Mixing
Smooth quartic surface in $mathbbP^3$ that contains a smooth curve of genus 2 and degree 6.
Clash Royale CLAN TAG#URR8PPP
up vote
2
down vote
favorite
I am reading through an article by Matsumura and Monsky on Automorphisms of Hypersurfaces in which they state that there exist quartic surfaces is $mathbbP^3$ which have infinite automorphism groups, this is Theorem 4 in the linked article.
They construct an example of such a surface but it is not explicit, they say;
"Let $F$ be a non-singular (smooth) quartic surface in $mathbbP^3$ containing a non-singular curve $C$ of genus 2 and of degree 6."
I was wondering if there was an explicit formula for such a surface?
I have read around and found that a possible construction of a non-singular qaurtic surface in $mathbbP^3$ is to take the intersection of 2 quadrics in $mathbbP^4$, and also a construction of a non-singular curve of genus 2 and degree 6 is to take a polynomial $f(x)$ in one variable with 6 distinct roots and then consider the hyper-elliptic curve $y^2=f(x)$. However I do not know how to combine these two constructions together to give the surface I am looking for.
Finally they say that the result is classical and cite 3 articles claiming they contain other examples/constructions, one due to G. Fano (Sopra alcune superficie del 4 ordine rappresentabili sal piano doppio (1906)), and two due to F. Severi (Complementi alla teoria della base per la totalita delle curve di una superficie algebric (1910), and Geometria dei sistemi algebrici (1958)) all of which are very old and also in Italian. I cannot get a hold of any of the three articles (nor can I read Italian), so these do not provide much help.
Any help about a possible construction or a pointer to an article that isn't in Italian would be greatly appreciated. Thanks.
algebraic-geometry algebraic-curves projective-geometry intersection-theory
asked Jul 16 at 11:17
user551642
503
add a comment |Â
up vote
2
down vote
favorite
I am reading through an article by Matsumura and Monsky on Automorphisms of Hypersurfaces in which they state that there exist quartic surfaces is $mathbbP^3$ which have infinite automorphism groups, this is Theorem 4 in the linked article.
They construct an example of such a surface but it is not explicit, they say;
"Let $F$ be a non-singular (smooth) quartic surface in $mathbbP^3$ containing a non-singular curve $C$ of genus 2 and of degree 6."
I was wondering if there was an explicit formula for such a surface?
I have read around and found that a possible construction of a non-singular qaurtic surface in $mathbbP^3$ is to take the intersection of 2 quadrics in $mathbbP^4$, and also a construction of a non-singular curve of genus 2 and degree 6 is to take a polynomial $f(x)$ in one variable with 6 distinct roots and then consider the hyper-elliptic curve $y^2=f(x)$. However I do not know how to combine these two constructions together to give the surface I am looking for.
Finally they say that the result is classical and cite 3 articles claiming they contain other examples/constructions, one due to G. Fano (Sopra alcune superficie del 4 ordine rappresentabili sal piano doppio (1906)), and two due to F. Severi (Complementi alla teoria della base per la totalita delle curve di una superficie algebric (1910), and Geometria dei sistemi algebrici (1958)) all of which are very old and also in Italian. I cannot get a hold of any of the three articles (nor can I read Italian), so these do not provide much help.
Any help about a possible construction or a pointer to an article that isn't in Italian would be greatly appreciated. Thanks.
algebraic-geometry algebraic-curves projective-geometry intersection-theory
asked Jul 16 at 11:17
user551642
503
1
"I have read around and found that a possible construction of a non-singular quartic surface in P^3 is to take the intersection of 2 quadrics in P^4..." But that gives a quartic surface in P^4, which typically will be nondegenerate (i.e. not contained in any P^3).
– Asal Beag Dubh
Jul 16 at 15:47
1
If you know how to find the ideal of a smooth genus-2 curve of degree 6 in $mathbf P^3$ the rest should be straightforward to compute: that ideal will contain lots of elements of degree 4, and you can ask e.g. Macaulay 2 to give you a random such element. With probability $1-epsilon$ the surface defined by such an element will be smooth. (cont'd)
– Asal Beag Dubh
Jul 16 at 15:54
1
For any hyperelliptic curve $C$, the "tricanonical" embedding defined by $|3K_C|$ will embed it as a curve of degree 6 in $mathbf P^4$. Projecting from a general point will then give a smooth curve of genus 2 and degree 6 as required. So the problem now is to compute the ideal of a hyperelliptic curve tricanonically embedded in $mathbf P^4$. I don't know how to do that right now.
– Asal Beag Dubh
Jul 16 at 15:59
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
I am reading through an article by Matsumura and Monsky on Automorphisms of Hypersurfaces in which they state that there exist quartic surfaces is $mathbbP^3$ which have infinite automorphism groups, this is Theorem 4 in the linked article.
They construct an example of such a surface but it is not explicit, they say;
"Let $F$ be a non-singular (smooth) quartic surface in $mathbbP^3$ containing a non-singular curve $C$ of genus 2 and of degree 6."
I was wondering if there was an explicit formula for such a surface?
I have read around and found that a possible construction of a non-singular qaurtic surface in $mathbbP^3$ is to take the intersection of 2 quadrics in $mathbbP^4$, and also a construction of a non-singular curve of genus 2 and degree 6 is to take a polynomial $f(x)$ in one variable with 6 distinct roots and then consider the hyper-elliptic curve $y^2=f(x)$. However I do not know how to combine these two constructions together to give the surface I am looking for.
Finally they say that the result is classical and cite 3 articles claiming they contain other examples/constructions, one due to G. Fano (Sopra alcune superficie del 4 ordine rappresentabili sal piano doppio (1906)), and two due to F. Severi (Complementi alla teoria della base per la totalita delle curve di una superficie algebric (1910), and Geometria dei sistemi algebrici (1958)) all of which are very old and also in Italian. I cannot get a hold of any of the three articles (nor can I read Italian), so these do not provide much help.
Any help about a possible construction or a pointer to an article that isn't in Italian would be greatly appreciated. Thanks.
algebraic-geometry algebraic-curves projective-geometry intersection-theory
asked Jul 16 at 11:17
user551642
503
I am reading through an article by Matsumura and Monsky on Automorphisms of Hypersurfaces in which they state that there exist quartic surfaces is $mathbbP^3$ which have infinite automorphism groups, this is Theorem 4 in the linked article.
They construct an example of such a surface but it is not explicit, they say;
"Let $F$ be a non-singular (smooth) quartic surface in $mathbbP^3$ containing a non-singular curve $C$ of genus 2 and of degree 6."
I was wondering if there was an explicit formula for such a surface?
I have read around and found that a possible construction of a non-singular qaurtic surface in $mathbbP^3$ is to take the intersection of 2 quadrics in $mathbbP^4$, and also a construction of a non-singular curve of genus 2 and degree 6 is to take a polynomial $f(x)$ in one variable with 6 distinct roots and then consider the hyper-elliptic curve $y^2=f(x)$. However I do not know how to combine these two constructions together to give the surface I am looking for.
Finally they say that the result is classical and cite 3 articles claiming they contain other examples/constructions, one due to G. Fano (Sopra alcune superficie del 4 ordine rappresentabili sal piano doppio (1906)), and two due to F. Severi (Complementi alla teoria della base per la totalita delle curve di una superficie algebric (1910), and Geometria dei sistemi algebrici (1958)) all of which are very old and also in Italian. I cannot get a hold of any of the three articles (nor can I read Italian), so these do not provide much help.
Any help about a possible construction or a pointer to an article that isn't in Italian would be greatly appreciated. Thanks.
algebraic-geometry algebraic-curves projective-geometry intersection-theory
asked Jul 16 at 11:17
user551642
503
asked Jul 16 at 11:17
user551642
503
asked Jul 16 at 11:17
user551642
503
asked Jul 16 at 11:17
user551642
503
503
1
"I have read around and found that a possible construction of a non-singular quartic surface in P^3 is to take the intersection of 2 quadrics in P^4..." But that gives a quartic surface in P^4, which typically will be nondegenerate (i.e. not contained in any P^3).
– Asal Beag Dubh
Jul 16 at 15:47
1
If you know how to find the ideal of a smooth genus-2 curve of degree 6 in $mathbf P^3$ the rest should be straightforward to compute: that ideal will contain lots of elements of degree 4, and you can ask e.g. Macaulay 2 to give you a random such element. With probability $1-epsilon$ the surface defined by such an element will be smooth. (cont'd)
– Asal Beag Dubh
Jul 16 at 15:54
1
For any hyperelliptic curve $C$, the "tricanonical" embedding defined by $|3K_C|$ will embed it as a curve of degree 6 in $mathbf P^4$. Projecting from a general point will then give a smooth curve of genus 2 and degree 6 as required. So the problem now is to compute the ideal of a hyperelliptic curve tricanonically embedded in $mathbf P^4$. I don't know how to do that right now.
– Asal Beag Dubh
Jul 16 at 15:59
add a comment |Â
1
"I have read around and found that a possible construction of a non-singular quartic surface in P^3 is to take the intersection of 2 quadrics in P^4..." But that gives a quartic surface in P^4, which typically will be nondegenerate (i.e. not contained in any P^3).
– Asal Beag Dubh
Jul 16 at 15:47
1
If you know how to find the ideal of a smooth genus-2 curve of degree 6 in $mathbf P^3$ the rest should be straightforward to compute: that ideal will contain lots of elements of degree 4, and you can ask e.g. Macaulay 2 to give you a random such element. With probability $1-epsilon$ the surface defined by such an element will be smooth. (cont'd)
– Asal Beag Dubh
Jul 16 at 15:54
1
For any hyperelliptic curve $C$, the "tricanonical" embedding defined by $|3K_C|$ will embed it as a curve of degree 6 in $mathbf P^4$. Projecting from a general point will then give a smooth curve of genus 2 and degree 6 as required. So the problem now is to compute the ideal of a hyperelliptic curve tricanonically embedded in $mathbf P^4$. I don't know how to do that right now.
– Asal Beag Dubh
Jul 16 at 15:59
1
1
"I have read around and found that a possible construction of a non-singular quartic surface in P^3 is to take the intersection of 2 quadrics in P^4..." But that gives a quartic surface in P^4, which typically will be nondegenerate (i.e. not contained in any P^3).
– Asal Beag Dubh
Jul 16 at 15:47
"I have read around and found that a possible construction of a non-singular quartic surface in P^3 is to take the intersection of 2 quadrics in P^4..." But that gives a quartic surface in P^4, which typically will be nondegenerate (i.e. not contained in any P^3).
– Asal Beag Dubh
Jul 16 at 15:47
1
1
If you know how to find the ideal of a smooth genus-2 curve of degree 6 in $mathbf P^3$ the rest should be straightforward to compute: that ideal will contain lots of elements of degree 4, and you can ask e.g. Macaulay 2 to give you a random such element. With probability $1-epsilon$ the surface defined by such an element will be smooth. (cont'd)
– Asal Beag Dubh
Jul 16 at 15:54
If you know how to find the ideal of a smooth genus-2 curve of degree 6 in $mathbf P^3$ the rest should be straightforward to compute: that ideal will contain lots of elements of degree 4, and you can ask e.g. Macaulay 2 to give you a random such element. With probability $1-epsilon$ the surface defined by such an element will be smooth. (cont'd)
– Asal Beag Dubh
Jul 16 at 15:54
1
1
For any hyperelliptic curve $C$, the "tricanonical" embedding defined by $|3K_C|$ will embed it as a curve of degree 6 in $mathbf P^4$. Projecting from a general point will then give a smooth curve of genus 2 and degree 6 as required. So the problem now is to compute the ideal of a hyperelliptic curve tricanonically embedded in $mathbf P^4$. I don't know how to do that right now.
– Asal Beag Dubh
Jul 16 at 15:59
For any hyperelliptic curve $C$, the "tricanonical" embedding defined by $|3K_C|$ will embed it as a curve of degree 6 in $mathbf P^4$. Projecting from a general point will then give a smooth curve of genus 2 and degree 6 as required. So the problem now is to compute the ideal of a hyperelliptic curve tricanonically embedded in $mathbf P^4$. I don't know how to do that right now.
– Asal Beag Dubh
Jul 16 at 15:59
add a comment |Â
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2853324%2fsmooth-quartic-surface-in-mathbbp3-that-contains-a-smooth-curve-of-genus%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
1
"I have read around and found that a possible construction of a non-singular quartic surface in P^3 is to take the intersection of 2 quadrics in P^4..." But that gives a quartic surface in P^4, which typically will be nondegenerate (i.e. not contained in any P^3).
– Asal Beag Dubh
Jul 16 at 15:47
1
If you know how to find the ideal of a smooth genus-2 curve of degree 6 in $mathbf P^3$ the rest should be straightforward to compute: that ideal will contain lots of elements of degree 4, and you can ask e.g. Macaulay 2 to give you a random such element. With probability $1-epsilon$ the surface defined by such an element will be smooth. (cont'd)
– Asal Beag Dubh
Jul 16 at 15:54
1
For any hyperelliptic curve $C$, the "tricanonical" embedding defined by $|3K_C|$ will embed it as a curve of degree 6 in $mathbf P^4$. Projecting from a general point will then give a smooth curve of genus 2 and degree 6 as required. So the problem now is to compute the ideal of a hyperelliptic curve tricanonically embedded in $mathbf P^4$. I don't know how to do that right now.
– Asal Beag Dubh
Jul 16 at 15:59