# 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

- Wolfram Decker, Lawrence Ein, Frank-Olaf Schreyer (1993)
- Sorin Popescu (1993)
- Wolfram Decker, Frank-Olaf Schreyer (2000)

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. Explicit examples to be included here are under construction.

To ease subsequent computations, all hard coded examples are defined over a finite prime field.

## 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.

#### A Rational Surface with $d=4$, $\pi=0$

`veronese`

— Method`veronese()`

Return a smooth rational surface in $\mathbb P^4$ with degree `4`

and sectional genus `0`

.

The returned surface is defined over a prime field of characteristic 31991.

#### 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.

#### A Rational Surface with $d=6$, $\pi=3$

`bordiga`

— Method`bordiga()`

Return a smooth rational surface in $\mathbb P^4$ with degree `6`

and sectional genus `3`

.

The returned surface is defined over a prime field of characteristic 31991.

#### 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.

#### A 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.

#### A 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.

#### A 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.

#### A 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.

#### A 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.

#### A 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.

#### A 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.

#### A 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.

#### A 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.

#### A 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.

## Ruled 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.

## Enriques 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.

#### An 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.

#### An 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.

#### An 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.

#### An 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.

## K3 Surfaces

#### A K3 Surface with $d=7$, $\pi=5$

`k3_d7_pi5`

— Function`k3_d7_pi5()`

Return a smooth K3 surface in $\mathbb P^4$ with degree `7`

and sectional genus `5`

.

The returned surface is defined over a prime field of characteristic 31991.

#### A K3 Surface with $d=8$, $\pi=6$

`k3_d8_pi6`

— Function`k3_d8_pi6()`

Return a smooth K3 surface in $\mathbb P^4$ with degree `8`

and sectional genus `6`

.

The returned surface is defined over a prime field of characteristic 31991.

#### A K3 Surface with $d=9$, $\pi=8$

`k3_d9_pi8`

— Function`k3_d9_pi8()`

Return a smooth K3 surface in $\mathbb P^4$ with degree `9`

and sectional genus `8`

.

The returned surface is defined over a prime field of characteristic 31991.

#### 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.

#### A 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.

#### A 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.

#### A 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.

#### A 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.

#### A 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.

#### A 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.

#### A 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.

#### A 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.

#### A 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.

## Bielliptic 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.

#### A 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.

## Abelian 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.

#### An 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.

#### An 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.

## Elliptic 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.

#### An 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.

#### An 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.

#### An 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.

#### An 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.

#### An 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.

#### An 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.

#### An 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.

#### An 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.

## Contact

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