f$? Given a general point $yin M$, it is easy to see that
$$
f^{-1}(y) = {xin X: x in overline{Ly}cap C}
$$
which consists of $d$ distinct points (where $d$ is the degree of $C$), because $yin M$ was general. Moreover, since $L$ and $M$ are general, the ramification points $f$ will have degree $1$ (i.e. simple ramification points). The ramification points of $f$ are exactly those points $xin C$ such that $L$ meets the tangent line $ell_{x}$ to $C$ at $x$. So Riemann-Hurwitz tells us
$$
2g - 2 = d(2cdot 0 - 2)+text{# ramification points}
$$
Thus, the number of tangent lines that meet $L$ is precisely:
$$
2g + 2d - 2 = 2(g+d-1)
$$
We conclude that
$$
[T(C)] = 2(g+d-1) sigma_{1,2}
$$
in the Chow ring of $mathbb{G}(1, 3)$.
answered Nov 8 '16 at 1:21
PrismPrism
5,05731981
$endgroup$
add a comment |
$begingroup$
I came across to the following beautiful argument in Notes on Grassmannians and Schubert Varieties by Chipalkatti (see page 27).
Given a general line $Lsubseteqmathbb{P}^3$ we want to compute the number of tangent lines to $C$ that intersect $L$. Pick a general line $M$ (which will be skew to $L$) and consider the projection away from $L$, i.e. consider the map
$$
f: C longrightarrow mathbb{P}^1 cong M
$$
given by sending $xin C$ to the intersection of $M$ and the plane $overline{Lx}$ (generated by $L$ and $x$). Now we will apply the Riemann-Hurwitz theorem!
What is the degree of $f$? Given a general point $yin M$, it is easy to see that
$$
f^{-1}(y) = {xin X: x in overline{Ly}cap C}
$$
which consists of $d$ distinct points (where $d$ is the degree of $C$), because $yin M$ was general. Moreover, since $L$ and $M$ are general, the ramification points $f$ will have degree $1$ (i.e. simple ramification points). The ramification points of $f$ are exactly those points $xin C$ such that $L$ meets the tangent line $ell_{x}$ to $C$ at $x$. So Riemann-Hurwitz tells us
$$
2g - 2 = d(2cdot 0 - 2)+text{# ramification points}
$$
Thus, the number of tangent lines that meet $L$ is precisely:
$$
2g + 2d - 2 = 2(g+d-1)
$$
We conclude that
$$
[T(C)] = 2(g+d-1) sigma_{1,2}
$$
in the Chow ring of $mathbb{G}(1, 3)$.
answered Nov 8 '16 at 1:21
PrismPrism
5,05731981
$endgroup$
I came across to the following beautiful argument in Notes on Grassmannians and Schubert Varieties by Chipalkatti (see page 27).
Given a general line $Lsubseteqmathbb{P}^3$ we want to compute the number of tangent lines to $C$ that intersect $L$. Pick a general line $M$ (which will be skew to $L$) and consider the projection away from $L$, i.e. consider the map
$$
f: C longrightarrow mathbb{P}^1 cong M
$$
given by sending $xin C$ to the intersection of $M$ and the plane $overline{Lx}$ (generated by $L$ and $x$). Now we will apply the Riemann-Hurwitz theorem!
What is the degree of $f$? Given a general point $yin M$, it is easy to see that
$$
f^{-1}(y) = {xin X: x in overline{Ly}cap C}
$$
which consists of $d$ distinct points (where $d$ is the degree of $C$), because $yin M$ was general. Moreover, since $L$ and $M$ are general, the ramification points $f$ will have degree $1$ (i.e. simple ramification points). The ramification points of $f$ are exactly those points $xin C$ such that $L$ meets the tangent line $ell_{x}$ to $C$ at $x$. So Riemann-Hurwitz tells us
$$
2g - 2 = d(2cdot 0 - 2)+text{# ramification points}
$$
Thus, the number of tangent lines that meet $L$ is precisely:
$$
2g + 2d - 2 = 2(g+d-1)
$$
We conclude that
$$
[T(C)] = 2(g+d-1) sigma_{1,2}
$$
in the Chow ring of $mathbb{G}(1, 3)$.
answered Nov 8 '16 at 1:21
PrismPrism
5,05731981
answered Nov 8 '16 at 1:21
PrismPrism
5,05731981
answered Nov 8 '16 at 1:21
PrismPrism
5,05731981
answered Nov 8 '16 at 1:21
PrismPrism
5,05731981
5,05731981
add a comment |
add a comment |
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