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# terraciniLocus -- compute the Terracini locus of a projective variety

## Description

There are two methods to compute the Terracini locus of a projective variety.

First, consider a rational variety parametrized by a polynomial map $f:\mathbb P^n\dashrightarrow\mathbb P^m$. In Macaulay2, this may be represented using a RingMap object from the coordinate ring of $\mathbb P^m$ to the coordinate ring of $\mathbb P^n$. We consider the twisted cubic in $\mathbb P^3$.

 i1 : R = QQ[s,t] o1 = R o1 : PolynomialRing i2 : S = QQ[x_0..x_3] o2 = S o2 : PolynomialRing i3 : f = map(R, S, {s^3, s^2*t, s*t^2, t^3}) 3 2 2 3 o3 = map (R, S, {s , s t, s*t , t }) o3 : RingMap R <-- S

In this case, the ideal of the preimage of the Terracini locus in $(\mathbb P^n)^r$ is returned. So in our twisted cubic example, if $r=2$, then we get the ideal of the pairs of points in $\mathbb P^1\times\mathbb P^1$ whose images under $f$ belong to the 2nd Terracini locus.

 i4 : terraciniLocus(2, f) o4 = ideal 1 o4 : Ideal of QQ[z ..z ] 0,0 1,1

We see that the Terracini locus is empty, which is true for all rational normal curves.

We may also consider varieties in $\mathbb P^n$ defined by an ideal. Let us continue with the twisted cubic example.

 i5 : I = ker f 2 2 o5 = ideal (x - x x , x x - x x , x - x x ) 2 1 3 1 2 0 3 1 0 2 o5 : Ideal of S

In this case, we may only use $r=2$. The ideal of the pairs of points in $\mathbb P^n\times\mathbb P^n$ belonging to the Terracini locus is returned. So for the twisted cubic, we get an ideal in the coordinate ring of $\mathbb P^3\times\mathbb P^3$.

 i6 : terraciniLocus(2, I) o6 = ideal 1 o6 : Ideal of QQ[z ..z ] 0,0 1,3

For more examples, see https://github.com/d-torrance/terracini-loci.

## Ways to use terraciniLocus :

• terraciniLocus(ZZ,Ideal)
• terraciniLocus(ZZ,Matrix,Ideal)
• terraciniLocus(ZZ,RingMap)

## For the programmer

The object terraciniLocus is .