Nongeneral Type Surfaces in $\mathbb P^4$
Every smooth, projective surface can be embedded in $\mathbb P^5$, but there are constraints on the numerical invariants of a smooth surface in $\mathbb P^4$: The invariants of each such surface $S$ satisfy the double point formula
\[d^2-5d-10(\pi-1)+2(\chi(\mathcal O_S)-K_S^2) = 0.\]
Here, $d$ is the degree of $S$, $\pi$ its sectional genus, $\chi(\mathcal O_S)$ its Euler-Poincare characteristic, and $K_S$ its canonical class. The double point formula is a key ingredient in the proof of a theorem of Ellingsrud and Peskine which states that there are only finitely many families of smooth surfaces in $\mathbb P^4$ which are not of general type. That is, the degree of such surfaces in bounded from above. The best bound known so far is $52$, while examples exist up to degree $15$.
For details, we refer to
and the references given there.
Below, we present functions which return one hard coded example for each family presented in the first two papers above. Based on these papers, the existence of further families has been shown. Hard coded OSCAR examples for these surfaces are under construction.
Rational Surfaces
A Rational Surface with $d=3$, $\pi=0$
cubic_scroll — Method
cubic_scroll()Return a smooth rational surface in $\mathbb P^4$ with degree 3 and sectional genus 0.
The returned surface is defined over a prime field of characteristic 31991.
sourceA Rational Surface with $d=4$, $\pi=0$
A Rational Surface with $d=5$, $\pi=2$
castelnuovo — Method
castelnuovo()Return a smooth rational surface in $\mathbb P^4$ with degree 5 and sectional genus 2.
The returned surface is defined over a prime field of characteristic 31991.
sourceA Rational Surface with $d=6$, $\pi=3$
A Rational Surface with $d=7$, $\pi=4$
rational_d7_pi4 — Method
rational_d7_pi4()Return a smooth rational surface in $\mathbb P^4$ with degree 7 and sectional genus 4.
The returned surface is defined over a prime field of characteristic 31991.
sourceA Rational Surface with $d=8$, $\pi=5$
rational_d8_pi5 — Method
rational_d8_pi5()Return a smooth rational surface in $\mathbb P^4$ with degree 8 and sectional genus 5.
The returned surface is defined over a prime field of characteristic 31991.
sourceA Rational Surface with $d=8$, $\pi=6$
rational_d8_pi6 — Method
rational_d8_pi6()Return a smooth rational surface in $\mathbb P^4$ with degree 8 and sectional genus 6.
The returned surface is defined over a prime field of characteristic 31991.
sourceA Rational Surface with $d=9$, $\pi=6$
rational_d9_pi6 — Method
rational_d9_pi6()Return a smooth rational surface in $\mathbb P^4$ with degree 9 and sectional genus 6.
The returned surface is defined over a prime field of characteristic 31991.
sourceA Rational Surface with $d=9$, $\pi=7$
rational_d9_pi7 — Method
rational_d9_pi7()Return a smooth rational surface in $\mathbb P^4$ with degree 9 and sectional genus 7.
The returned surface is defined over a prime field of characteristic 31991.
sourceA Rational Surface with $d=10$, $\pi=8$
rational_d10_pi8 — Method
rational_d10_pi8()Return a smooth rational surface in $\mathbb P^4$ with degree 10 and sectional genus 8.
The returned surface is defined over a prime field of characteristic 31991.
sourceA Rational Surface with $d=10$, $\pi=9$ which is Contained in one Quartic
rational_d10_pi9_quart_1 — Method
rational_d10_pi9_quart_1()Return a smooth rational surface in $\mathbb P^4$ with degree 10 and sectional genus 9 which is contained in precisely one quartic.
The returned surface is defined over a prime field of characteristic 31991.
sourceA Rational Surface with $d=10$, $\pi=9$ which is Contained in a Pencil of Quartics
rational_d10_pi9_quart_2 — Method
rational_d10_pi9_quart_2()Return a smooth rational surface in $\mathbb P^4$ with degree 10 and sectional genus 9 which is contained in a pencil of quartics.
The returned surface is defined over a prime field of characteristic 31991.
sourceA Rational Surface with $d=11$, $\pi=11$, and no 6-Secant
rational_d11_pi11_ss_0 — Method
rational_d11_pi11_ss_0()Return a smooth rational surface in $\mathbb P^4$ with degree 11, sectional genus 11, and no 6-secant.
The returned surface is defined over a prime field of characteristic 31991.
sourceA Rational Surface with $d=11$, $\pi=11$, and one 6-Secant
rational_d11_pi11_ss_1 — Method
rational_d11_pi11_ss_1()Return a smooth rational surface in $\mathbb P^4$ with degree 11, sectional genus 11, and one 6-secant.
The returned surface is defined over a prime field of characteristic 31991.
sourceA Rational Surface with $d=11$, $\pi=11$, and Infinitely Many 6-Secants
rational_d11_pi11_ss_inf — Method
rational_d11_pi11_ss_inf()Return a smooth rational surface in $\mathbb P^4$ with degree 11, sectional genus 11, and infinitely many 6-secants.
The returned surface is defined over a prime field of characteristic 31991.
sourceRuled Surfaces
A Ruled Surface with $d=5$, $\pi=1$
quintic_elliptic_scroll — Method
quintic_elliptic_scroll()Return a smooth ruled surface in $\mathbb P^4$ with degree 5 and sectional genus 1.
The returned surface is defined over a prime field of characteristic 31991.
sourceEnriques Surfaces
An Enriques Surface with $d=9$, $\pi=6$
enriques_d9_pi6 — Method
enriques_d9_pi6()Return a smooth Enriques surface in $\mathbb P^4$ with degree 9 and sectional genus 6.
The returned surface is defined over a prime field of characteristic 31991.
sourceAn Enriques Surface with $d=10$, $\pi=8$
enriques_d10_pi8 — Method
enriques_d10_pi8()Return a smooth Enriques surface in $\mathbb P^4$ with degree 10 and sectional genus 8.
The returned surface is defined over a prime field of characteristic 31991.
sourceAn Enriques Surface with $d=11$, $\pi=10$
enriques_d11_pi10 — Method
enriques_d11_pi10()Return a smooth Enriques surface in $\mathbb P^4$ with degree 11 and sectional genus 10.
The returned surface is defined over a prime field of characteristic 43.
sourceAn Enriques Surface with $d=13$, $\pi=16$
enriques_d13_pi16 — Method
enriques_d13_pi16()Return a smooth Enriques surface in $\mathbb P^4$ with degree 13 and sectional genus 16.
The returned surface is defined over a prime field of characteristic 31991.
sourceAn Enriques Surface with $d=13$, $\pi=16$
enriques_d13_pi16_two — Method
enriques_d13_pi16_two()Return a smooth Enriques surface in $\mathbb P^4$ with degree 13 and sectional genus 16.
The returned surface is defined over a prime field of characteristic 31991.
sourceK3 Surfaces
A K3 Surface with $d=7$, $\pi=5$
A K3 Surface with $d=8$, $\pi=6$
A K3 Surface with $d=9$, $\pi=8$
A K3 Surface with $d=10$, $\pi=9$ which is Contained in one Quartic
k3_d10_pi9_quart_1 — Method
k3_d10_pi9_quart_1()Return a smooth K3 surface in $\mathbb P^4$ with degree 10 and sectional genus 9 which is contained in precisely one quartic.
The returned surface is defined over a prime field of characteristic 31991.
sourceA K3 Surface with $d=10$, $\pi=9$ which is Contained in a Pencil of Quartics
k3_d10_pi9_quart_2 — Method
k3_d10_pi9_quart_2()Return a smooth K3 surface in $\mathbb P^4$ with degree 10 and sectional genus 9 which is contained in a pencil of quartics.
The returned surface is defined over a prime field of characteristic 31991.
sourceA K3 Surface with $d=11$, $\pi=11$ and no 6-Secant
k3_d11_pi11_ss_0 — Method
k3_d11_pi11_ss_0()Return a smooth K3 surface in $\mathbb P^4$ with degree 11, sectional genus 11, and no 6-secant.
The returned surface is defined over a prime field of characteristic 31991.
sourceA K3 Surface with $d=11$, $\pi=11$ and one 6-Secant
k3_d11_pi11_ss_1 — Method
k3_d11_pi11_ss_1()Return a smooth K3 surface in $\mathbb P^4$ with degree 11, sectional genus 11, and one 6-secant.
The returned surface is defined over a prime field of characteristic 31991.
sourceA K3 Surface with $d=11$, $\pi=11$ and two 6-Secants
k3_d11_pi11_ss_2 — Method
k3_d11_pi11_ss_2()Return a smooth K3 surface in $\mathbb P^4$ with degree 11, sectional genus 11, and two 6-secants.
The returned surface is defined over a prime field of characteristic 31991.
sourceA K3 Surface with $d=11$, $\pi=11$ and three 6-Secants
k3_d11_pi11_ss_3 — Method
k3_d11_pi11_ss_3()Return a smooth K3 surface in $\mathbb P^4$ with degree 11, sectional genus 11, and three 6-secants.
The returned surface is defined over a prime field of characteristic 31991.
sourceA K3 Surface with $d=11$, $\pi=12$
k3_d11_pi12 — Method
k3_d11_pi12()Return a smooth K3 surface in $\mathbb P^4$ with degree 11 and sectional genus 12.
The returned surface is defined over a prime field of characteristic 31991.
sourceA K3 Surface with $d=12$, $\pi=14$
k3_d12_pi14 — Method
k3_d12_pi14()Return a smooth K3 surface in $\mathbb P^4$ with degree 12 and sectional genus 14.
The returned surface is defined over a prime field of characteristic 31991.
sourceA K3 Surface with $d=13$, $\pi=16$
k3_d13_pi16 — Method
k3_d13_pi16()Return a smooth K3 surface in $\mathbb P^4$ with degree 13 and sectional genus 16.
The returned surface is defined over a prime field of characteristic 31991.
sourceA K3 Surface with $d=14$, $\pi=19$
k3_d14_pi19 — Method
k3_d14_pi19()Return a smooth K3 surface in $\mathbb P^4$ with degree 14 and sectional genus 19.
The returned surface is defined over a prime field of characteristic 31991.
sourceBielliptic Surfaces
A Bielliptic Surface with $d=10$, $\pi=6$
bielliptic_d10_pi6 — Method
bielliptic_d10_pi6()Return a smooth bielliptic surface in $\mathbb P^4$ with degree 10 and sectional genus 6.
The returned surface is defined over a prime field of characteristic 911.
sourceA Bielliptic Surface with $d=15$, $\pi=21$
bielliptic_d15_pi21 — Method
bielliptic_d15_pi21()Return a smooth bielliptic surface in $\mathbb P^4$ with degree 15 and sectional genus 21.
The returned surface is defined over a prime field of characteristic 911.
sourceAbelian Surfaces
An Abelian Surface with $d=10$, $\pi=6$
abelian_d10_pi6 — Method
abelian_d10_pi6()Return a smooth abelian surface in $\mathbb P^4$ with degree 10 and sectional genus 6.
The returned surface is defined over a prime field of characteristic 31991.
sourceAn Abelian Surface with $d=15$, $\pi=21$ which is Contained in a Net of Quintics
abelian_d15_pi21_quintic_3 — Method
abelian_d15_pi21_quintic_3()Return a smooth abelian surface in $\mathbb P^4$ with degree 15 and sectional genus 21 which is contained in a net of quintics.
The returned surface is defined over a prime field of characteristic 31991.
sourceAn Abelian Surface with $d=15$, $\pi=21$ which is Contained in one Quintic
abelian_d15_pi21_quintic_1 — Method
abelian_d15_pi21_quintic_1()Return a smooth abelian surface in $\mathbb P^4$ with degree 15 and sectional genus 21 which is contained in precisely one quintic.
The returned surface is defined over a prime field of characteristic 31991.
sourceElliptic Surfaces
An Elliptic Surface with $d=7$, $\pi=6$
elliptic_d7_pi6 — Method
elliptic_d7_pi6()Return a smooth elliptic surface in $\mathbb P^4$ with degree 7 and sectional genus 6.
The returned surface is defined over a prime field of characteristic 31991.
sourceAn Elliptic Surface with $d=8$, $\pi=7$
elliptic_d8_pi7 — Method
elliptic_d8_pi7()Return a smooth elliptic surface in $\mathbb P^4$ with degree 8 and sectional genus 7.
The returned surface is defined over a prime field of characteristic 31991.
sourceAn Elliptic Surface with $d=9$, $\pi=7$
elliptic_d9_pi7 — Method
elliptic_d9_pi7()Return a smooth elliptic surface in $\mathbb P^4$ with degree 9 and sectional genus 7.
The returned surface is defined over a prime field of characteristic 31991.
sourceAn Elliptic Surface with $d=10$, $\pi=9$
elliptic_d10_pi9 — Method
elliptic_d10_pi9()Return a smooth elliptic surface in $\mathbb P^4$ with degree 10 and sectional genus 9.
The returned surface is defined over a prime field of characteristic 31991.
sourceAn Elliptic Surface with $d=10$, $\pi=10$
elliptic_d10_pi10 — Method
elliptic_d10_pi10()Return a smooth elliptic surface in $\mathbb P^4$ with degree 10 and sectional genus 10.
The returned surface is defined over a prime field of characteristic 31991.
sourceAn Elliptic Surface with $d=11$, $\pi=12$
elliptic_d11_pi12 — Method
elliptic_d11_pi12()Return a smooth elliptic surface in $\mathbb P^4$ with degree 11 and sectional genus 12.
The returned surface is defined over a prime field of characteristic 31991.
sourceAn Elliptic Surface with $d=12$, $\pi=13$
elliptic_d12_pi13 — Method
elliptic_d12_pi13()Return a smooth elliptic surface in $\mathbb P^4$ with degree 12 and sectional genus 13.
The returned surface is defined over a prime field of characteristic 31991.
sourceAn Elliptic Surface with $d=12$, $\pi=14$ and no 6-Secant
elliptic_d12_pi14_ss_0 — Method
elliptic_d12_pi14_ss_0()Return a smooth elliptic surface in $\mathbb P^4$ with degree 12, sectional genus 14, and no 6-secant.
The returned surface is defined over a prime field of characteristic 31991.
sourceAn Elliptic Surface with $d=12$, $\pi=14$, and Infinitely Many 6-Secants
elliptic_d12_pi14_ss_inf — Method
elliptic_d12_pi14_ss_inf()Return a smooth elliptic surface in $\mathbb P^4$ with degree 12, sectional genus 14, and infinitely many 6-secants.
The returned surface is defined over a prime field of characteristic 31991.
sourceContact
Please direct questions about this part of OSCAR to the following people:
You can ask questions in the OSCAR Slack.
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